# Roadmap: perfectoid rings and spaces

**Summary.** Perfectoid spaces are the adic spaces on which Frobenius can be inverted: locally they
are adic spectra of perfectoid Tate rings, complete uniform Tate rings with a pseudouniformiser `ϖ`
such that `ϖ^p` divides `p` and Frobenius is an isomorphism `R°/ϖ → R°/ϖ^p`. This roadmap builds
their theory in that generality, with no perfectoid base field and with products of mixed- and
equal-characteristic rings allowed: almost mathematics over an arbitrary basic setup; perfectoid
Tate rings, tilts, Fontaine's `θ` and the classification of untilts by primitive ideals; the tilting
homeomorphism, perfectoid rational localisation, the sheaf theorem with almost acyclicity of `O⁺`,
and perfectoid spaces with fibre products; almost purity and the étale site; injections,
immersions, Zariski closed immersions and separatedness; cofiltered limits and finite-stage descent
of étale objects; `κ`-small perfectoid spaces and the pro-étale calculus; and tilde-limits and
Frobenius-controlled towers, sousperfectoid frames and spaces, analytic approximation,
perfectoid torsor presentations and the affinoid perfectoid basis of the rigid pro-étale site. The end results are the tilting equivalence
`Perfd/X ≃ Perf/X♭` with `X_ét ≃ X♭_ét`, the almost purity theorem (for a perfectoid Tate ring
`R`, `FÉt(R) ≃ FÉt(R♭)`, every finite étale `R`-algebra is perfectoid, and `S°` is almost finite
étale over `R°`), and the pro-étale calculus on which DiamondsAndVStacks builds its topologies.

Everything rests on Tau Ceti's Huber rings and adic spaces (the Foundations of adic spaces roadmap,
`TauCetiRoadmap/AdicSpaces/README.md`, called AdicSpaces below, and its implementation under
`TauCeti.RingTheory.Huber` and `TauCeti.AlgebraicGeometry.AdicSpace`) and on Mathlib's Witt vectors,
perfection, `PreTilt` and `WittVector.fontaineTheta`. The outputs are the geometric inputs of
pro-étale descent and diamonds (DiamondsAndVStacks) and of perfectoid Shimura varieties
and period spaces downstream. The analytic étale interfaces come from AdicEtaleGeometry;
this roadmap proves their perfectoid specializations.

| Layer | Construction |
| --- | --- |
| P0 | Almost mathematics over a basic setup |
| P1 | Perfectoid Tate rings, tilts, untilts and perfectoidness of split Tate rings |
| P2 | Rational localisation, the sheaf theorem and perfectoid spaces |
| P3 | Almost purity, sousperfectoid rings and spaces, polydiscs and the étale site |
| P4 | Injections, immersions and separatedness |
| P5 | Cofiltered limits, tubular approximation and finite-stage affinoid étale descent |
| P6 | `κ`-small perfectoid spaces, pro-étale morphisms and perfectoid torsor presentations |
| P7 | Tilde-limits, Frobenius-controlled towers and the rigid pro-étale basis |

Layers P0 and P1 can be developed in parallel up to the almost setup of a perfectoid ring, where
they meet.

## Scope and ownership

The tower-quotient and torsor-descent layers P8–P9 form the roadmap [PerfectoidTowers](../PerfectoidTowers/README.md).
Each layer's Dependencies paragraph names its mathematical suppliers by section.

**Dependencies.** DerivedModuleAlgebra §§0.1–0.11, 0.13–0.14, 0.25, 1.T.1, 1.1–1.9; AdicSpaces §§0–5, 6.1;
AdicSpacesPartII R0.0–R0.20.2, R1; AdicSpacesPartIII R2.4, R3.1, R3.5, R3.8, R3.9;
AdicEtaleGeometry A0, A1.0–A1.16, A1.2a, A1.6a, A2, A3.1–A3.2, A4.2–A4.5,
A4.2a, A4.7; PivotalSpherical Layers 2 and 7.

PerfectoidTowers imports both PerfectoidSpaces and DiamondsAndVStacks; diamond objects,
spatial presentations, and coefficient descent using diamonds belong to those two roadmaps. The
v-topology on perfectoid spaces belongs to DiamondsAndVStacks: P7 uses only the raw compact-open
covering condition
(Mathlib `IsCompactOpenCovered`), while the v-site built on that condition belongs to
DiamondsAndVStacks, its precoverage and topology to D2.2 and its subcanonicity and v-descent to
D2.7. The ordinary cotangent complex of a ring map and its square-zero deformation theory belong to
DerivedModuleAlgebra §§0.4–0.11, 0.13 and 0.25, consistently with SchemeAndStackFoundations §4.6,
and are consumed in P0.19; this roadmap owns only
its almost counterpart (P0.20). DerivedModuleAlgebra §§0.1–0.4, 0.14 and 1.1–1.9 own
the generic derived operations, ideal completion and bounded-torsion p-complete module descent.
Its §1.T.1 owns ordinary module towers, their product difference map, limits, six-term sequence
and module Milnor sequence; P0.27–P0.28 use these directly for almost annihilation and completion.
P4.15.1 consumes their carriers and maps; its finite-ideal simplicial descent extension and
transfinite compatibility feed only the prismatic construction of P4.15.2–P4.15.5.

The roadmap owns, layer by layer:

- P0: basic setups `(V, m)` (an idempotent ideal with `m ⊗_V m` flat) and their examples (the root
  setup of an element with compatible `p`-power roots, the setup `(K°, K°°)` of a valuation ring
  with non-principal maximal ideal), firm modules, the Serre quotient `V^a`-Mod of almost modules
  with almost elements and the left adjoint `M_!`, almost algebras and change of setup, almost
  finitely generated, presented, flat and projective modules with the `ε`-criteria, almost finite
  étale algebras with the trace-form criterion, the almost cotangent complex, nilpotent deformations
  of almost algebras and finite étale lifting, and almost sheaves with their almost cohomology.
- P1: perfectoid Tate rings and pairs, Frobenius on `R°/ϖ`, the tilt `R♭` with the sharp map and
  its functoriality, perfectoid fields and their tilts (compared with Mathlib's `Tilt`), Fontaine's
  map `θ` compared with `WittVector.fontaineTheta`, untilts and primitive ideals with the
  primitive-kernel theorem, the tilting equivalence for perfectoid Tate rings, completed filtered
  colimits and perfected Tate algebras, and the comparison with AdicSpaces' perfect field `F`.
- P2: the tilting homeomorphism `Spa(R, R⁺) ≅ Spa(R♭, R♭⁺)`, perfectoid rational localisation, the
  sheaf theorem with almost acyclicity of `O⁺`, affinoid perfectoid and perfectoid spaces, the tilting
  equivalence of perfectoid spaces, completed tensor products of perfectoid pairs and fibre
  products of perfectoid spaces.
- P3: the perfectoid-field application of henselian completion, almost purity in characteristic `p`, the Fontaine–Wintenberger
  theorem, the almost purity theorem, étale morphisms of perfectoid spaces with the comparison to
  AdicEtaleGeometry's étale morphisms, and the étale site of a perfectoid space with its tilting
  equivalence.
- P4: injections and immersions of perfectoid spaces, Zariski closed immersions and universal
  perfectoid Zariski closed subspaces, separated maps and the valuative criterion for separatedness.
- P5: the completed direct limit of perfectoid Tate pairs, cofiltered limits of affinoid perfectoid
  spaces with their cardinal bounds, `ω₁`-cofiltered limits, and finite-stage descent of étale objects.
- P6: `κ`-small and uniformly `κ`-small perfectoid spaces, affinoid pro-étale morphisms, pro-étale
  morphisms, products with profinite sets, the pro-category equivalence and the stability of
  pro-étale morphisms.
- P7: perfectoid tilde-limits, Frobenius-controlled integral towers, uniqueness and gluing of
  tilde-limits, the perfectoid Frobenius criterion for towers, and limits of perfectoid towers.


P1.1 owns the perfectoid Tate-ring theory and its equivalent Frobenius-modulo-`p`
criterion; P2.11 owns glued tilting. P3.17a's
sousperfectoid frames use P1.1's predicate `Perfectoid.IsPerfectoidTateRing` directly, so no
second perfectoid interface is compared with it; the modulo-`p` reformulation of P1.1 holds on
every complete Hausdorff uniform Tate ring.
The glued-tilting construction is `PerfectoidSpace.tiltFunctor` on this roadmap's
`Perfd` carrier, with the comparisons and slice equivalence specified in P2.11.

The boundaries with the neighbouring roadmaps: AdicEtaleGeometry A1.2–A1.3 and A1.6 supply the
finite étale and étale morphisms of analytic adic spaces and their sites on `TauCeti.AdicSpace`, and
A1.9 their forms `YonedaAdicSpace.IsFiniteEtale`, `IsEtaleLocalDescription` and `smallEtale` on the
analytic Yoneda carrier;
P3 restricts these notions to perfectoid spaces and compares them with its own. P6 constructs pro-étale morphisms
and the `κ`-small calculus. P7 uses Mathlib's compact-open covering condition
`IsCompactOpenCovered` directly and builds no site from it.
DiamondsAndVStacks builds the v-site on that condition (precoverage and topology in D2.2,
subcanonicity and v-descent in D2.7), uses it for its pro-étale and v-sheaf theory
and defines diamonds; PerfectoidTowers P8 proves
the quotient statements that identify the represented v-sheaf of a finite quotient. AdicSpacesPartII
owns the completed tensor product of Tate rings (R0.3–R0.4), fibre products of adic spaces (R0.7),
their closed immersions (R0.8) and separatedness (R0.10); AdicSpacesPartIII R3.9 owns the Gel'fand
spectrum of a Banach ring with Berkovich's maximum formula; AdicEtaleGeometry A1.6a owns vector
bundles on the étale site of a sheafy affinoid with a stable basis. AdicSpacesPartIV R5.6–R5.7
owns the coefficient theory on sousperfectoid spaces.
P2 proves that completed tensor products of
perfectoid rings are perfectoid, P1 adds only the unit criterion on the Gel'fand spectrum, and PerfectoidTowers P9
proves the torsor descent with coefficients. AdicSpacesPartII R0.20.1 owns the intrinsic tilde-limit relation, the completed-affinoid criterion and base change.
P7 compares its analytic Yoneda carrier with that theory and adds the perfectoid
specializations and restriction to opens. Separatedness of a morphism of perfectoid spaces (P4) is the notion
that DiamondsAndVStacks layer D3 restricts to on perfectoid spaces. Through
`YonedaAdicSpace.ofAdicSpace`, P4.19 agrees with AdicSpacesPartII R0.10's `AdicSpace.IsSeparated`
for morphisms locally of weakly finite type (R0.6) on adic representatives satisfying R0's stably
sheafy chart hypotheses; PerfectoidTowers P8.15's analytic separatedness is a different condition,
and its comparison with P4.19 belongs to P8.15. P4.2 compares with R0.6's `AdicSpace.QuasiCompact`,
and P4.16 with R0.8's cover-local `AdicSpace.IsClosedImmersion`: a map of perfectoid spaces is such
a closed immersion iff some affinoid perfectoid cover of the target has strongly Zariski closed
restrictions, a local quotient condition, not every topological closed immersion, and not
PerfectoidTowers P8.14's condition over every affinoid perfectoid open.
P4.15.1 imports derived completion and complete flatness from DerivedModuleAlgebra
§§0.1–0.4, 0.14 and 1.1–1.9, with the finite-ideal simplicial descent and transfinite
compatibilities specified in P4.15.1. P4.15.2–P4.15.5 own the prismatic and quasisyntomic
prerequisites for semiperfectoid perfectoidization: universal prisms and their perfection,
the affine Hodge–Tate comparison, and André's root-adjoining cover. Prismatic
cohomology of general spaces, the general quasisyntomic site and perfectoidization of arbitrary
animated rings are outside the scope. PerfectoidTowers P8 owns perfectoidization of integral
extensions of integral perfectoid rings. The locally analytic distribution theory on the coefficient spaces of PerfectoidTowers P9
and the Shimura-variety applications of P7 and PerfectoidTowers P8–P9 belong to the roadmaps that consume this one.

Tau Ceti already has, and this roadmap uses without restating: Huber rings, Tate rings,
pseudouniformisers, pairs of definition, the power-bounded subring, boundedness, uniformity and
stable uniformity, rings of integral elements and Huber pairs with their morphisms
(`TauCeti.Huber.IsHuberRing`, `IsTateRing`, `IsPseudoUniformizer`, `PairOfDefinition`,
`powerBoundedSubring`, `IsBounded`, `IsStablyUniform`, `IsRingOfIntegralElements`, `Pair`,
`Pair.Hom`); the open mapping theorem for Tate rings (`IsTateRing.isOpenMap`); restricted power
series and strongly noetherian Tate rings; the valuation spectrum, `Cont`, `Spa` with its rational
subsets and spectrality (`TauCeti.ValuationSpectrum.spectralSpace_spa_of_pairOfDefinition`);
completed rational localisation with its universal property; the structure presheaf, its stalks and
residue-field valuations; pre-adic spaces, open immersions and gluing; sheafiness for strongly
noetherian Tate rings (`TauCeti.Huber.isSheafyRing_of_isStronglyNoetherian`) and the two halves of
the sheaf condition for the Laurent cover of a stably uniform pair
(`TauCeti/AlgebraicGeometry/AdicSpace/Spa/StructurePresheaf/LaurentCover/StableUniform.lean`); the category of adic spaces
(`TauCeti.AdicSpace`, Wedhorn's `(Adic)` inside `TauCeti.PreAdicSpace`, with `PreAdicSpace.Hom` and
`PreAdicSpace.IsOpenImmersion`); quotient Huber pairs with their closed embeddings of adic spectra
(`TauCeti.Huber.Pair.quotient`, `TauCeti.Huber.Pair.quotient_plus`, `Pair.Hom.isClosedEmbedding_spaComap_quotientHom`,
`range_spaComap_quotientHom`, `ValuationSpectrum.isClosedEmbedding_spaComap_quotientMk`); the
recovery of a pair from the global sections of its adic spectrum
(`ValuationSpectrum.isIso_toPresentationLimit_top`, `mem_iff_forall_vle_one`); and the perfect-ring
input `W(R)` with its `(p, [ϖ])`-adic topology of the Fargues–Fontaine curve
(`TauCeti.FarguesFontaine.spaY`, AdicSpaces layers 0–6). The Buzzard–Verberkmoes theorem (stably
uniform implies sheafy) is AdicSpaces layer 4.2, a target of that roadmap, and Čech acyclicity of
the structure sheaf in every degree for a sheafy Tate pair is AdicSpacesPartIII R3.1; P2 applies
both to perfectoid pairs and reproves neither. Uniform Tate rings are reduced
(`TauCeti.Huber.IsUniform.isReduced`, used by P1 for perfectoid rings), and finite étale
algebras descend along algebraic filtered colimits (AdicEtaleGeometry A1.0.3).
P3 uses AdicEtaleGeometry A1.0.4–A1.0.5’s smooth lifting and completion comparisons
on Mathlib’s `HenselianRing` carrier. Supporting results below that have a library core build on it: the
`(p, [ϖ])`-adic completeness and separatedness of `W(R⁺)` for a perfect `ϖ`-adically complete `R⁺`
(`WittVector.isAdicComplete_span_p_teichmuller`, `isHausdorff_span_p_teichmuller`,
`completeSpace_and_t2Space_adicTopology_span_p_teichmuller`,
`TauCeti.Huber.WittVector.isHuberRing_adicTopology_span_p_teichmuller`), the homeomorphism `Spa` of a completion
`≅ Spa` (`TauCeti.ValuationSpectrum.spaCompletionHomeomorph`), spectrality of `spaComap` for Tate
sources (`TauCeti.Huber.Pair.Hom.isSpectralMap_spaComap_of_isTateRing`), lifting of idempotents
along henselian pairs (`TauCeti.HenselianRing.exists_isIdempotentElem_sub_mem`), henselianity of
adically complete pairs (`IsAdicComplete.henselianRing`), the cofinality of a directed family of
closed subgroups of a compact group (`Subgroup.exists_le_of_iInf_le_of_directed`), the multiplicative
perfection equivalence (`Perfection.quotientMulEquiv`). Where a target here is a variant of a library statement, its entry says in one
clause how it differs: the tilt `R♭` is defined for every perfectoid Tate ring and compared with
`PreTilt`, which is defined on the integral level, and with `Tilt`, Mathlib's tilt of a valued field
(the perfectoid-field case, P1); perfectoid Tate pairs are Tau Ceti Huber pairs whose ring is
perfectoid, not a new carrier; the sheafiness of perfectoid pairs is the stably-uniform case
of the Buzzard–Verberkmoes theorem of AdicSpaces layer 4.2 and their Čech acyclicity the sheafy
case of AdicSpacesPartIII R3.1 (*sheafy-cech-acyclicity*), both applied and not reproved; Mathlib
has no statement about the
kernel of `fontaineTheta` (its `BDeRhamPlus` and `BDeRham` are built from `θ` without it), so the
primitive-kernel theorem of P1 is new; closed immersions of affinoid adic spaces through quotient
Huber pairs are the library's (`TauCeti.Huber.Pair.quotient`), and P4's Zariski closed and strongly
Zariski closed immersions are their perfectoid case and say so.

External inputs and their applications in the layers:

- the surjectivity of Fontaine's `θ` for a perfectoid pair — Mathlib
  `surjective_fontaineTheta` (through the identification of P1.17; P1.18 keeps the
  kernel statement);
- the `(p, [ϖ])`-adic completeness, separatedness and Huber-ring structure of `W(R⁺)` for a perfect
  `ϖ`-adically complete `R⁺` — `WittVector.isAdicComplete_span_p_teichmuller`,
  `isHausdorff_span_p_teichmuller`, `completeSpace_and_t2Space_adicTopology_span_p_teichmuller`,
  `TauCeti.Huber.WittVector.isHuberRing_adicTopology_span_p_teichmuller` (P1.16, P1.33);
- the `ϖ`-adic topology and completeness of rings of definition of a complete Tate ring —
  AdicEtaleGeometry A1.0.2(i), `PairOfDefinition.isAdicComplete_of_completeSpace`
  (*power-bounded-henselian-along-pseudouniformizer*); P1.2
  (*integral-elements-adically-complete*) applies it to `R⁺`, using uniformity to make this
  open bounded subring a ring of definition, and proves complements (b)–(d);
- the Tate norm datum and the spectral seminorm of a Tate ring — AdicSpacesPartII R0.4
  (*tate-ring-norm* `Huber.TateNormDatum`, *spectral-seminorm* `Huber.spectralSeminorm`), of which
  P1's spectral gauge is
  the `ℤ[1/p]`-exponent form;
- the Gel'fand spectrum of a Banach ring: compactness, nonemptiness and the maximum formula —
  AdicSpacesPartIII R3.9 (*gelfand-spectrum-banach-ring*) (P1.12 keeps the unit
  criterion);
- the global sections of `Spa(R, R⁺)` — `TauCeti.ValuationSpectrum.isIso_toPresentationLimit_top`,
  `mem_iff_forall_vle_one` (P2.9); the category of adic spaces — `TauCeti.AdicSpace` (P2.9, P2.10);
- the rational-localisation functor on `Rat(X)^op` with its transitivity, presentation
  independence and universal property — `TauCeti.Huber.PairOfDefinition.completionLocObj`,
  `completedPlusSubring`, `pairHomOfRationalSubsetSubset`, `presentationRingEquivOfEq`,
  `existsUnique_continuous_ringHom_of_forall_comap_mem_rationalSubset` (P2.3, P2.4); the
  uncompleted residue field with its valuation — `TauCeti.ValuationSpectrum.residueFieldValuation`
  (P2.5); stable uniformity — `TauCeti.Huber.IsStablyUniform` (P2.6);
- the universal property of the completed tensor product — AdicSpacesPartII R0.3
  (*completed-tensor-product-universal-property*)
  (P2.13);
- adically complete pairs are henselian — Mathlib `IsAdicComplete.henselianRing`; henselian pairs
  under filtered colimits and along a pseudouniformiser — AdicEtaleGeometry A1.0.1–A1.0.2
  (*henselian-pairs-filtered-colimit*, *power-bounded-henselian-along-pseudouniformizer*);
  smooth lifting over a henselian pair is AdicEtaleGeometry A1.0.4’s `henselian_smooth_lifting`; finite étale
  lifting for a complete pair is the classical setup of P0.28’s
  `Almost.Algebra.finiteEtaleBaseChange_isEquivalence`;
- finite étale algebras over an algebraic filtered colimit of rings — AdicEtaleGeometry
  A1.0.3 (*finite-etale-algebras-filtered-colimit*) (P3.2, P5.9); the general completion
  equivalence is AdicEtaleGeometry A1.0.5's `finiteEtale_completedColimit`, and A4.5's
  `completion_finiteEtale_equivalence_continuousMap` (*finite-etale-invariance-along-tower*) is its
  torsor-tower specialization; the finite étale pair `Huber.Pair.finiteEtale` — AdicEtaleGeometry A1.2
  (*finite-etale-affinoid-algebra*) (P3.3, P3.10);
- composition, base change, cancellation and openness of (finite) étale morphisms of analytic adic
  spaces — AdicEtaleGeometry A1.5 (*etale-composition*, *etale-base-change*,
  *etale-diagonal*) (P3.17, P3.18); rational descent of finite étaleness and the étale structure
  sheaf — AdicEtaleGeometry A1.2 and A1.6 (*finite-etale-rational-descent*, *etale-structure-sheaf*)
  (P3.19); vector bundles on the étale site of a sheafy affinoid
  — AdicEtaleGeometry A1.6a (*stable-basis-etale-vector-bundles*) (P3.19);
- the quotient Huber pair and its closed embedding of adic spectra — `TauCeti.Huber.Pair.quotient`,
  `TauCeti.Huber.Pair.quotient_plus`, `Pair.Hom.isClosedEmbedding_spaComap_quotientHom`, `range_spaComap_quotientHom`,
  `ValuationSpectrum.isClosedEmbedding_spaComap_quotientMk`, with the open-mapping half
  `TauCeti.Huber.IsTateRing.isQuotientMap` (P4.12, P4.13, P4.16);
- spectrality of cofiltered limits of spectral spaces — AdicSpacesPartII R0.19.5;
  generic pro-categories and cofiltered limits — AdicSpacesPartII R0.19.9 on Mathlib's
  opposite `Ind` carrier; perfectoid realization, bounded presentations and flattening — P6.9–P6.12;
  cardinality of a completion with a bounded dense subset — P6.2
  (`completion_cardinality_bound`). DiamondsAndVStacks D0.13 and D0.24 consume these
  results; its cutoff-cardinal applications use [ECD] Lemma 4.1, pp. 19–20;
- affinoid étale objects at a finite stage and the compatibility of the finite-stage equivalences
  — P5.13.4 (*affinoid-etale-finite-stage-6-4-iv*,
  *etale-descent-to-finite-stage-6-4*) (P6.5); profinite-set objects of Scholze's pro-étale site —
  AdicEtaleGeometry A1.12 (*profinite-set-objects*), of which P6.8 is the perfectoid-space
  form;
- the cofinality of a decreasing family of closed subgroups of a profinite group —
  `Subgroup.exists_le_of_iInf_le_of_directed` and Mathlib
  `ProfiniteGrp.exist_openNormalSubgroup_sub_open_nhds_of_one` (P7.9, P7.11); base change of
  tilde-limits along morphisms locally of weakly finite type — AdicSpacesPartII R0.20.2
  (*tilde-limit-base-change*) (P7.12);
- the comonadicity of faithfully flat extension of scalars and the degree
  `≤ 1` Amitsur statement — Mathlib `comonadicExtendScalars`,
  `CommRingCat.isLimitForkPushoutSelfOfFaithfullyFlat` (P0.25); the honest lifting of idempotents
  — `TauCeti.HenselianRing.exists_isIdempotentElem_sub_mem`,
  `TauCeti.IsAdicComplete.exists_isIdempotentElem_eq` (P0.26).

## Conventions

Fix a prime `p`. A perfectoid Tate ring is complete and Hausdorff, uniform, and has a
pseudouniformiser `ϖ` with `ϖ^p | p` in `R°` such that Frobenius `R°/ϖ → R°/ϖ^p` is an isomorphism;
injectivity of Frobenius is automatic and the condition does not depend on `ϖ`. No perfectoid base
field is fixed; statements that hold only over a perfectoid field say so. A perfectoid Tate pair
`(R, R⁺)` is a Tau Ceti Huber pair with `R` perfectoid; `R°` is its maximal ring of integral elements
and `R°°` the topologically nilpotent elements. `Perfd` denotes perfectoid spaces of all
characteristics, a full subcategory of Tau Ceti's `PreAdicSpace` (sheafy pre-adic spaces covered by
affinoid perfectoid opens, P2.10), and `Perf` its characteristic-`p` subcategory, with `p = 0` in all
sections; `X♭` is the tilt, `♯` the sharp map,
which is multiplicative and continuous but not additive. Almost mathematics is always with respect
to a stated basic setup `(V, m)`: `m` idempotent with `m ⊗_V m` flat; for a perfectoid Tate ring the
setup is `(R°, R°°)`, equivalently the root ideal `(ϖ^{1/p^∞})` of a pseudouniformiser with compatible
`p`-power roots, and the two setups on `R⁺ ⊆ R°` are compared explicitly. "Almost zero", "almost
isomorphism", "almost finite étale" and "almost acyclic" are relative to that setup; an almost
statement is never silently upgraded to an actual one.

Rings of integral elements are open, integrally closed and contained in `R°`; completed colimits of
Tate pairs complete the colimit of the plus rings `ϖ`-adically and invert `ϖ`. "qcqs" means
quasicompact and quasiseparated; morphisms of perfectoid spaces are morphisms of adic spaces.
Tilde-limits carry both the topological condition and the affinoid density condition. Cardinals:
`κ`-smallness is tested on affinoid covers, and uniformly `κ`-small means `κ'`-small for a single
`κ' < κ` (P6.2); the cutoff cardinals are strong limits and may be singular.

Each target below is written with its hypotheses in the sentence, its source locator in parentheses
(short keys are resolved in the References), its API as named lemmas and its unit tests as `Checks`;
`Suggested.lean` gives typed forms for a subset of these targets and Checks. Existing definitions keep their library namespaces.
The names of definitions, lemmas and Checks in the target paragraphs are the names proposed for the
formalisation. A subsection whose targets are carried by declarations of `Suggested.lean` ends with
a `Lean:` line listing them, and the layer's Examples paragraph lists its Lean `example`s; a
target, definition or Check not named on a `Lean:` line is specified in this README only and has no
Lean declaration yet. Names of other roadmaps' declarations are theirs.

## Exact supplier contracts

Tables are ordered by the consuming layer within each supplier group.

### From Mathlib

#### P0

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Ordinary algebra extensions and split extensions | `Algebra.Extension`, `Mathlib/RingTheory/Extension/Basic.lean`; `TrivSqZeroExt`, `TrivSqZeroExt.inrHom`, `Mathlib/Algebra/TrivSqZeroExt/Basic.lean` | P0.21 adds the specified square-zero kernel and morphisms fixing it; the polynomial automorphism check uses the existing split algebra. |
| Exact functors out of almost modules | `CategoryTheory.ObjectProperty.SerreClassLocalization.essImage_whiskeringLeft`, `.fullyFaithfulWhiskeringLeft`, `.exactFunctor_comp_iff`, `Mathlib/CategoryTheory/Abelian/SerreClass/Localization.lean:494,517,534`; `CategoryTheory.ObjectProperty.isoModSerre_isInvertedBy_iff`, `Mathlib/CategoryTheory/Abelian/SerreClass/MorphismProperty.lean:177` | Apply to `L = Almost.toAlmost S` and `P = S.almostZero`: an exact functor to an abelian category factors through `L` exactly when it kills `P`; full faithfulness gives uniqueness compatible with the factorization. |
| Idempotence equation | `IsIdempotentElem.eq`, `Mathlib/Algebra/Group/Idempotent.lean` | Apply directly to `S.isIdempotentElem`; no local equation theorem. |
| Serre quotient | `CategoryTheory.ObjectProperty.SerreClassLocalization.abelian`, `CategoryTheory.ObjectProperty.SerreClassLocalization.isZero_obj_iff`, `CategoryTheory.ObjectProperty.SerreClassLocalization.isIso_map_iff`, `CategoryTheory.ObjectProperty.SerreClassLocalization.preservesFiniteLimits`, `CategoryTheory.ObjectProperty.SerreClassLocalization.preservesFiniteColimits`, `CategoryTheory.ObjectProperty.SerreClassLocalization.exactFunctor_comp_iff`, `CategoryTheory.ObjectProperty.SerreClassLocalization.essImage_whiskeringLeft`, all in `Mathlib/CategoryTheory/Abelian/SerreClass/Localization.lean` | The almost-zero Serre property, small-Hom firm model, and almost adjoints. |
| Localisation from adjoints | `CategoryTheory.Abelian.isLocalization_isoModSerre_kernel_of_leftAdjoint`, `Mathlib/CategoryTheory/Abelian/SerreClass/Bousfield.lean` | Verify its hypotheses for the firm reflector. |
| Almost tensor and algebra carriers | `CategoryTheory.LocalizedMonoidal`, `CategoryTheory.Localization.Monoidal.toMonoidalCategory`, `Mathlib/CategoryTheory/Localization/Monoidal/Basic.lean`; `CategoryTheory.CommMon`, `Mathlib/CategoryTheory/Monoidal/CommMon_.lean`; `CategoryTheory.Mod`, `CategoryTheory.Mod.regular`, `CategoryTheory.Mod.Hom.mk'`, `Mathlib/CategoryTheory/Monoidal/Mod.lean:227,278,336` | Specialize to the almost-zero Serre class and pin almost elements by the unit and multiplication equations. |
| Unitalization | `Unitization`, `Unitization.inl`, `Unitization.inr`, `Mathlib/Algebra/Algebra/Unitization.lean`; `Algebra.TensorProduct.instNonUnitalRing`, `Mathlib/RingTheory/TensorProduct/Basic.lean` | Quotient by `(xy, −(x ⊗ y) ⊗ 1)` for lower shriek-shriek. |
| Exterior powers | `ExteriorAlgebra.exteriorPower`, `Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean` | Almostify the exterior powers of the module of almost elements. |
| Full cotangent model | `CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy`, `Mathlib/AlgebraicTopology/ExtraDegeneracy.lean`; `AlgebraicTopology.alternatingFaceMapComplex`, `Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean`; `ComplexShape.embeddingUpIntLE`, `Mathlib/Algebra/Homology/Embedding/Basic.lean`; `DerivedCategory.Q`, `Mathlib/Algebra/Homology/DerivedCategory/Basic.lean` | Polynomial resolutions with a set-level contraction and termwise augmented differentials; full cotangent homology rather than the existing degree-one truncation. The ordinary model is the consumed carrier of P0.19, owned by DerivedModuleAlgebra §§0.1–0.4; P0.20 uses the same resolutions for the almost complex. |
| Derived module functors | `CategoryTheory.Functor.leftDerived`, `Mathlib/CategoryTheory/Abelian/LeftDerived.lean`; `CategoryTheory.Functor.rightDerived`, `Mathlib/CategoryTheory/Abelian/RightDerived.lean` | Ordinary Tor and module-valued Ext, followed by almostification; these do not identify categorical projectivity with almost projectivity. |
| Finite decompositions | `CategoryTheory.Abelian.hasFiniteBiproducts`, `Mathlib/CategoryTheory/Abelian/Basic.lean` | Rank projectors and their constant-rank exterior-power conditions. |
| Tensor–Hom adjunction | `ModuleCat.monoidalClosedHomEquiv`, `Mathlib/Algebra/Category/ModuleCat/Monoidal/Closed.lean` | Restrict it to the firm category. |
| Cone triangle sign | `CochainComplex.mappingCone.triangle`, `CochainComplex.mappingCone.inl_v_triangle_mor₃_f`, `Mathlib/Algebra/Homology/HomotopyCategory/Pretriangulated.lean` | Apply the existing minus-projection convention to cotangent transitivity. |
| Derived comparison maps | `CategoryTheory.Functor.mapDerivedCategory`, `mapDerivedCategoryFactors`, `mapDerivedCategorySingleFunctor`, `Mathlib/Algebra/Homology/DerivedCategory/ExactFunctor.lean`; `CategoryTheory.ShiftedHom.map`, `Mathlib/CategoryTheory/Shift/ShiftedHom.lean`; `ModuleCat.extendRestrictScalarsAdj`, `Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean`; `Algebra.Extension.ofSurjective`, `Algebra.Extension.Cotangent.mk`, `Mathlib/RingTheory/Extension/Basic.lean` | P0 specifies their application to almostification and the signed cotangent extension classifier; it does not redefine the derived functor or the conormal module. |
| Differentials and first cotangent homology | `KaehlerDifferential.DLinearMap_apply`, `Mathlib/RingTheory/Kaehler/Basic.lean`; `KaehlerDifferential.polynomialEquiv`, `Mathlib/RingTheory/Kaehler/Polynomial.lean`; `Algebra.Extension.cotangentComplex`, `Algebra.H1Cotangent`, `Mathlib/RingTheory/Extension/Cotangent/Basic.lean` | The two lowest degrees of the cotangent complex, which the consumed carrier of P0.19 (owned by DerivedModuleAlgebra §§0.1–0.4) extends, and the almost comparisons of P0.20. |
| Trace on finite free algebras | `Algebra.trace`, `Algebra.trace_eq_zero_of_not_exists_basis`, `Mathlib/RingTheory/Trace/Defs.lean` | P3.12's `Perfectoid.FiniteEtale.trace` is the finite **projective** trace, including algebras of nonconstant rank; P0's almost trace supplies its classical-setup construction. |
| Descent of ordinary modules | `comonadicExtendScalars`, `Mathlib/Algebra/Category/ModuleCat/Descent.lean`; `CommRingCat.isLimitForkPushoutSelfOfFaithfullyFlat`, `Mathlib/Algebra/Category/Ring/EqualizerPushout.lean` | Identify comonad coalgebras with descent data and extend to almost algebras. |
| Cotangent exact sequence | `Algebra.Generators.Cotangent.exact`, `Mathlib/RingTheory/Kaehler/JacobiZariski.lean:95` | The exact sequence for composable polynomial presentations underlies P0’s full cotangent-complex comparison. |
| Polynomial translations and dual numbers | `Polynomial.algEquivAevalXAddC`, `Polynomial.algEquivAevalXAddC_symm`, `Mathlib/Algebra/Polynomial/AlgebraMap.lean`; `TrivSqZeroExt.fstHom`, `Mathlib/Algebra/TrivSqZeroExt/Basic.lean` | P0.22 consumes DerivedModuleAlgebra §0.25's `CotangentComplex.polynomialLift_sign`, which uses these automorphisms and quotient maps to compute the deformation sign. |

#### P0/P7

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Modules killed by an ideal | `Module.IsTorsionBySet.module`, `Module.IsTorsionBySet.mk_smul`, `Module.isTorsionBySet_quotient_ideal_smul`, `Mathlib/Algebra/Module/Torsion/Basic.lean` | P0 applies the quotient action to `J`, `JM`, and `M/JM` in the square-zero flatness criterion. |
| Canonical scalar-extension comparisons | `ModuleCat.extendRestrictScalarsAdj`, `ModuleCat.extendScalarsId`, `ModuleCat.extendScalarsComp`, `Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean`; `CategoryTheory.Under.pushoutId`, `CategoryTheory.Under.pushoutComp`, `Mathlib/CategoryTheory/Comma/Over/Pullback.lean` | P0 specifies the almost-module comparisons on adjunction-unit generators and uses the supplied almost-algebra pushout comparisons directly. |
| Locally discrete pseudofunctors | `CategoryTheory.pseudofunctorOfIsLocallyDiscrete`, `Mathlib/CategoryTheory/Bicategory/Functor/LocallyDiscrete.lean` | P0 and P7 provide the canonical identity and composition isomorphisms; the constructor supplies the action on 2-morphisms. |

#### P1

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Multiplication of tilt coordinates | `map_mul`, `Mathlib/Algebra/Group/Hom/Defs.lean` | Apply to P1’s `Perfectoid.tilt.coeff`. Check `tilt.coeff_mul` uses the monoid homomorphism law. |
| Perfect plus ring and strict Witt ring | `WittVector.eq_zero_of_p_mul_eq_zero` and `WittVector.isAdicCompleteIdealSpanP`, `Mathlib/RingTheory/WittVector/Complete.lean` | P1.2 gives the perfect plus ring (every `R⁺` of a characteristic-`p` perfectoid Tate ring is perfect). Use P1.2 for perfectness of `R⁺`, then Mathlib for `p`-torsion freeness and `p`-adic completeness of `W(R⁺)`. The statement of *witt-vectors-of-perfect-plus-ring* (P1.16) supplies the remaining completeness, nonzerodivisor and quotient statements. |
| Completion and monomials | `AdicCompletion`, `Mathlib/RingTheory/AdicCompletion/Basic.lean`; `AdicCompletion.evalₐ`, `Mathlib/RingTheory/AdicCompletion/Algebra.lean`; `AddMonoidAlgebra.single`, `Mathlib/Algebra/MonoidAlgebra/Defs.lean`; `Localization.Away`, `Mathlib/RingTheory/Localization/Away/Basic.lean` | The perfectoid completion topology, its universal property and tilting comparison. |
| Multiplicative perfection | `Perfection.coeffMonoidHom`, `Perfection.coeffMonoidHom_pow_p'`, `Perfection.extMonoid`, `Perfection.mapMonoidHom`, `PerfectionMap.id`, `PerfectionMap.equiv`, `PerfectionMap.comp_symm_equiv`, all in `Mathlib/RingTheory/Perfection.lean` | Limiting addition and topological Tate-ring comparison; characteristic-`p` root existence and uniqueness use the supplier equivalence directly. |
| Restriction to integral subrings | `RingHom.restrict`, `Mathlib/Algebra/Ring/Subsemiring/Basic.lean` | Supply `map_mem_powerBoundedSubring`; use the restriction directly. |
| Localised quotient | `IsLocalization.map`, `IsLocalization.map_surjective_of_surjective`, `Mathlib/RingTheory/Localization/Defs.lean`; `IsLocalization.ker_map`, `Mathlib/RingTheory/Localization/Algebra.lean` | The primitive-ideal topology and perfectoid theorem; no local surjectivity/kernel theorem. |
| Integral tilt and Fontaine map | `PreTilt`, `PreTilt.coeff`, `Tilt`, `PreTilt.val`, `PreTilt.isDomain`, `Mathlib/RingTheory/Perfection.lean`; `PreTilt.untilt`, `PreTilt.mk_untilt_eq_coeff_zero`, `PreTilt.untilt_iterate_frobeniusEquiv_symm_pow`, `Mathlib/RingTheory/Perfectoid/Untilt.lean`; `WittVector.fontaineTheta`, `WittVector.fontaineTheta_teichmuller`, `WittVector.mk_fontaineTheta`, `surjective_fontaineTheta`, `Mathlib/RingTheory/Perfectoid/FontaineTheta.lean` | Supply the nonzero integral-ring completeness and nonunit hypotheses, and identify the Tate tilt. |

#### P2

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Binary-product projections and uniqueness | `CategoryTheory.Limits.prod`, `CategoryTheory.Limits.prod.fst`, `CategoryTheory.Limits.prod.snd`, `CategoryTheory.Limits.prod.lift`, `CategoryTheory.Limits.prod.{lift_fst, lift_snd, hom_ext}`, `Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts/BinaryProducts.lean` | Apply to `PerfectoidSpace.Perf p` using `PerfectoidSpace.hasBinaryProducts_perf`; the projection equations and uniqueness are supplied for every perfectoid test object in characteristic p. |
| Restriction of a homeomorphism to a subset | `Homeomorph.image`, `Mathlib/Topology/Homeomorph/Lemmas.lean` | Apply to `spaTiltHomeomorph`; no separate existence theorem. |
| Specialisation under an embedding | `Topology.IsInducing.specializes_iff`, `Mathlib/Topology/Inseparable.lean` | Apply to the inducing map of `spaTiltHomeomorph`. |
| Pullback projection equations and uniqueness | `CategoryTheory.Limits.pullback.lift_fst`, `.lift_snd`, `.hom_ext`, `Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean` | Apply to perfectoid test spaces using `PerfectoidSpace.hasPullbacks`; `pullback.isPullback_toAdicSpace` supplies the universal property for arbitrary adic test spaces. |
| Sheaf and Čech cohomology | `CategoryTheory.Sheaf.H`, `Sheaf.H.map`, `Sheaf.cohomologyPresheafFunctor`; `Mathlib/CategoryTheory/Sites/SheafCohomology/Basic.lean`; `CategoryTheory.cechComplexFunctor`, `Mathlib/CategoryTheory/Sites/SheafCohomology/Cech.lean`; `CategoryTheory.Abelian.Ext.mapExactFunctor`, `Mathlib/Algebra/Homology/DerivedCategory/Ext/Map.lean`; `CategoryTheory.HasExt.standard`, `Mathlib/Algebra/Homology/DerivedCategory/Ext/Basic.lean` | Perfectoid vanishing and scalar-annihilation targets on the supplied cohomology; P7.15 uses the supplied Ext maps for the finite-stage and limit pullbacks in `residueEtaleCohomologyColimit`. |
| Reduction of modules | `ModuleCat.extendScalars`, `ModuleCat.restrictScalars`; `Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean`; `IsHausdorff`, `IsAdicComplete`, `Mathlib/RingTheory/AdicCompletion/Basic.lean` | Termwise reduction and almost lifting with exact separation and completeness hypotheses. |
| Polynomial presentations and adic completion | `MvPolynomial.eval₂Hom`, `Mathlib/Algebra/MvPolynomial/Eval.lean`; `AdicCompletion`, `Mathlib/RingTheory/AdicCompletion/Basic.lean` | Root-adjoined rational models and their completion comparison. |

#### P3

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Localized completion map | `Localization.awayMap`, `Mathlib/RingTheory/Localization/Away/Basic.lean` | Use `Localization.awayMap (completionMap t I) t` in scalar extension, section approximation and orbit comparisons. |
| Equivalences preserve limits | `CategoryTheory.Adjunction.isEquivalencePreservesLimits`, `Mathlib/CategoryTheory/Adjunction/Limits.lean` | `etaleTilt_preservesFiniteLimits` applies this instance to `etaleTiltFunctor_isEquivalence`; P3 supplies the geometric equivalence. |
| Sheaves under a site equivalence | `CategoryTheory.Equivalence.sheafCongr`, `Mathlib/CategoryTheory/Sites/Equivalence.lean` | `etaleTiltSheafEquiv` applies this construction to `etaleTiltFunctor` with covering equivalence `etaleTilt_covers`. |
| Finite étale categories, base change and geometric fibre | `CommAlgCat.FiniteEtale`, `CommAlgCat.FiniteEtale.of`, `CommAlgCat.FiniteEtale.ofHom`, `CommAlgCat.FiniteEtale.baseChange`, `CommAlgCat.FiniteEtale.baseChangeSelfIso`, `CommAlgCat.FiniteEtale.fiber`, `CommAlgCat.FiniteEtale.equivOfIsSepClosed`, all in `Mathlib/RingTheory/Etale/Finite.lean` | Untilting and almost purity; the algebraically closed fibre equivalence is used directly. |
| Complete henselian pairs | `IsAdicComplete.henselianRing`, `Mathlib/RingTheory/Henselian.lean` | AdicEtaleGeometry A1.0.4–A1.0.5 owns generic henselian/Elkik approximation and the completed filtered-colimit comparison. P3 uses its perfectoid-field specialization; A4 uses A1 directly. |
| Restriction to power-bounded rings | `RingHom.restrict`, `Mathlib/Algebra/Ring/Subsemiring/Basic.lean` | Apply it with P1's preservation of power-bounded elements and P3's finite-étale uniformity theorem. |
| Jacobian minors and colon ideals | `MvPolynomial.pderiv`, `Mathlib/Algebra/MvPolynomial/PDeriv.lean`; `Submodule.colon`, `Mathlib/RingTheory/Ideal/Colon.lean` | AdicEtaleGeometry A1.0.4's `jacobianIdeal` combines all minors and colon ideals for arbitrary finite polynomial presentations; P3 uses this supplied approximation theory. |
| Tensor coherence and product projections | `Algebra.TensorProduct.cancelBaseChange`, `Mathlib/RingTheory/TensorProduct/Maps.lean`; `AlgHom.fst`, `AlgHom.snd`, `AlgHom.prodMap`, `Mathlib/Algebra/Algebra/Prod.lean` | P3.5 uses these canonical maps to characterize untilting comparisons and their identity and composition laws. |
| Moduli functors and integral completions | `CategoryTheory.coyoneda`, `Mathlib/CategoryTheory/Yoneda.lean`; `Ideal.Quotient.factor`, `Mathlib/RingTheory/Ideal/Quotient/Defs.lean`; `Ideal.adicTopology`, `Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean` | AdicEtaleGeometry A1.0.4 owns the filtration `tⁿI` and its section topologies; A1.0.5 owns algebra laws on a fixed projective module and the smooth groupoid comparison. P3 applies these supplied constructions. |
| Finite product étaleness and discrete finite sets | `Algebra.FormallyEtale.pi_iff` and the finite-product `Algebra.Etale` instance, `Mathlib/RingTheory/Etale/Pi.lean`; `FintypeCatDiscrete` scoped instances, `Mathlib/Topology/Category/FinTopCat.lean` | Transport product étaleness to binary products; use continuous actions on finite discrete geometric fibres. |
| Field structure on the original tilt ring | `MulEquiv.isField`, `Mathlib/Algebra/Field/Equiv.lean`; `IsField.toField`, `Mathlib/Algebra/Field/IsField.lean` | Transport the supplied field property while retaining the tilt ring operations for geometric fibres. P1 supplies `Perfectoid.tiltField.toTilt`. |
| Étale cohomology and effective descent | `CategoryTheory.Sheaf.H`, `Mathlib/CategoryTheory/Sites/SheafCohomology/Basic.lean`; `CategoryTheory.Pseudofunctor.IsStack`, `Mathlib/CategoryTheory/Sites/Descent/IsStack.lean` | P3 supplies the actual perfectoid étale sheaves, their scalar endomorphisms, and the finite étale pullback prestack. |
| Slice pullback coherence | `CategoryTheory.Over.pullbackId`, `CategoryTheory.Over.pullbackComp`, `Mathlib/CategoryTheory/Comma/Over/Pullback.lean`; `CategoryTheory.LocallyDiscrete.eq_of_hom`, `Mathlib/CategoryTheory/Bicategory/LocallyDiscrete.lean` | P3 and P5 restrict these comparisons to the finite étale, qcqs étale and quasicompact separated étale slices of `Perfd p`; the new component equations specify those restrictions. |

#### P4

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Monomorphisms and diagonals | `CategoryTheory.Mono`, `CategoryTheory.mono_comp`, `CategoryTheory.mono_of_mono`, `Mathlib/CategoryTheory/Category/Basic.lean`; `CategoryTheory.Limits.pullback.isIso_diagonal_iff`, `Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean` | Affinoid tests, residue-field criteria and the topological pullback comparison; use generic categorical consequences directly. |

#### P5

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Pseudofunctor Grothendieck category | `CategoryTheory.Pseudofunctor.Grothendieck`, `Mathlib/CategoryTheory/Bicategory/Grothendieck.lean` | Localize at arrows with invertible fibre component; identify morphisms by common cospan refinement. |
| Coherent pseudococones | `CategoryTheory.Pseudofunctor.StrongTrans`, `Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Pseudo.lean` | Compare the induced functor with the specified cocone, including uniqueness relative to its comparison data. |
| Categorical localization | `CategoryTheory.MorphismProperty.Localization`, `CategoryTheory.MorphismProperty.Q`, `Mathlib/CategoryTheory/Localization/Construction.lean` | Apply the supplied category construction to filtered pseudofunctors; prove the finite-stage Hom description. |
| Module base change | `ModuleCat.extendScalars`, `Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean`; `CategoryTheory.typeToCat`, `Mathlib/CategoryTheory/Category/Cat.lean` | Restrict scalar extension to finitely presented modules; compute module and discrete examples of the filtered 2-colimit. |
| Constant pseudofunctor | `CategoryTheory.Functor.const`, `Mathlib/CategoryTheory/Functor/Const.lean`; `CategoryTheory.Functor.toPseudofunctor'`, `Mathlib/CategoryTheory/Bicategory/Functor/LocallyDiscrete.lean` | Construct the filtered 2-colimit, using these suppliers directly for constant diagrams. |
| Adic completion and finite quotients | `AdicCompletion.isAdicComplete`, `AdicCompletion.pow_smul_top_eq_ker_eval`, `Mathlib/RingTheory/AdicCompletion/Completeness.lean`; `AdicCompletion.evalₐ`, `AdicCompletion.surjective_evalₐ`, `Mathlib/RingTheory/AdicCompletion/Algebra.lean` | Perfectoidness of the completed colimit and its plus-ring comparison. |
| Completion injectivity and surjectivity | `AdicCompletion.of_injective_iff`, `AdicCompletion.of_surjective_iff`, `Mathlib/RingTheory/AdicCompletion/Basic.lean` | Prove separatedness and completeness of the uncompleted colimit under `ℵ₁`-filteredness. |
| Cofiltered tails and ordinary reindexing | `CategoryTheory.Over.initial_forget`, `Mathlib/CategoryTheory/Filtered/Final.lean`; `CategoryTheory.Functor.Final.colimitIso`, `Mathlib/CategoryTheory/Limits/Final.lean` | Compatibility with plus rings and completion; the 2-colimit equivalence in P5.8. |

#### P6

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Restricting an equivalence to full subcategories | `CategoryTheory.Equivalence.congrFullSubcategory`, `Mathlib/CategoryTheory/ObjectProperty/Equivalence.lean` | P6's `kappaSmallSliceTilt` applies it to P2's slice tilt using the tilting invariance and the isomorphism invariance of `IsKappaSmall` (P6.4, P6.1). |
| Pro-category cofiltered limits | Mathlib's `HasFilteredColimits (Ind Cᵒᵖ)` instance in `Mathlib/CategoryTheory/Limits/Indization/Category.lean:131`; `CategoryTheory.Limits.has_cofiltered_limits_op_of_has_filtered_colimits` in `Mathlib/CategoryTheory/Limits/Shapes/Opposites/Filtered.lean:34` | For `C : Type u` with `Category.{v} C` and a small cofiltered `I : Type v`, these supply `HasLimitsOfShape I (Ind Cᵒᵖ)ᵒᵖ` by instance synthesis, a Check specified in this README. P6.12 uses them directly. All small limits additionally require finite limits in the base category. |
| Adjunction triangle identities | `CategoryTheory.Adjunction.left_triangle_components`, `.right_triangle_components`, `Mathlib/CategoryTheory/Adjunction/Basic.lean` | `proFlatten_obj` applies both equations to the adjunction supplied by `proFlatten_adjunction` on the opposite Ind carriers. |
| Bounded pro-objects | `CategoryTheory.Ind.lim`, `CategoryTheory.Ind.yoneda`, `Mathlib/CategoryTheory/Limits/Indization/Category.lean`; `CategoryTheory.ObjectProperty.FullSubcategory`, `Mathlib/CategoryTheory/ObjectProperty/FullSubcategory.lean` | P6.10 owns `PerfectoidSpace.HasKappaSmallPresentation`, `.of_diagram` and `PerfectoidSpace.ProKappa`: an existential bound on the total arrow set of a presentation, on the supplied Ind carrier. |
| Cardinal completion | `UniformSpace.Completion`, `Mathlib/Topology/UniformSpace/Completion.lean`; `FirstCountableTopology`, `Mathlib/Topology/Bases.lean`; `IsUniformAddGroup`, `Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean` | P6.2 owns the universe-polymorphic additive-group bound `completion_cardinality_bound`. |
| Pro-object flattening | `CategoryTheory.Ind.presentation`, `Mathlib/CategoryTheory/Limits/Indization/Category.lean`; `CategoryTheory.Limits.limit`, `Mathlib/CategoryTheory/Limits/HasLimits.lean` | P6.12 owns `proFlatten`, `proFlatten_adjunction` and `proFlatten_obj`. |
| Empty and singleton cardinalities | `Cardinal.mk_eq_zero`, `Cardinal.mk_eq_one`, `Mathlib/SetTheory/Cardinal/Defs.lean` | Compute the point set and section ring of the empty affinoid; no new cardinal definitions. |
| Strong limits and power-set bounds | `Cardinal.IsStrongLimit`, `Cardinal.mk_set`, `Cardinal.mk_le_of_injective`, `Cardinal.power_mul`, `Cardinal.power_le_power_right`, `Mathlib/SetTheory/Cardinal/Order.lean` | Apply to valuative relations and countable completion; the perfectoid size predicate is additional. |
| Union and cofinality bounds | `Cardinal.mk_iUnion_le`, `Mathlib/SetTheory/Cardinal/Basic.lean`; `Cardinal.iSup_lt_of_lt_cof_ord`, `Mathlib/SetTheory/Cardinal/Cofinality/Ordinal.lean` | The uniform bound in the singular-cardinal cover criterion. |
| Profinite presentations | `Profinite.asLimit`, `Mathlib/Topology/Category/Profinite/AsLimit.lean` | Realize finite quotients as disjoint unions of perfectoid spaces. |
| Scheme pro-étale objects | `AlgebraicGeometry.Scheme.ProEt`, `Mathlib/AlgebraicGeometry/Sites/Proetale.lean` | A comparison boundary only: its weakly étale scheme maps are different from ECD's perfectoid maps. |
| Small categorical models | `CategoryTheory.SmallModel`, `CategoryTheory.equivSmallModel`, `Mathlib/CategoryTheory/EssentiallySmall.lean`; `CategoryTheory.Skeleton`, `Mathlib/CategoryTheory/Skeletal.lean` | Replace the affinoid étale and canonical factorization categories by small models; count isomorphism classes with a skeleton. |
| Canonical factorization category | `CategoryTheory.StructuredArrow`, `StructuredArrow.mk`, `StructuredArrow.homMk`, `StructuredArrow.proj`, `Mathlib/CategoryTheory/Comma/StructuredArrow/Basic.lean` | The category of arrows from the given perfectoid space into the affinoid étale slice. |
| Pro-object carrier | `CategoryTheory.Ind`, `Ind.yoneda`, `Ind.lim`, `Mathlib/CategoryTheory/Limits/Indization/Category.lean` | Use `(Ind Cᵒᵖ)ᵒᵖ`; the perfectoid realization and its cardinal restriction are additional. |
| One-object index categories | `CategoryTheory.SingleObj`, `SingleObj.mapHom`, `Mathlib/CategoryTheory/SingleObj.lean` | The absorbing-zero example and its faithful nonfull two-arrow subcategory. |
| Continuous profinite maps | `CompHausLike.ofHom`, `Mathlib/Topology/Category/CompHausLike/Basic.lean` | Bundle constant maps and the two-point transposition used by the Hom variance Checks. |
| Finite module presentations | `Module.Finite.exists_fin'`, `Mathlib/RingTheory/Finiteness/Cardinality.lean` | Supplies the finite free surjection used in the finite-algebra cardinal bound. |
| Roots and continuous algebra maps | `rootsOfUnity`, `Mathlib/RingTheory/RootsOfUnity/Basic.lean`; `PadicInt.toZModPow`, `Mathlib/NumberTheory/Padics/RingHoms.lean`; `ContinuousAlgHom`, `Mathlib/Topology/Algebra/Algebra.lean` | Supply the finite roots, character exponents and completed Fourier-map carrier. |

#### P7

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Generic-fibre topology | `IsTopologicalAddGroup.rightUniformSpace`, `Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean`; `Localization.Away`, `Mathlib/RingTheory/Localization/Away/Basic.lean` | Parameter-power neighbourhood bases and compatible maps for the completed tower and strong completion. |
| Regularity transport and descent | `IsRegularLocalRing.of_ringEquiv`, `Mathlib/RingTheory/RegularLocalRing/Defs.lean`; `Submodule.IsNoetherian.of_isNoetherian_tensorProduct_of_faithfullyFlat`, `Mathlib/RingTheory/Flat/FaithfullyFlat/Basic.lean` | Cohen presentations, specified coefficient-extension completions and the regular tower conclusions. |
| Initiality of a cofiltered family | `CategoryTheory.Functor.initial_of_exists_of_isCofiltered`, `CategoryTheory.Over.initial_forget`, `Mathlib/CategoryTheory/Filtered/Final.lean`; `CategoryTheory.Functor.Initial.limitIso`, `Mathlib/CategoryTheory/Limits/Final.lean` | Apply to the compact-open level family and transport the density clause of tilde-limits. |
| Completion and its finite quotients | `AdicCompletion`, `Mathlib/RingTheory/AdicCompletion/Basic.lean`; `AdicCompletion.pow_smul_top_eq_ker_eval`, `Mathlib/RingTheory/AdicCompletion/Completeness.lean` | Complete the integral tower and identify its generic fibre. |

- coordinate compatibility and extensionality for the inverse limit of powers —
  `Perfection.coeffMonoidHom_pow_p'` and `Perfection.extMonoid` in
  `Mathlib/RingTheory/Perfection.lean`. The tilt uses these on its underlying multiplicative
  perfection; its addition and canonical additive uniformity are the additional constructions.

The integral ring of the strong completion in P7 is `AdicCompletion` in
`Mathlib/RingTheory/AdicCompletion/Basic.lean`; it does not need a separate type synonym.

`WittVector`, `WittVector.teichmuller`, `WittVector.isAdicCompleteIdealSpanP`, `Perfection`,
`Perfection.quotientMulEquiv`, `Perfection.teichmuller`, `Perfection.coeffMonoidHom`,
`Perfection.mapMonoidHom`, `PreTilt` with `PreTilt.coeff`, `PreTilt.untilt`,
`PreTilt.mk_untilt_eq_coeff_zero`, `PreTilt.untilt_iterate_frobeniusEquiv_symm_pow`, `PreTilt.val`,
`PreTilt.isDomain`, `Tilt`, `WittVector.fontaineTheta` with `surjective_fontaineTheta`,
`fontaineTheta_teichmuller` and `mk_fontaineTheta`, `HenselianRing` and `HenselianLocalRing`,
`IsAdicComplete.henselianRing`, `Algebra.Etale`, `CommAlgCat.FiniteEtale` with
`FiniteEtale.baseChange` and `FiniteEtale.equivOfIsSepClosed`, `Module.Flat`,
`Module.FaithfullyFlat` with `Module.FaithfullyFlat.bijective_of_tensorProduct`,
`comonadicExtendScalars`, `CommRingCat.isLimitForkPushoutSelfOfFaithfullyFlat`, `IsAdicComplete`,
`AdicCompletion`, `KaehlerDifferential`, `Algebra.Extension.Cotangent`,
`Algebra.Generators.Cotangent.exact`, `Algebra.H1Cotangent`, the Serre-class localisation of abelian
categories (`CategoryTheory.ObjectProperty.isoModSerre`, `MorphismProperty.Localization'`,
`SerreClassLocalization.abelian`, `isZero_obj_iff`, `isIso_map_iff`, `preservesFiniteLimits`,
`isoModSerre_isInvertedBy_iff`, `LocalizedMonoidal`, `CommMon`), `IsIdempotentElem`,
`ValuationRing`, `CategoryTheory.Sheaf.H` and the Čech machinery, `MulRingSeminorm`,
`ProfiniteGrp.exist_openNormalSubgroup_sub_open_nhds_of_one`, `CategoryTheory.Ind`, and the
cardinal arithmetic of `Cardinal.IsRegular` and `Cardinal.IsStrongLimit`. `Subgroup.exists_le_of_iInf_le_of_directed`
is Tau Ceti's, listed below.

### From Tau Ceti

#### P0

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Ordinary idempotent lifting | `TauCeti.HenselianRing.exists_isIdempotentElem_sub_mem`, `TauCeti/RingTheory/Henselian/Idempotent.lean`; `TauCeti.IsAdicComplete.exists_isIdempotentElem_eq`, `TauCeti/RingTheory/AdicCompletion/Idempotent.lean` | Lifting for tight henselian almost pairs. |

#### P1

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Reducedness | `TauCeti.Huber.IsUniform.isReduced`, `TauCeti/RingTheory/Huber/Uniform.lean` | Apply to the bounded power-bounded subring of a perfectoid Tate ring; no second reducedness theorem. |
| Ordinary restricted series | `TauCeti.Huber.restrictedMvPowerSeriesCompletion`, `TauCeti/RingTheory/Huber/WeightedRestrictedSeries/Completion.lean`; `TauCeti.Huber.weightedX`, `TauCeti/RingTheory/Huber/WeightedRestrictedSeries/Basic.lean` | The inclusion in the perfected algebra and the continuous linear projection onto integral exponents. |

#### P2

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Stable uniformity and sheafiness | `TauCeti.Huber.IsStablyUniform`, `TauCeti.Huber.isStablyUniform_iff_forall_isUniform_completionLocObj`, `TauCeti/RingTheory/Huber/StableUniform.lean`; `TauCeti.Huber.IsSheafyRing`, `IsSheafyRing.isSheafyForEveryPresentation`, `TauCeti/AlgebraicGeometry/AdicSpace/Spa/StructurePresheaf/SheafyRing.lean` | `Perfectoid.isStablyUniform` supplies `Perfectoid.PairCat.isStablyUniform` and `Perfectoid.isSheafyRing` supplies sheafiness for every plus ring. |
| Structure and integral presheaves | `TauCeti.ValuationSpectrum.presentationLimitPresheafInCommRingCat`, `TauCeti/AlgebraicGeometry/AdicSpace/Spa/StructurePresheaf/Stalk/Basic.lean`; `presentationLimitPlusSubring`, `presentationLimitPlusPresheaf`, `TauCeti/AlgebraicGeometry/AdicSpace/Spa/StructurePresheaf/Plus.lean`; `presentationLimitPresheafIsoRationalSubsetLimitPresheaf`, `TauCeti/AlgebraicGeometry/AdicSpace/Spa/StructurePresheaf/SubsetLimit.lean` | `rationalSectionsEquivPresentationLimit` identifies the compatible-family model with these supplied carriers, naturally in opens and preserving the integral subrings. |
| Completed rational-localization carrier | `TauCeti.Huber.PairOfDefinition.completionLocObj`, `TauCeti/RingTheory/Huber/LocalizationTopology/CompleteSeparated/Basic.lean` | `RationalLoc` is an abbreviation of this object’s carrier; its unbundled instances use the supplied completion uniformity and ring structure. |
| Rational universal property | `TauCeti.Huber.PairOfDefinition.existsUnique_continuous_ringHom_completion_locTopology`, `TauCeti.Huber.PairOfDefinition.completion_locTopology_ringHom_ext_of_continuous`, `TauCeti/RingTheory/Huber/LocalizationTopology/Completion.lean`; `TauCeti.ValuationSpectrum.existsUnique_continuous_ringHom_of_forall_comap_mem_rationalSubset`, `TauCeti/AlgebraicGeometry/AdicSpace/Spa/Localization/UniversalProperty.lean` | Use the supplied extension, its structure-map identity and uniqueness directly. `rationalLocalization.lift_mem_plus` adds preservation of the completed plus ring. |
| Tate structure and integral pair | `TauCeti.Huber.PairOfDefinition.isTateRing_completion_locTopology_of_isTateRing`, `TauCeti/RingTheory/Huber/LocalizationTopology/Completion.lean`; `TauCeti.Huber.PairOfDefinition.toCompletionLoc`, `TauCeti.Huber.PairOfDefinition.continuous_toCompletionLoc` in the same file; `TauCeti.Huber.PairOfDefinition.completedPlusSubring`, `TauCeti.Huber.PairOfDefinition.isRingOfIntegralElements_completedPlusSubring`, `TauCeti.Huber.PairOfDefinition.toCompletionLoc_mem_completedPlusSubring`, `TauCeti/RingTheory/Huber/LocalizationTopology/Plus.lean` | Bundle the supplied structures and map, with no new Tate or integral-elements theorem. |
| Strict finite type | `TauCeti.Huber.IsStrictlyTopologicallyFiniteType`; `TauCeti/RingTheory/Huber/TopologicallyFiniteType.lean` | Finite-type source in a p-finite model, with its topology retained. |

#### P4

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Quotient Huber pairs | `TauCeti.Huber.Pair.quotient`, `TauCeti.Huber.Pair.quotient_plus`, `TauCeti.Huber.Pair.quotientHom`, `TauCeti/RingTheory/Huber/Pair.lean`; `TauCeti.Huber.Pair.Hom.range_spaComap_quotientHom`, `TauCeti.Huber.Pair.Hom.isClosedEmbedding_spaComap_quotientHom`, `TauCeti/AlgebraicGeometry/AdicSpace/Spa/HuberPair.lean` | Recognise which quotients are perfectoid and construct the universal perfectoid quotient. |
| Finite inequality presentations | `TauCeti.Huber.PairOfDefinition.Presentation`, `TauCeti/RingTheory/Huber/LocalizationTopology/Presentation.lean`; `TauCeti.Huber.PairOfDefinition.hasDenominatorPower_of_pow_le_span`, `TauCeti/RingTheory/Huber/LocalizationTopology/Basic.lean`; `TauCeti.ValuationSpectrum.mem_rationalSubset_iff`, `TauCeti/AlgebraicGeometry/AdicSpace/Spa/RationalSubset/Basic.lean` | The chosen `({1} ∪ F)/1` presentation; use the existing denominator-power proof and rational-subset membership API. |

#### P5

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Integral closure under topological completion | `UniformSpace.Completion.isIntegrallyClosedIn_topologicalClosure_map_coeRingHom`, `TauCeti/Topology/Algebra/Nonarchimedean/Completion/Basic.lean` | Identify the algebraic principal-adic completion inside its localization (P5.1). |
| Valuations on directed unions of images | `TauCeti.ValuationSpectrum.ofDirected`, `TauCeti.ValuationSpectrum.comap_ofDirected`, `TauCeti.ValuationSpectrum.eq_ofDirected`, `TauCeti/AlgebraicGeometry/AdicSpace/ValuationSpectrum/OfDirected.lean` | Continuity and rational-subset descent for the generic Tate-pair spectrum comparison of AdicSpacesPartII R0.19.7, applied in P5.4. |
| Completion on spectra | `TauCeti.ValuationSpectrum.spaCompletionHomeomorph`, `TauCeti/AlgebraicGeometry/AdicSpace/Spa/Completion/Homeomorph.lean`; `TauCeti.ValuationSpectrum.spaCompletionHomeomorph_preimage_mem_spaRationalFamily_iff`, `TauCeti/AlgebraicGeometry/AdicSpace/Spa/Completion/RationalSubset.lean` | Identify the uncompleted spectrum with the inverse limit, then apply these suppliers. |

#### P6

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Valuative relations and empty spectra | `TauCeti.ValuationSpectrum.ext'` and the subsingleton-ring `IsEmpty` instance, `TauCeti/AlgebraicGeometry/AdicSpace/ValuationSpectrum/Basic.lean` | Inject the spectrum into binary relations; use the existing empty-spectrum instance. |
| Completed rational localizations | `TauCeti.Huber.PairOfDefinition.completionLocObj`, `TauCeti/RingTheory/Huber/LocalizationTopology/CompleteSeparated/Basic.lean` | The cardinal predicate uses the supplied carrier directly. |

#### P7

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Completion on adic spectra | `TauCeti.ValuationSpectrum.spaCompletionHomeomorph`, `TauCeti/AlgebraicGeometry/AdicSpace/Spa/Completion/Homeomorph.lean` | Relate the completed spectrum to the compatible finite-level valuations. |

- the rational structure map and plus ring — `TauCeti.Huber.PairOfDefinition.toCompletionLoc`
  and `continuous_toCompletionLoc` in `TauCeti/RingTheory/Huber/LocalizationTopology/Completion.lean`,
  and `completedPlusSubring` in `TauCeti/RingTheory/Huber/LocalizationTopology/Plus.lean`.
  `RationalLoc` abbreviates the carrier of `PairOfDefinition.completionLocObj`; the pair
  construction bundles these suppliers as a `Huber.Pair` and `Pair.Hom`;
- spectrality of maps of adic spectra with Tate source ring —
  `TauCeti.Huber.Pair.Hom.isSpectralMap_spaComap_of_isTateRing` in
  `TauCeti/AlgebraicGeometry/AdicSpace/Spa/MorphismSpectral.lean`. A general Huber tilde-limit does not
  make its projections spectral: the rational open `T ≠ 0` of the discrete ring `𝔽_p[T]`
  pulls back along `𝔽_p[T] → K⟨T^{1/p^∞}⟩` to the non-quasicompact punctured perfectoid disc;

Stable uniformity is `TauCeti.Huber.IsStablyUniform` in
`TauCeti/RingTheory/Huber/StableUniform.lean`, with the exact criterion
`TauCeti.Huber.isStablyUniform_iff_forall_isUniform_completionLocObj` in that file.
The new P2 assertion is that perfectoid rings satisfy this library criterion.


The Huber-ring theory (`TauCeti.Huber.IsHuberRing`, `IsTateRing`, `IsTateRing.isOpenMap`,
`IsPseudoUniformizer`, `PairOfDefinition`, `powerBoundedSubring`, `IsBounded`, `IsUniform`,
`IsUniform.isReduced`, `IsUniform.of_normedDivisionRing`, `IsStablyUniform` with
`TauCeti.Huber.isStablyUniform_iff_forall_isUniform_presentationLimit` and
`PairOfDefinition.isStablyUniform_completion_locTopology`, `IsRingOfIntegralElements`, `Pair`,
`Pair.Hom`, `Pair.quotient` with `TauCeti.Huber.Pair.quotient_plus`, `Padic.isTateRing`,
`Huber.WittVector.isHuberRing_adicTopology_span_p_teichmuller` and
`isTateRing_adicTopology_span_p_teichmuller_iff`, `PairOfDefinition.completionLocObj`,
`completedPlusSubring`, `pairHomOfRationalSubsetSubset`, `presentationRingEquivOfEq`,
`TauCeti.Huber.isSheafyRing_of_isStronglyNoetherian`, `IsSheafyRing`, the Laurent-cover halves
`injective_presentationLimitMap_inf_laurentCoverOpen_of_isStablyUniform` and
`exists_presentationLimitMap_eq_of_inf_laurentCoverOpen_of_isStablyUniform`), the adic spectrum
(`ValuationSpectrum.spa`, `rationalSubset`, `spectralSpace_spa_of_pairOfDefinition`,
`spaCompletionHomeomorph`, `spaLocalizationHomeomorph`, `isSpectralMap_spaComap_of_isTateRing`,
`Cont`, `isIso_toPresentationLimit_top`, `mem_iff_forall_vle_one`,
`existsUnique_continuous_ringHom_of_forall_comap_mem_rationalSubset`, `residueFieldValuation`,
`Pair.Hom.isClosedEmbedding_spaComap_quotientHom`, `range_spaComap_quotientHom`,
`isClosedEmbedding_spaComap_quotientMk`), the category of adic spaces (`TauCeti.AdicSpace`,
`TauCeti.PreAdicSpace`, `PreAdicSpace.Hom`, `PreAdicSpace.IsOpenImmersion`), Witt-vector
completeness (`WittVector.isAdicComplete_span_p_teichmuller`,
`isHausdorff_span_p_teichmuller`, `completeSpace_and_t2Space_adicTopology_span_p_teichmuller`), the
Fargues–Fontaine input (`TauCeti.FarguesFontaine.spaY`, `frobeniusHomeomorph`, `windowU`, `windowV`,
`spaX`), henselian idempotent lifting (`TauCeti.HenselianRing.exists_isIdempotentElem_sub_mem`,
`TauCeti.IsAdicComplete.exists_isIdempotentElem_eq`), the cofinality lemma for compact groups
(`Subgroup.exists_le_of_iInf_le_of_directed`), the inverse-system compactness lemma
(`TauCeti.exists_forall_map_eq_of_compact_t2`), and the uniform-group subring topology
(`TauCeti.Topology.Algebra.IsUniformGroup.Subring`).

### From other roadmaps

#### P0

| Object | Exact supplier | Convention/consumer |
| --- | --- | --- |
| Categorical duality trace | PivotalSpherical §Layer 2 / Left and right trace, `leftTrace`, `rightTrace`, common spherical `trace` | P0.12 retains `Almost.Module.trace` as its local internal realization for `Almost.Algebra.trace` and trace-form consumers; prove almost duality and use the supplier's standard symmetric pivotal/spherical structure (§Layer 7), with the tensor-unit/scalar comparison. |
| Cross-object cyclicity | PivotalSpherical §Layer 2 / Cyclicity, `leftTrace_comm` and common trace cyclicity | P0.12 transports the supplied law through the internal trace comparison for two almost finite projective objects; translate composition order to Lean's `≫`. |
| Tensor trace law | PivotalSpherical §Layer 2 / The left trace is monoidal, `leftTrace_tensor` and common trace monoidality | P0.12 transports the tensor-unit/unitor formula to multiplication in `A` through the chosen internal tensor and dual comparisons. |
| Classical projective trace comparison | AdicSpacesPartIII §R3.8 / *projective-module-trace*, `LinearMap.projectiveTrace`, `LinearMap.projectiveTrace_eq_trace` | P0.12's `trace_test_free_endomorphism` compares the supplied ordinary finite-projective trace with the local internal trace after almostification; the ordinary carrier belongs to AdicSpacesPartIII, the almost realization to this roadmap. |
| Ordinary module towers and Milnor maps | DerivedModuleAlgebra §1.T.1 / *lim and lim¹ of a tower*, *The six-term sequence of lim and lim¹*, *The Milnor sequence for towers of complexes* | P0.27 uses `ℕᵒᵖ ⥤ ModuleCat V`, `ModuleTower.Product`, `ModuleTower.shift`, `ModuleTower.lim`, `ModuleTower.limOne`, `ModuleTower.limMap` and `ModuleTower.limOneMap` directly. P0.28 uses the same natural six-term boundary and `V`-linear Milnor injection and projection. |

### From TauCetiRoadmap.AdicSpaces

Layers 0–1: Tate rings, pseudouniformisers, bounded sets, pairs of definition, completion,
valuation spectra and continuous valuations. Layers 2–3: Huber pairs, `Spa`, rational subsets,
completed rational localisation, structure presheaf, stalks, residue fields, pre-adic spaces. Layers
4–5: stable uniformity and the Buzzard–Verberkmoes theorem (layer 4.2, stably uniform implies
sheafy), adic spaces, open and closed subspaces, gluing. Layer 6: the perfect field `F` with its
`A_inf`, compared with the tilt of a perfectoid field in P1.

### From TauCetiRoadmap.AdicEtaleGeometry, AdicSpacesPartII and AdicSpacesPartIII

Supplier versions: [AdicSpacesPartII (`ee74c10`)](https://github.com/TauCetiProject/TauCetiRoadmap/tree/ee74c10ea2552c2cd75177bbe7082f3d125b709f/TauCetiRoadmap/AdicSpacesPartII)
and [AdicSpacesPartIII (`46632e7`)](https://github.com/TauCetiProject/TauCetiRoadmap/tree/46632e784d5f2d0817411fe6eaa03f35ca2d0884/TauCetiRoadmap/AdicSpacesPartIII).

#### P1

P1.2 uses AdicEtaleGeometry A1.0.2(i)'s ring-of-definition completeness theorem for its
integral-elements corollary. P1 uses AdicSpacesPartII R0.4 for Tate norm data and the spectral
seminorm (P1.3), and AdicSpacesPartIII R3.9 for the Gel'fand spectrum (P1.12–P1.13). P1.32 uses
AdicEtaleGeometry A1.0.0, A1.0.3a and A4.2–A4.5's split-ring and completion theory. The perfectoid frame predicate is P1.1's own
`Perfectoid.IsPerfectoidTateRing`, which P3.17a uses directly;
the modulo-`p` formulation they need is `IsPerfectoidTateRing.iff_frobenius_surjective_modP`.

#### P2

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Analyticity and affinoid sheafiness | `TauCetiRoadmap.AdicEtaleGeometry.YonedaAdicSpace.IsAnalytic`, `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.16; the sheafiness predicate `R.Sheafy` of an object `R` of `CAff`, same README, Geometric signatures and Checks | P2 and P7 use the supplied Tate-chart analyticity predicate. `AdicSpace.IsHonest` uses `R.Sheafy` for its affinoid charts, as do `Perfectoid.FrobeniusTower.isTildeLimit`, `PerfectoidSpace.GoodAffinoid.isUniformCompletion` and `PerfectoidSpace.IsPreperfectoid.closedSubspace`. |
| Ordinary and Čech acyclicity | AdicSpacesPartIII *sheafy-cech-acyclicity* (ordinary and Čech acyclicity of the structure sheaf on sheafy Tate affinoids), `TauCetiRoadmap/AdicSpacesPartIII/README.md`, R3.1 | Apply to `TauCetiRoadmap.AdicSpacesPartII.AdicSpace.Chart`, with coefficient ring `c.affinoid.ring` and spectrum `c.affinoid.spa`, made sheafy by `Perfectoid.isSheafyRing`, and transport through `rationalPresheaf_iso_presentationLimit`; these supply ordinary acyclicity of `rationalSheaf` and Čech acyclicity of `rationalPresheaf`. |
| Yoneda-adic carrier and affinoid spectra | `TauCetiRoadmap.AdicEtaleGeometry.YonedaAdicSpace`, `YonedaAdicSpace.ofSpaY`, `YonedaAdicSpace.spaYFunctor`, `YonedaAdicSpace.IsOpenImmersion`, `YonedaAdicSpace.space`; `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.16 | The analytic carrier of the comparisons of P2.9–P2.10; the full perfectoid subcategory itself is cut out of Tau Ceti's `PreAdicSpace` by sheafiness and open perfectoid affinoid covers (P2.10). |
| Classical adic comparison | `TauCetiRoadmap.AdicEtaleGeometry.YonedaAdicSpace.ofAdicSpace`, `ofAdicSpace_spa`, `ofAdicSpace_space` (natural in the adic space), and *adic-spaces-in-yoneda-adic-spaces* (`ofAdicSpace` is full and faithful and preserves and reflects open immersions); `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.16 | The comparisons of P2.9 and P2.10 identify the Yoneda forms of the perfectoid predicates with the classical presentations and open covers, which are the Lean definitions. |
| Maps from arbitrary classical adic spaces | `TauCetiRoadmap.AdicEtaleGeometry.YonedaAdicSpace.mapsFromAdicSpaceEquiv`, `mapsFromAdicSpaceEquiv_sheafy`, `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.16; `TauCetiRoadmap.AdicEtaleGeometry.AdicSpace.Oplus`, same README, Geometric signatures and Checks | `spaHomEquiv_adic` transports the supplied equivalence to morphisms into the Yoneda spectrum of a perfectoid pair; the supplier supplies the topology of global sections. |
| Complete affinoid category | `TauCetiRoadmap.AdicEtaleGeometry.CAff`, whose morphisms are Tau Ceti's `Huber.Pair.Hom`, and `CAff.of`; `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.16 | Underlying Huber pairs and their continuous plus-preserving maps. |
| Topology on completed tensors and spectral rings | `TauCetiRoadmap.AdicSpacesPartII.Huber.CompletedTensor.isUniformAddGroup`, `.isTopologicalRing`, `.isHuberRing`, `Pair.completedTensor.instIsTateRing`, `SpectralTop.isUniformAddGroup`, `.isTopologicalRing`, `.isTateRing`; `TauCetiRoadmap/AdicSpacesPartII/README.md`, R0.3–R0.4 | Register these supplied theorems as instances for the perfectoid tensor statements. |
| Completed tensors and uniformization | `TauCetiRoadmap.AdicSpacesPartII.Huber.CompletedTensor`, `Uniformization`, `Pair.completedTensor` with `completedTensor.tmul`, `completedTensor.inl` and `completedTensor.inr` (in `Huber.Pair`), `Pair.uniformCompletedTensor.fromCompletedTensor`; `TauCetiRoadmap/AdicSpacesPartII/README.md`, R0.3–R0.4 | P2 adds perfectoidness, the almost integral comparison and its computed tensor conventions. |
| Compatible points over Tate bases | `TauCetiRoadmap.AdicEtaleGeometry.Huber.exists_spaPoint_of_isCompletedPushout`; `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A0.1 (*tate-completed-tensor-points*) | P2.15(d) and P2.16 apply point lifting on complete perfectoid Tate charts; P2.17 applies it on complete Tate charts before uniformization. |

#### P3

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Analytic carrier, affinoid charts and open immersions | `TauCetiRoadmap.AdicEtaleGeometry.YonedaAdicSpace`, `.ofSpaY`, `.spaYFunctor`, `.IsOpenImmersion`, `.space`; `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.16 | P3 compares its perfectoid-chart conditions with A1’s covering-chart predicates on this same carrier. |
| Analytic finite étale and étale predicates, sites and coverings | `TauCetiRoadmap.AdicEtaleGeometry.YonedaAdicSpace.IsFiniteEtale`, `.IsEtaleLocalDescription`, `.smallEtale`, `.smallFiniteEtale`, `.smallEtaleTopology`, `.smallFiniteEtaleTopology`, `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.9 | Use these predicates and sites directly on the Yoneda carrier, through `YonedaAdicSpace.ofAdicSpace`. `IsFiniteEtale.iff_adic` and `IsEtale.iff_adic` unfold the restricted predicates; `IsFiniteEtale.iff_forall_affinoidPerfectoid` supplies the perfectoid-chart characterization; `smallEtaleAdicEquiv` and `smallFiniteEtaleAdicEquiv` identify the sites, including arrows and covers. |
| Strong sheafiness and integral functions | `TauCetiRoadmap.AdicEtaleGeometry.Huber.IsStronglySheafy`, `AdicSpace.IsLocallyStronglySheafy`, `TauCetiRoadmap/AdicEtaleGeometry/Suggested.lean`, A1.1; `AdicSpace.Oplus` (Geometric signatures and Checks), `AdicSpace.structureSheafEtale`, `AdicSpace.structureSheafPlusEtale` (A1.6), `YonedaAdicSpace.ofAdicSpace` (A1.16), `TauCetiRoadmap/AdicEtaleGeometry/README.md` | `stronglySheafy` proves the supplied ring predicate. `locallyStronglySheafy` supplies an adic representative of a perfectoid Yoneda space, and `globalSections_adicComparison` identifies functions, integral functions, and pullbacks. |
| Natural finite-algebra topology and integral closure | `TauCetiRoadmap.AdicEtaleGeometry.Huber.FiniteEtale`, `.Pair.finiteEtale`, `.Pair.finiteEtale.hom`, `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.2 | P3 proves perfectoidness and compares the actual geometric tilt and Witt untilt. |
| Rational vector bundles | `TauCetiRoadmap.AdicSpacesPartIII.AdicSpace.vectorBundleEquivalence`, `TauCetiRoadmap/AdicSpacesPartIII/README.md`, R3.5 | Apply the equivalence to `TauCetiRoadmap.AdicSpacesPartII.AdicSpace.Chart`, with the Tate hypothesis on `c.affinoid.ring` and vector bundles on `c.affinoid.spa`; P3.19 uses this rational-site comparison ([KL15] Theorem 2.7.7, p. 58). |
| Étale vector bundles | AdicEtaleGeometry *stable-basis-etale-vector-bundles*, `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.6a | P3.19 uses the equivalence with finite locally free modules on the étale site, with a cover-dense affinoid basis stable under rational localization and finite étale maps; perfectoid affinoids supply this basis ([KL15] Theorem 8.2.22(c)–(d), p. 165; Definition 8.3.1, p. 166). |

#### P4

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Adic morphism properties | `TauCetiRoadmap.AdicSpacesPartII.AdicSpace.QuasiCompact`, `.IsClosedImmersion`, `.IsSeparated`, `.IsSeparated.iff_valuativeCriterion`, `TauCetiRoadmap/AdicSpacesPartII/Suggested.lean`, R0.6, R0.8 and R0.10 | `quasiCompact_iff_adicCompactOpens`, `isAdicClosedImmersion_iff_adicCharts` and `isSeparated_iff_adicDiagonal` transport the geometric conditions through `YonedaAdicSpace.ofAdicSpace`. R0's separatedness additionally requires local weak finite type and stably sheafy source charts; P4's topological closed immersions are more general than R0's quotient-defined ones. |
| Quasi-compact étale slices and finite towers; separated rigid quotients | `TauCetiRoadmap.AdicSpacesPartII.AdicSpace.QuasiCompact`, `.IsSeparated`, `TauCetiRoadmap/AdicSpacesPartII/Suggested.lean`, R0.6 and R0.10; `TauCetiRoadmap.AdicEtaleGeometry.YonedaAdicSpace.IsEtaleLocalDescription`, `.IsFiniteEtale`, `.smallEtale`, `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.9 | `qcqsEtale` tests the map and its diagonal using R0 on `toAdicSpace`; `finite_tower_over_perfectoid_tower` uses R0 on the limit map. `analyticQcqsEtale` restricts A1’s site. |
| Banach tensors over an analytic field | `TauCetiRoadmap.AdicSpacesPartII.Huber.BanachModule.CompletedTensor`, `.injective_tensorProduct_of_field`, `.ker_map_of_field`, `.injective_kerMap_of_field`, `.isStrictMap_map_of_field`, `TauCetiRoadmap/AdicSpacesPartII/README.md`, R0.5; `BanachModule.productSeminorm`, same README, R0.18 | Use the completed tensor for the nonarchimedean maximum-cost seminorm and the full kernel theorem without a strictness hypothesis. These are admitted sibling-roadmap suppliers. |

#### P6

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Affinoid étale finite-stage descent | This roadmap P5.13.4 *affinoid-etale-finite-stage-6-4-iv* (its essential surjectivity, named `affinoid_etale_finite_stage_essSurj`, and its Hom-colimit bijection) | P5.13.4 supplies essential surjectivity and the Hom-colimit comparison in ECD 6.4(iv). P6.5 cites both and states the common-stage model comparison and relative structure maps on `Perfd`. |
| Classical pro-étale objects and étale maps | `TauCetiRoadmap.AdicEtaleGeometry.AdicSpace.ProEtaleSite`, `AdicSpace.ProEtaleSite.forget`, `YonedaAdicSpace.smallEtale`, `YonedaAdicSpace.IsEtaleLocalDescription`, `YonedaAdicSpace.IsFiniteEtale`; `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.9 and A1.12 | P6.14 uses the supplied classical site and the supplied étale predicates. `ScholzeAffinoidTower.isAffinoidProEtale_scholze2013` concerns the completed perfectoid realizations of a cofiltered system of its affinoid presentations. |
| Complete pair and space carriers | `TauCetiRoadmap.AdicEtaleGeometry.CAff`, `CAff.IsRationalLocalization`, `YonedaAdicSpace`, `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.16 | The carriers of the affinoid tower comparison and of the comparison of the global perfectoid objects with the analytic carrier. |

#### P7

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Affinoid tilde-limit comparison | `TauCetiRoadmap.AdicSpacesPartII.AdicSpace.IsTildeLimit`, `.IsTildeLimit.affinoid`, `.IsTildeLimit.baseChange`, `TauCetiRoadmap/AdicSpacesPartII/Suggested.lean`, R0.20.1–R0.20.2; the completed-direct-limit criterion of *tilde-limits-affinoid-criterion*, `TauCetiRoadmap/AdicSpacesPartII/README.md`, R0.20.1 | `AdicSpace.IsTildeLimit.ofAdicSpace_iff` identifies the analytic Yoneda relation with AdicSpacesPartII R0.20.1 on honest analytic spaces with qcqs transitions. `PerfectoidSpace.IsCompletedColimitOfTower.isTildeLimit_affinoid` applies the completed-direct-limit criterion with one generator on a cofinal tail of complete sheafy pairs; `PerfectoidSpace.IsTildeLimit_affinoid.isTildeLimit` transports the single-affinoid conditions to the analytic Yoneda diagram; `PerfectoidSpace.IsCompletedColimitOfTower.isTildeLimit` adds a perfectoid apex. The intrinsic tilde-limit theory belongs to AdicSpacesPartII R0.20.1–R0.20.2; `PerfectoidSpace.IsTildeLimit.toHuber` compares the analytic Yoneda form with Huber’s arbitrary-open conditions on qcqs spaces. |
| Finite étale algebras under uniformization | `TauCetiRoadmap.AdicEtaleGeometry.Huber.Pair.uniformization.baseChange_finiteEtale_equivalence`, `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A4.4; [KL15] Proposition 2.8.16(a), p. 64 | P7.4's `GoodAffinoid.finiteEtaleEquiv` composes this comparison with AdicEtaleGeometry A1.0.5's `finiteEtale_completedColimit` and the good chart's uniform-completion identification. |
| Finite-stage étale sites | `TauCetiRoadmap.AdicEtaleGeometry.YonedaAdicSpace.smallEtale`, `.IsEtaleLocalDescription`, `.IsFiniteEtale`, `.smallEtaleTopology`, `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.9 | P7.15 restricts the supplied site to qcqs structural maps in `analyticQcqsEtale`; `residueEtaleComparison_finite` uses the supplied finite étale predicate on both carriers, and `inseparable_tower_etale_topos` uses the supplied topology with finite-stage analyticity. |
| Affinoid point carrier | `TauCetiRoadmap.AdicEtaleGeometry.YonedaAdicSpace.space_ofSpaY`, `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.16 | Choose a family of the supplied homeomorphisms with a naturality equation for every map of pairs; use it to identify the points in the residue-density condition. |

AdicEtaleGeometry, by subsection: A0 (fibre products of analytic adic spaces) and A1.16 (the
analytic Yoneda carrier `YonedaAdicSpace`, `CAff`, `ofAdicSpace`, global functions); A1.1–A1.3, A1.5,
A1.6 and A1.8: strongly sheafy rings, the finite étale pair (*finite-etale-affinoid-algebra*), étale
and finite étale morphisms of adic spaces with their stability properties (*etale-composition*,
*etale-base-change*, *etale-diagonal*), the étale site with its structure sheaf and rational descent,
and the Galois category; A1.9: the finite étale and étale predicates and sites on the analytic
Yoneda carrier (*etale-site-generalized*); A1.12: the corrected pro-étale site (*pro-etale-site-corrected*) with its
profinite-set objects, for the comparisons of P6; A1.0.1–A1.0.2: henselian pairs under
filtered colimits and along a pseudouniformiser; A1.0.3–A1.0.5: algebraic finite étale
colimits, henselian approximation, section topologies and completion comparisons;
A1.2a: uniformity of finite étale algebras; A1.6a: vector bundles on a stable étale basis;
A4.4: finite étale algebras under uniformization. The generic finite étale Galois torsor towers of AdicEtaleGeometry A4.2–A4.5 and
A4.7 supply the completed pairs, self-products and orbit homeomorphisms used by P1.32 and P6.15–P6.17. AdicSpacesPartII, by
subsection: R0.1 (adic ring homomorphisms), R0.3 and R0.4 (completed tensor products with their
universal property (*completed-tensor-product-universal-property*) and uniform completed tensor
products, Tate norm data and the spectral seminorm (*tate-ring-norm*, *spectral-seminorm*)), R0.5
(Banach tensors), R0.6–R0.10 (adic morphisms, fibre products, closed immersions, separatedness and
the valuative criterion), R0.13 (finite morphisms), and the analytification of R1. AdicSpacesPartIII,
by subsection: R3.1 (Čech acyclicity of the structure sheaf on sheafy Tate affinoids,
*sheafy-cech-acyclicity*), R3.5 (vector bundles on sheafy Tate affinoids), R3.8 (finite projective
modules and trace forms over sheafy Tate rings) and R3.9 (the Gel'fand spectrum of a Banach ring,
*gelfand-spectrum-banach-ring*). AdicSpacesPartII R0.19.5–R0.19.10 supplies spectral limits,
Tate-pair spectra, finite-stage opens, generic Pro-categories and topos qcqs theory;
R0.20.1–R0.20.2 supplies intrinsic tilde-limits, the affinoid criterion and base change.
AdicSpacesPartIV R5.5–R5.6 consume P3.17a–P3.17b for smooth products and coefficient algebras.
DiamondsAndVStacks layers D0 and D2–D6 consume P5.4's perfectoid limit comparisons and P6's pro-categories and
completion bound. Its cutoff cardinals specialize the bounds using [ECD] Lemma 4.1, pp. 19–20. Its
pro-étale structure sheaves over a perfectoid base (D2.5), Čech-to-derived comparison, v-descent of
functions (D2.7, on its own v-site, D2.2), immersions and separatedness of pro-étale stacks (D3.8),
locally profinite torsors (D3.10), spatial diamonds and `Spd` are downstream interpretations of this
roadmap's perfectoid results and represented-functor comparisons.

### Combined library and roadmap suppliers

#### P1

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Characteristic-p `A_inf` comparison | `WittVector.map`, `WittVector.map_coeff`, `WittVector.map_injective`, `WittVector.map_surjective`, `Mathlib/RingTheory/WittVector/Basic.lean`; `TauCeti.FarguesFontaine.spaY`, `TauCeti/AlgebraicGeometry/AdicSpace/FarguesFontaine/Y.lean`; `TauCeti.FarguesFontaine.windowU`, `TauCeti/AlgebraicGeometry/AdicSpace/FarguesFontaine/Window.lean` | `Perfectoid.tilt.ainfEquiv` applies the Witt functor to the integral restriction of `tilt.equivOfCharP`; it identifies the input rings and their two-generator adic topologies, without redefining the curve or its windows. |

#### P2

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Residue valuations and their topology | `TauCeti.ValuationSpectrum.residueFieldValuation`; `TauCeti/AlgebraicGeometry/AdicSpace/ResidueField/Basic.lean`; `continuous_algebraMap_residueFieldValuation`, `TauCeti/AlgebraicGeometry/AdicSpace/ResidueField/Valued.lean`; `UniformSpace.Completion.coeRingHom`, `Mathlib/Topology/Algebra/UniformRing.lean` | Complete the supplied valued field and prove its perfectoid and tilting properties. |

#### P3

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Étale ring and almost coefficients | `AdicSpace.structureSheafEtale`, `AdicSpace.structureSheafPlusEtale`, `AdicSpace.smallEtale`, `AdicSpace.smallEtaleTopology` in `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.6; `CommRingCat` in `Mathlib/Algebra/Category/Ring/Basic.lean`, `CommAlgCat` in `Mathlib/Algebra/Category/CommAlgCat/Basic.lean`, `CategoryTheory.Presheaf.IsSheaf` in `Mathlib/CategoryTheory/Sites/Sheaf.lean` | `etaleORing` and `etaleOPlusRing` evaluate P2's perfectoid global-section carriers; `etaleRing_forget` and `etaleSheaves_adicComparison` identify the additive realizations. `etaleAdicBaseChange` has a perfectoid Yoneda source and uses the supplied adic target site. `etaleOPlusAlmost` uses the existing almost algebra category. P0 supplies `Almost.Algebra.toAlmost`. |

#### P4

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Open immersions on adic representatives | `TauCeti.PreAdicSpace.IsOpenImmersion`, `TauCeti/AlgebraicGeometry/AdicSpace/PreAdicSpace/OpenImmersion.lean`; `TauCetiRoadmap.AdicEtaleGeometry.YonedaAdicSpace.ofAdicSpace`, which preserves and reflects open immersions (*adic-spaces-in-yoneda-adic-spaces*), `TauCetiRoadmap/AdicEtaleGeometry/README.md`, A1.16 | `openImmersion_iff_adic` compares `Perfd` with the Yoneda notion; `openImmersion_iff_preAdic` compares compatible adic representatives. |
| Ordinary annulus and Gauss norms | `TauCetiRoadmap.AdicEtaleGeometry.Huber.tateSeries`, `.tateSeries.X`, `TauCetiRoadmap/AdicEtaleGeometry/Suggested.lean`; `TauCeti.ValuationSpectrum.gaussPoint`, `.gaussPoint_vle_iff`, `.supp_gaussPoint`, `TauCeti/AlgebraicGeometry/AdicSpace/Spa/Polydisc/GaussPoint.lean` | The ordinary annulus is a quotient of the supplied restricted-series ring. P4's `gaussPoint` concerns the completed root-polynomial algebra, rather than the supplier's ordinary disc. |

#### P7

| Object | Exact supplier and file | Use |
| --- | --- | --- |
| Algebraic torsor descent | `comonadicExtendScalars`, `Mathlib/Algebra/Category/ModuleCat/Descent.lean`; `Algebra.Etale.of_etale_tensorProduct_of_faithfullyFlat`, `Mathlib/RingTheory/Etale/Descent.lean`; `TauCetiRoadmap.AdicEtaleGeometry.Algebra.galoisTensorMap`, `TauCetiRoadmap/AdicEtaleGeometry/Suggested.lean` | AEG's finite-group formulation uses the Galois tensor isomorphism over arbitrary commutative rings and faithfully flat descent. AdicEtaleGeometry A4.2a owns this module descent. |
| Compact-open coverings | `IsCompactOpenCovered`, `Mathlib/Topology/Sets/CompactOpenCovered.lean` | P7 uses the raw compact-open covering condition `IsCompactOpenCovered`. |
| Represented inverse limits | `TauCetiRoadmap.AdicSpacesPartII.AdicSpace.IsTildeLimit`, `.IsTildeLimit.baseChange`, `TauCetiRoadmap/AdicSpacesPartII/Suggested.lean`, R0.20.1–R0.20.2, with the completed-direct-limit criterion of *tilde-limits-affinoid-criterion*, `TauCetiRoadmap/AdicSpacesPartII/README.md`, R0.20.1; `TauCetiRoadmap.AdicSpacesPartII.Huber.Pair.uniformization`, `.Pair.toUniformization`, `.Pair.uniformization.lift`, `TauCetiRoadmap/AdicSpacesPartII/README.md`, R0.4; `CategoryTheory.Limits.limit`, `Mathlib/CategoryTheory/Limits/HasLimits.lean`, and limits of functors | P7.11 adds limits of the represented functors `representedFunctor` on perfectoid test spaces, projection-compatible point comparisons, and HJ 5.11 finite-level descriptions; no topology is used. |
| Valued residue completion | `TauCeti.ValuationSpectrum.residueFieldValuation`, `TauCeti/AlgebraicGeometry/AdicSpace/ResidueField/Basic.lean`; `UniformSpace.Completion.coeRingHom`, `Mathlib/Topology/Algebra/UniformRing.lean` | Density of finite-level residue fractions in a geometric cone. |
| Compact-group cofinality | `Subgroup.exists_le_of_iInf_le_of_directed`, `TauCeti/Topology/Algebra/Group/Profinite/Basic.lean`; `ProfiniteGrp.exist_openNormalSubgroup_sub_open_nhds_of_one`, `ProfiniteGrp.closedSubgroup_eq_sInf_open`, `Mathlib/Topology/Algebra/ClopenNhdofOne.lean` | Construct the chosen indices in `CofinalityWitness`; no new compactness theorem. |

### Sousperfectoid and analytic approximation suppliers

| Object | Owner and section | Use |
| --- | --- | --- |
| Perfectoid frames, sousperfectoid rings and spaces | This roadmap P3.17a.1–P3.17a.4 | Polydiscs, tubes, closed étale loci and rational or finite étale stability. |
| Perfectoid times polydisc | This roadmap P3.17b | Coordinate-root obstruction and relative ball charts. |
| Generic pseudocoherent and standard-presentation APIs | AdicEtaleGeometry A3.1–A3.2 | Sheafy Tate-base Čech exactness, strict Laurent multiplication, restricted series, Newton iteration and standard presentations. |
| Tube and closed-locus approximation | This roadmap P5.13.1–P5.13.4 | Affinoid finite-stage descent and P6.5's relative-carrier adapter. |
| Generic split rings, torsor towers and completions | AdicEtaleGeometry A1.0.0/A1.0.3a, A4.2–A4.5 and A4.7 | Finite étale splittings, completed self-products, descent and orbit spectra. |
| Perfectoidness of split Tate rings | This roadmap P1.32, [split Tate rings](#p1-32-split-tate-rings) | Perfectoid completed tower pairs. |
| Continuous function rings and torsor presentations | This roadmap P6.15–P6.17 | Perfectoid action-groupoid objects and analytic presentation interfaces. |
| Rigid pro-étale site and corrected coverings | AdicEtaleGeometry A1.10–A1.12 | The site carrier, topology and morphism to the étale site. |
| Affinoid perfectoid basis on a rigid base | This roadmap P7.16, [rigid pro-étale site](#p7-16-rigid-proetale-site) | Basis, completed pairs and points, with the mixed-characteristic field hypotheses. |

## Internal dependencies

These are constructions owned by this roadmap, ordered by the consuming layer. The second column
names the earlier target; where a construction is a declaration of `Suggested.lean`, the
column says so, and otherwise the construction is specified in this README only, under the
proposed name given.

#### P0

| Object | Earlier target | Use |
| --- | --- | --- |
| Flat idempotent ideal multiplication | P0.2 (*almost-zero-iff-tilde-tensor-zero*), the bijectivity of `BasicSetup.tildeToIdeal : m̃ → m` for flat `m` (README only) | For a basic setup with flat ideal, multiplication `m ⊗_V m → m` is bijective; the classical, zero and two-term Checks of `tildeToIdeal` give concrete instances. The Lean `BasicSetup.tildeMul : m̃ → V` composes it with `m ↪ V`. |

#### P1

| Object | Earlier target | Use |
| --- | --- | --- |
| Uniformity and compatible roots | P1.1 and P1.3: the field `Perfectoid.IsPerfectoidTateRing.isBounded_powerBoundedSubring` and the theorem `Perfectoid.PseudoUniformizerRoots.nonempty`, both in `TauCetiRoadmap/PerfectoidSpaces/Suggested.lean`; `TauCeti.Huber.IsUniform`, `TauCeti/RingTheory/Huber/Uniform.lean` | `IsPerfectoidTateRing.isUniform` applies the defining field, and `PseudoUniformizerRoots.nonempty` gives the existence of compatible roots. |

#### P2

| Object | Earlier target | Use |
| --- | --- | --- |
| Rational restriction functor | P2.4: `Perfectoid.rationalPairFunctor`, its action on morphisms and the base-coordinate restriction identity (README only) | P2 uses this functor throughout. Its whole and empty values are checked by `rationalPairFunctor_whole` and `rationalPairFunctor_empty`; the base-coordinate restriction identity is checked by evaluating at a ring element. |
| Glued tilting | P2.11: `PerfectoidSpace.tiltFunctor`, `tiltHomeomorph`, `tiltSpaIso`, `tiltRestrictIso`, `sliceTiltFunctor_isEquivalence` (README only) | P2.11 owns these constructions. |
| Absolute-product chart universal property | P2.18: the field `PerfectoidSpace.ProductChart.universal` (README only) | Apply this field to `prodChart p A B ϖ ψ hA hB m n`: maps to a complete Tate pair correspond to maps from both factors whose two bounding ratios lie in the target plus ring. The existence of absolute products is the Lean `PerfectoidSpace.hasBinaryProducts_perf`. |

#### P3

| Object | Earlier target | Use |
| --- | --- | --- |
| Compatible pullback points | P2.16: `pullback_points` (README only); the pullbacks themselves are the Lean `PerfectoidSpace.hasPullbacks` | Apply this to étale maps as well; point surjectivity requires no étaleness hypothesis. |
| Almost finite étale algebras and slice untilting | `Almost.Algebra.FiniteEtale`, `.toAlmost`, `.elements` (P0.14, P0.8), `Perfectoid.PairCat.sliceUntiltFunctor` (P1.21) (README only) | P3 inverts a pseudouniformiser in almost elements and restricts the concrete Witt-vector slice untilt to finite étale algebras. |
| Global perfectoid objects and functions | `PerfectoidSpace.Perfd` (P2.10, in `TauCetiRoadmap/PerfectoidSpaces/Suggested.lean`); `.globalSections` and `.globalPlus` are abbreviations of the structure-sheaf sections `Γ(X, O_X)`, `Γ(X, O⁺_X)` of the underlying `PreAdicSpace`; `.globalPullback`, `.globalSharp`, `.sliceTiltFunctor` (P2, README only) | The new sites and sheaves use Tau Ceti's structure-sheaf sections as the carriers, with the chart descriptions as comparison lemmas. |

#### P4

| Object | Earlier target | Use |
| --- | --- | --- |
| Rational evaluation in a completed residue field | `Perfectoid.rationalResidueMap` with its specification (P3) and `.rationalPair_presentation` (P2) (README only) | Evaluate a perfectoid rational pair through any presentation comparison and the P3 map on `CAff.of R.pair`; the integrality statement of P4.5 supplies the integral-image condition. |
| Perfectoid geometry and small quotients | `PerfectoidSpace.Perfd` (P2.10, in `TauCetiRoadmap/PerfectoidSpaces/Suggested.lean`); `.pullback_points`, `.spaSpaceEquiv`, `Perfectoid.rationalPair`, `.rationalRes`, `.residuePair`, `.isPerfectoid_of_denseRange` in P2; `Perfectoid.tilt.sharpQuotEquiv`, `.sharpQuotEquiv_apply`, `Perfectoid.perfectedTateAlgebra` and `.laurent` in P1 (README only) | Global P4 morphism properties, rational-intersection and ideal constructions specialize these carriers; the existing tensor, residue and tilt constructions are not redefined. |

The normed-field Tate structure is supplied by Tau Ceti's Huber/Normed API,
`TauCeti.Huber.IsTateRing.of_nontriviallyNormedField`
([source at `83245abb`](https://github.com/TauCetiProject/TauCeti/blob/83245abb3e4a4377e8d779ac045363a90c7e3a86/TauCeti/RingTheory/Huber/Normed.lean#L244-L250)).
At the dependency pin `a91d3aaf`, the Tate instance is available only for `ℚ_p`;
`Suggested.lean` carries a compatibility signature with the supplier's exact name.
Dependency versions containing the supplied instance use its import in place of this signature.
`Perfectoid.IsTateRing.instProd` remains a target of P1.

### Generic approximation and descent suppliers

| Object | Owner and section | Use |
| --- | --- | --- |
| Henselian approximation and section topologies | AdicEtaleGeometry A1.0.4 | Smooth lifting and the density and openness inputs to finite étale completion. |
| Finite étale henselian completion and completed colimits | AdicEtaleGeometry A1.0.5 | P3.2's perfectoid-field application and P5/P7's completed-limit comparisons use `finiteEtale_completedColimit`. |
| Finite étale uniformity and power-bounded integral closure | AdicEtaleGeometry A1.2a | P3's almost-purity development uses `Huber.isBounded_powerBoundedSubring_of_finiteEtale`. |
| Finite-group algebraic module descent | AdicEtaleGeometry A4.2a | Effective faithfully flat descent for the Galois tensor isomorphism over arbitrary commutative rings. |

### Cotangent and deformation suppliers

| Object | Owner and section | Use |
| --- | --- | --- |
| Polynomial resolutions, derived tensor and the full cotangent model | DerivedModuleAlgebra §§0.1–0.2 and 0.4 | P0.19 consumes the ordinary carrier and its functorial models for P0.20's almost construction. |
| Square-zero extension classes and cotangent calculus | DerivedModuleAlgebra §§0.5–0.11 and 0.13 | P0.19 consumes extension classification, the universal property, low-degree comparisons, transitivity, the simplicial diagonal, base change, filtered colimits and regular-quotient computations. |
| Square-zero obstruction and deformation laws | DerivedModuleAlgebra §0.25 | P0.19 and P0.22 consume ordinary obstruction classes, flat deformations, morphism lifting, the signed transitivity boundary and the normalized torsor and automorphism comparisons. |

P0.25 constructs descent for almost modules and algebras from `comonadicExtendScalars` and
`CommRingCat.isLimitForkPushoutSelfOfFaithfullyFlat`; P0.28 constructs finite étale lifting for
complete almost pairs, including the classical setup. The classical henselian approximation,
algebra-structure moduli and completion comparisons are supplied by AdicEtaleGeometry A1.0.4–A1.0.5.
The ordinary cotangent complex of a ring map and its square-zero obstruction class are supplied
by DerivedModuleAlgebra §§0.1–0.2, 0.4–0.11, 0.13 and 0.25, consistently with
SchemeAndStackFoundations §4.6;
P0.19 consumes them and P0.20 owns the almost cotangent complex.

## How to read the build

P0 and P1 are independent until the almost setup of a perfectoid ring: P0 builds almost mathematics
over an arbitrary basic setup up to almost finite étale algebras, the almost cotangent complex and
finite étale lifting; P1 builds perfectoid Tate rings, tilts, perfectoid fields, `θ`, untilts and
the tilting equivalence of rings. P2 turns the ring theory into geometry (the tilting homeomorphism,
rational localisation, the sheaf theorem, perfectoid spaces and their fibre products), P3 proves
almost purity and builds the étale site, P4 the injections, immersions and separatedness, P5 the
cofiltered limits and finite-stage descent, P6 the `κ`-small calculus and pro-étale morphisms, P7 the
tilde-limits and Frobenius-controlled towers. Build order is by declarations: construct the
almost-algebra and module categories and the algebra adjunction in P0.8 before P0.16's
change-of-setup equivalence, then prove P0.8's `moduleEquiv`, its comparisons and the closed
monoidal structure. Construct P0.9's unitization before P0.25; its faithful-flatness comparison
uses P0.11 and is proved in P0.25. Construct `uniformTilt` in P1.7 before P1.4's perfectoid
specialization, and its comparisons with that specialization afterwards. Construct P1.24's
integral predicate and Frobenius criterion before P1.20's integral-to-Tate criterion.
A target's `*Needs:*` clause names the prerequisite declarations, the subsections of other
roadmaps and the library declarations it rests on, and each layer's Dependencies paragraph collects
the subsections of other roadmaps that the layer uses. P4.4 and the targets built on it use the
completed direct limit of P5.1–P5.3, which depends only on P1 and P2. Each subsection ends with the
Lean declarations that carry it, when there are any; targets without Lean are stated here only,
their carriers being formalisation tasks in their own right.

Each definition requires three discriminating Checks on its own carrier, including a
computed value and a degenerate case or non-example. Arithmetic identities, facts solely
about imported objects, repetitions of a target, and laws true of every morphism of the
same kind are supplementary calculations; they do not count toward these three Checks.
In particular, the following named calculations are supplementary wherever they occur:

- `cech_test_two_terms`, `naiveModel_test_two_exponents`, `lim1_test_sign`,
  `completion_test_torsion`;
- `trace_test_product`, `trace_test_dual_numbers`, `trace_test_unramified`,
  `differentials_test_char_two`, `cotangent_test_quotient`, `deformation_test_not_flat`,
  `differentials_test_sign`, `cotangent_test_unit_ideal`, `finiteEtale_test_swap`,
  `traceForm_test_product_diagonal`, `traceForm_test_dual_numbers_quotient`,
  `trace_test_quadratic_order`, `flat_squareZero_regular`, `not_isPerfectoidField_trivial`,
  `almostExact_test_nonseparated`, `cartan_zero_scalar`, `cartan_unit_scalar`,
  `cartan_two_term_additivity`;
- `filteredPrincipal_test_roots`, the root case of the powers-generate Checks of P0.1, `test_hom_formula`,
  `tight_test_principal`, `IsFiniteEtale_test_quadratic`, `kaehler_test_mathlib`,
  `map_id_test`, `test_constant_residue`, `restriction_disc_two_radii`,
  `rationalLoc_test_supplier`, `tiltHomeomorph_rationalSubset`, `pullback_tilt`,
  `perfectedTateAlgebra_fin_test`;
- `adjunction_test_two_terms`, `firmAdj_test_two_terms`, `sharp₀_zero`, `sharp₀_one`,
  `sharp₀_multiplication`, `global_sharp_zero_one`, `tiltEquiv_test_zero_one`,
  `residue_tilt_zero_one`, `rational_tilt_zero_one`, `tensor_tilt_two_constants`,
  `spaTiltHomeomorph_test_empty`, `spaTiltHomeomorph_test_whole`, `spa_space_empty`,
  `spa_space_whole`, `slice_identity`, `slice_affinoid_arrow`, `absolute_product_symmetry`,
  `pullback_identity_projection`, `homogeneous_zero`, `homLinearEquiv_zero`.

For example, `completion_test_torsion` only computes reduction from `ℤ/4` to `ℤ/2`;
it does not instantiate the completion complex. A complex comparison needs a Check
using that complex and its specified comparison map.

## Layer P0: almost mathematics over a basic setup

Almost mathematics is developed over an abstract basic setup `(V, m)`, so that it can be
instantiated later for `(R°, R°°)` and for the root ideal of a pseudouniformiser without
circularity. The layer builds the almost category as the Serre quotient of `V`-modules by almost
zero modules, with the functor of almost elements and the left adjoint `M_!`; the closed symmetric
monoidal structure; almost algebras with the left adjoint `B_!!`; almost finite generation and
presentation, flatness, projectivity, duals and traces, and the rank decomposition; flat, unramified,
étale and finite étale morphisms with the trace-form criterion and the comparison with Mathlib's
finite étale algebras; inversion of an element of `m`; almost Kähler differentials, the cotangent
complex of a ring map and the almost cotangent complex with its deformation theory; faithfully flat
descent; tight ideals, henselian pairs and modules over complete almost algebras; finite étale
lifting modulo a tight ideal; and almost sheaves with almost sheaf cohomology and almost Čech
complexes on a site.

### P0.1 basic setup `(V, m)`

The layer opens with the basic setup itself, its two standing examples and the firm-module model of
the almost category.

**Basic setup `(V, m)`: an idempotent ideal with flat `m ⊗ m`.** (*almost-basic-setup*) Let `V` be a
commutative ring (`[CommRing V]` in universe `u`) and `m` an ideal of `V`. A basic setup is a pair
`S = (V, m)` such that (i) `m` is idempotent, `m·m = m` (Mathlib `IsIdempotentElem m` in the
semiring `Ideal V`), and (ii) the `V`-module `m̃ := m ⊗_V m` is flat. Condition (i) is Gabber–Ramero's
basic setup, the only hypothesis they use before (2.5.14); it makes almost zero modules closed under
extensions ([GR] §2.1, (2.1.1), p. 7). Condition (ii) is the flatness they assume from (2.5.14) on
and use earlier for the exactness of `M ↦ M_!` in Corollary 2.2.22; it is imposed here once for all
almost mathematics of this roadmap, and every setup used here satisfies it: root ideals and maximal
ideals of valuation rings with non-finitely-generated maximal ideal satisfy condition (A) below, and
`m = V` is flat ([GR] §2.1, (2.1.1), p. 7, and (2.5.14), p. 32). Conventions: a `V`-module
`M` is almost zero if `m·M = 0`; a `V`-linear map is an almost isomorphism if its kernel and
cokernel are almost zero; `m̃` always denotes the tensor product `m ⊗_V m` (it maps onto
`m·m = m`, isomorphically when `m` is flat); the classical limit is `m = V`, which recovers ordinary
commutative algebra ([GR] §2.1, Example 2.1.2(i)-(ii), p. 7). Condition (A): `m` is a filtered union
of principal ideals; condition (B): for every `k ≥ 1` the `k`-th powers of elements of `m` generate
`m`. For an idempotent `m`, (A) implies (B): `x ∈ m = m^k` is a sum of products of `k` elements of
`m`, which lie in a common principal ideal `(z)`, so each product lies in `(z^k)`; without
idempotence (A) does not imply (B), as `(t) ⊆ 𝔽_p[t]` shows. (A) together with `m·m = m` implies
(ii) by *filtered-principal-union-tilde-flat* (P0.2). A basic setup is data, not a type class, because one
ring carries several setups: for a perfectoid Tate ring `R`, `(R°, R°°)` and `(R⁺, the root ideal of a pseudouniformiser)`, compared in layer P1. Bhatt's variant of the setup, with `m` itself flat, is a
special case ([Bhatt] §4.1, Assumption 4.1.2, p. 17). In `Almost`, define `BasicSetup` as the
structure with fields `ideal : Ideal V`, `isIdempotentElem : IsIdempotentElem ideal` and
`flat_tilde : Module.Flat V (ideal ⊗[V] ideal)`; `BasicSetup.tilde` as the `V`-module
`m̃ = S.ideal ⊗[V] S.ideal` with its multiplication map onto `m`; `BasicSetup.IsAlmostZero` (for a
`V`-module `M`: `∀ ε ∈ S.ideal, ∀ x : M, ε • x = 0`); `BasicSetup.IsAlmostIso` (kernel and cokernel
of `f : M →ₗ[V] N` almost zero); `BasicSetup.almostZero` (the object property of `ModuleCat V`
given by `IsAlmostZero`); `BasicSetup.classical` (the setup `(V, ⊤)`); `BasicSetup.zero` (the
setup `(V, ⊥)`: `⊥` is idempotent and `m̃ = 0` is flat); `BasicSetup.baseChange` (for
an algebra `W` over `V`, the setup `(W, S.ideal.map (algebraMap V W))`, *basic-setup-base-change*);
`BasicSetup.IsFilteredPrincipal` (condition (A)) and `BasicSetup.PowersGenerate` (condition (B)).
Use `IsIdempotentElem.eq S.isIdempotentElem` for `S.ideal * S.ideal = S.ideal`. Prove
`BasicSetup.isAlmostZero_classical_iff` (`(classical V).IsAlmostZero M ↔ Subsingleton M`),
`BasicSetup.IsFilteredPrincipal.powersGenerate` ((A) implies (B)),
`BasicSetup.powersGenerate` (flat `m̃` already implies (B), [GR] Proposition 2.1.7(ii), p. 9) and
`BasicSetup.isAlmostZero_iff_tensor_eq_zero` (`IsAlmostZero M ↔ Subsingleton (S.tilde ⊗[V] M)`,
the first clause of *almost-zero-iff-tilde-tensor-zero* (P0.2)). *Needs:* Mathlib `IsIdempotentElem`, `Module.Flat`,
`Ideal.span`.

**Checks.**

- Degenerate: for every commutative ring `V`, `BasicSetup.classical V`
  (ideal `⊤`) is a basic setup and `(classical V).IsAlmostZero M ↔ Subsingleton M`.
- Degenerate: for every `V`, `(V, ⊥)` is a basic setup
  (`⊥·⊥ = ⊥`, `⊥ ⊗ ⊥ = 0` is flat) and every `V`-module is almost zero for it.
- Non-example: for `V = ℤ_p` and `m` its maximal ideal, `m * m ≠ m`, so
  there is no `BasicSetup` with this ideal; accordingly `ℤ/p²`, an extension of `ℤ/p` by `ℤ/p` (both
  killed by `m`), is not killed by `m`, so a definition accepting non-idempotent ideals would lose
  closure of almost zero modules under extensions.
- Computation: for `V` the valuation ring of `ℂ_p` (or of the
  completion of `ℚ_p(p^{1/p^∞})`) and `m` its maximal ideal, the residue field `V/m` is almost zero,
  while `V/pV` is not, because `p^{1/p} • 1 ≠ 0` in `V/pV` (`p^{1/2}` would not do for the second
  field when `p` is odd: `1/2 ∉ ℤ[1/p]`, so `p^{1/2} ∉ V`).
- Non-example: the principal ideal `(t) ⊆ 𝔽_p[t]`
  satisfies (A) but not (B): the squares of its elements generate `(t²) ≠ (t)`; so "(A) implies
  (B)" needs `m·m = m`.

- Over `(ℤ, ℤ)`, `tildeMul(2 ⊗ 3 + 5 ⊗ 7) = 41`.
- The same tensor ideal is linearly equivalent to `ℤ`.
- `tildeMul(1 ⊗ 1) = 1` over `(ℤ, ℤ)`;
  for `m = 0`, the entire linear map `tildeMul` is zero.
- `test_tilde_zero` — for `m = 0`, `m̃ = 0`; on the zero ring the classical and zero ideals coincide.
- Both `IsAlmostZero` and its object property kill `V/m` and the zero module; over `(ℤ, ℤ)` they reject `ℤ`.
- Scalar extension preserves the classical setup, makes the ideal zero over `V/m`, and makes it the unit ideal over `V[1/ε]` for `ε ∈ m` (including `ε = 0`).

- `filteredPrincipal_test_classical`, `filteredPrincipal_test_zero`, `filteredPrincipal_test_roots` —
  the unit ideal (common divisor `1`), zero ideal (common divisor `0`), and any compatible-root ideal satisfy (A).
- Positive powers generate the unit ideal, zero ideal, and root ideal respectively; for roots, pass far enough down the chain.
- `almostIso_test_classical` — classically, an almost isomorphism is exactly a bijective linear map.
- `almostIso_test_ideal_inclusion` — `m ↪ V` is an almost isomorphism, although it need not be surjective.
- `almostIso_test_zero_setup` — every linear map is an almost isomorphism when `m = 0`.
- `almostIso_test_classical_double` — multiplication by `2` on `ℤ` is not an almost isomorphism in the classical setup.
- `tildeToIdeal_test_classical`, `tildeToIdeal_test_zero`, `tildeToIdeal_test_two_terms` —
  multiplication onto the ideal is bijective classically and for the zero ideal, and sends `2 ⊗ 3 + 5 ⊗ 7` to `41` over `ℤ`.

Lean: `Almost.BasicSetup`, `BasicSetup.tilde`, `tildeMul`, `IsAlmostZero`, `almostZero`,
`IsAlmostIso`, `classical`, `zero`, `baseChange`, `IsFilteredPrincipal`, `PowersGenerate`,
`IsFilteredPrincipal.powersGenerate`, `powersGenerate`, `flat_tilde_top`,
`isAlmostZero_classical_iff`, `BasicSetup.isAlmostIso_classical_iff`,
`isIdempotentElem_map`, `flat_tilde_map`,
`isAlmostZero_iff_tensor_eq_zero`.

### P0.2 the root setup of compatible p-power roots

**The basic setup of an element with compatible p-power roots.** (*root-ideal-basic-setup*) Let `V`
be a commutative ring, `p ≥ 2` an integer and `(ϖ_n)_{n≥0}` a sequence in `V` with
`ϖ_{n+1}^p = ϖ_n` for all `n`; put `ϖ := ϖ_0` and `m := ⋃_n ϖ_n V = Ideal.span {ϖ_n}`. No topology
and no perfectoid hypothesis enter: the construction is purely algebraic. Then `(V, m)` is a basic
setup, the root setup of `(ϖ_n)`: `m` is the increasing union of the principal ideals `ϖ_n V`,
`m·m = m`, and `m̃` is flat by condition (A). A `V`-module `M` is almost zero for it iff `ϖ_n M = 0`
for all `n` ([Bhatt] §4.1, Example 4.1.3, p. 18; cf. [ECD] §3, Definition 3.21, p. 19).
If `ϖ` is a nonzerodivisor of `V` then every
`ϖ_n` is a nonzerodivisor, each `ϖ_n V` is free of rank one, `m` is flat and countably generated,
and the multiplication `m̃ → m` is an isomorphism ([Bhatt] §4.1, Example 4.1.3, p. 18); the
nonzerodivisor hypothesis is used only for this clause. Replacing `(ϖ_n)` by the shifted sequence
`(ϖ_{n+k})_n`, or by any sequence `(ϖ'_n)` with `ϖ'_n V = ϖ_{n+c} V` for all large `n`, gives the
same setup. This is the input layer P1 instantiates with a pseudouniformiser admitting compatible
`p`-power roots. Define `BasicSetup.ofCompatibleRoots` (given `p ≥ 2` and `ϖ : ℕ → V` with
`ϖ (n+1) ^ p = ϖ n`, the setup with ideal `Ideal.span (Set.range ϖ)`); the ideal is `Ideal.span (Set.range ϖ)` by
definition. Prove `BasicSetup.mem_ofCompatibleRoots_ideal` (`x ∈ ideal ↔ ∃ n, ϖ n ∣ x`),
`BasicSetup.ofCompatibleRoots_isAlmostZero_iff` (`IsAlmostZero M ↔ ∀ n (x : M), ϖ n • x = 0`),
`BasicSetup.isIdempotentElem_span_roots` and `BasicSetup.flat_tilde_span_roots` (idempotence and
flatness of `m̃` for `Ideal.span (Set.range ϖ)`, the two fields of the structure), and, as
statements of this README with no Lean declaration yet: the simp rule `ϖ n ∈ ideal`; condition (A)
for the root setup; if `IsSMulRegular V (ϖ 0)`, flatness `Module.Flat V ideal` with `tilde → ideal` a
linear equivalence; invariance under shifting the sequence, or changing it by a cofinal equivalence
of principal ideals; and base change,
`(ofCompatibleRoots p ϖ h).baseChange W = ofCompatibleRoots p (algebraMap V W ∘ ϖ) _`. *Needs:* P0.1; Mathlib `Ideal.span`, `Module.Flat.of_free`,
`IsSMulRegular`.

Two supporting results. Almost zero modules and almost isomorphisms are detected by `m̃ ⊗ −`
(*almost-zero-iff-tilde-tensor-zero*): for a basic setup `S` and a `V`-module `M`, the conditions
`m·M = 0`, `m ⊗_V M = 0` and `m̃ ⊗_V M = 0` are equivalent; a `V`-linear `f : M → N` is an almost
isomorphism iff `1 ⊗ f : m̃ ⊗_V M → m̃ ⊗_V N` is an isomorphism; and if `m` is flat the multiplication
`BasicSetup.tildeToIdeal : m̃ → m` is bijective ([GR] §2.1, Remark 2.1.4(i), p. 8; [GR] §2.1,
(2.1.3), p. 7) — for
`V = 𝒪_{ℂ_p}` and `M = V/m`, `m ⊗ M = m/m² = 0`, while for `m = (p)` in `ℤ_p` one has
`m ⊗ ℤ_p/p = (p)/(p²) ≠ 0` (this uses Mathlib `Module.Flat.lTensor_exact`). Condition (A) implies that
`m ⊗ m` is flat (*filtered-principal-union-tilde-flat*): for an ideal `m` with `m·m = m` which is
the directed union `⋃_α V·x_α` of principal ideals, `m̃ = m ⊗_V m` is a flat `V`-module, so `(V, m)`
is a basic setup; the proof is the factorisation through `λ_{αβ}` of Mathlib
`Module.Flat.iff_forall_exists_factorization` ([GR] §2.1, (2.1.6), p. 8, and
Proposition 2.1.7(i) with proof, pp. 8–9). It applies to a root setup without assuming
`ϖ` to be a nonzerodivisor, and to a valuation ring with non-principal maximal ideal.

**Checks.**

- Computation: for
  `V = 𝔽_p[t^{1/p^∞}]` (the perfection of `𝔽_p[t]`, or the monoid algebra over `ℤ[1/p]_{≥0}`) and
  `ϖ_n = t^{1/p^n}`: `V/(t)` is not almost zero (`t^{1/p} ≠ 0` in it) while `V/m ≅ 𝔽_p` is almost
  zero.
- Degenerate: if `ϖ_0 = 1` (e.g. all `ϖ_n = 1`) then
  `m = V` and the root setup is `BasicSetup.classical V`.
- Non-example: the principal ideal `(t) ⊆ 𝔽_p[t]` (a
  single element without roots) is not idempotent, `(t)·(t) = (t²)`; so the root setup cannot be
  replaced by the ideal generated by `ϖ` alone.
- Compatibility: for `V` a valuation ring of rank
  one, `ϖ ∈ m` nonzero, equipped with compatible roots `ϖ_n` of `ϖ` for all `n` (e.g. `𝒪_{ℂ_p}`,
  `ϖ_n = p^{1/p^n}`), `ofCompatibleRoots` and `ofValuationRing` (P0.3) have the same ideal, the
  maximal ideal: `|x| < 1` forces `|x| ≤ |ϖ|^{1/p^n}` for some `n` because `|ϖ|^{1/p^n} → 1`; for
  `ϖ` a unit the root ideal is `V`, and for rank `> 1` (value group `ℤ[1/p] × ℤ[1/p]`
  lexicographic, `ϖ` of value `(1, 0)`, roots of value `(1/p^n, 0)`) the root ideal misses the
  elements of value `(0, 1)`, which lie in the maximal ideal.
- Degenerate: if every `ϖ_n = 0` the root setup is the
  zero setup `(V, ⊥)`; `ϖ_0 = 0` alone does not force this: for `V = 𝔽_p[t^{1/p^∞}]/(t)` and
  `ϖ_n = t^{1/p^n}`, `ϖ_0 = 0` but `m = (t^{1/p^n} : n ≥ 1) ≠ 0` is an idempotent ideal of
  nilpotent elements, and `V` itself is not almost zero (`t^{1/p} • 1 ≠ 0`).

- `roots_test_quotients` — for a compatible root chain with `t₁ ∉ (t₀)`, `V/(t₀)` is not almost zero while `V/m` is.
- `roots_test_unit`, `roots_test_unit_initial`, `roots_test_zero`, `roots_test_nilpotent` —
  the all-one chain is classical, a unit initial term forces the unit ideal, the all-zero chain gives zero,
  and `t₀ = 0` with `t₁ ≠ 0` forces a nonzero root ideal.
- `roots_test_no_roots` — `(X)` in `𝔽_p[X]` is not idempotent.

Lean: `BasicSetup.ofCompatibleRoots`, `mem_ofCompatibleRoots_ideal`,
`ofCompatibleRoots_isAlmostZero_iff`, `isIdempotentElem_span_roots`, `flat_tilde_span_roots`,
`flat_tilde_of_isFilteredPrincipal` (*filtered-principal-union-tilde-flat*),
`isAlmostZero_iff_tensor_eq_zero` (first clause of *almost-zero-iff-tilde-tensor-zero*).

### P0.3 the setup `(K°, K°°)` of a valuation ring

**The basic setup `(K°, K°°)` of a valuation ring with non-principal maximal ideal.**
(*valuation-ring-basic-setup*) Let `V` be a valuation ring (a domain in which the ideals are
totally ordered; Mathlib `ValuationRing V` with `IsDomain V`) whose maximal ideal `m` is not
finitely generated (equivalently not principal; for a rank-one valuation: the valuation is
nondiscrete, e.g. `V = K°` for a perfectoid field `K`). No completeness and no rank-one hypothesis is
needed. Then `(V, m)` is a basic setup: `m·m = m`, `m` is torsion-free hence flat, and `m̃ ≅ m`
([GR] §2.1, Example 2.1.2(i), p. 7; [Sch12] §4, Lemma 4.2 and its proof, p. 18). A `V`-module is
almost zero iff it is killed by `m`, iff it is a vector space over the residue field `k = V/m`; in
particular `k` is almost zero ([ECD] §3, Example 3.22(i), p. 19). This is the setup of [Sch12] §4.
Define `BasicSetup.ofValuationRing` (for `[ValuationRing V] [IsDomain V]` with
`¬ (IsLocalRing.maximalIdeal V).FG`, the setup with ideal the maximal ideal); the ideal is `IsLocalRing.maximalIdeal V` by definition. Prove
`BasicSetup.isIdempotentElem_maximalIdeal_iff` (for a valuation domain which is not a field,
`maximalIdeal V ≠ ⊥ → (IsIdempotentElem (maximalIdeal V) ↔ ¬ (maximalIdeal V).FG)`; for a field
the maximal ideal `⊥` is idempotent and finitely generated, so the hypothesis is needed),
`BasicSetup.isIdempotentElem_maximalIdeal_of_not_fg` (the idempotence of a maximal ideal that is not
finitely generated, fields included), `BasicSetup.flat_tilde_maximalIdeal` (`m ⊗ m` is flat, `m`
being torsion-free), the statements of this README with no Lean declaration yet that
`IsAlmostZero M ↔ maximalIdeal V • ⊤ = ⊥`, that the residue field is almost zero, that
`Module.Flat V (maximalIdeal V)` with `tilde ≃ₗ maximalIdeal`, and that condition (A) holds, and
`BasicSetup.ofValuationRing_eq_ofCompatibleRoots` (equality with the root setup when `ϖ_0` is a nonzero element of the maximal ideal and every
element of that ideal is divisible by some `ϖ_n`; this is automatic in rank one since
`|ϖ_n| = |ϖ_0|^{1/p^n} → 1`;
for `ϖ` a unit the root ideal is `V` and for `ϖ = 0` it is `0`, neither the maximal ideal). *Needs:* P0.1; Mathlib `ValuationRing`,
`Module.Flat.flat_iff_torsion_eq_bot_of_isBezout`, `IsBezout`.

**Checks.**

- Computation: for `V = 𝒪_{ℂ_p}`: the residue field `𝔽̄_p`
  is almost zero, `V/pV` is not almost zero, and the ideal is the maximal ideal.
- Non-example: for `V = ℤ_p` the maximal ideal `pℤ_p` is
  finitely generated and `(pℤ_p)² ≠ pℤ_p`, so `ofValuationRing` does not apply; its hypothesis is
  necessary.
- For `V = ℤ_p`, multiplication `m ⊗_V m → m` has image `(p²)` and misses `p`.
  Both `m` and its tensor square are flat; multiplication identifies the tensor square with
  `m` only when `m² = m`.
- Degenerate: for `V` a field the maximal ideal is `0`,
  which is finitely generated; the construction does not apply, while the zero setup `(V, 0)` of
  P0.1 is the degenerate basic setup in which every module is almost zero. The ideal `0` is
  idempotent and finitely generated, which is why `isIdempotentElem_maximalIdeal_iff` assumes
  `maximalIdeal V ≠ ⊥`.
- Compatibility: for `V = 𝒪_K` with `K` perfectoid and
  `ϖ` a nonzero nonunit with compatible `p`-power roots: `ofValuationRing V = ofCompatibleRoots p (fun n ↦ ϖ^{1/p^n}) _`
  as basic setups.

- `valuation_test_quotients` — for any valuation domain `O` with non-finitely-generated maximal
  ideal, and `δ ∈ m` with `ε ∤ δ`, `O/m` is almost zero and `O/(ε)` is not. This is the
  algebraic form of the residue-field and root Check; it uses the same generality as the Lean construction.

- `valuation_test_field`, `valuation_test_dvr` — a field has zero finitely generated maximal ideal;
  `ℤ_p` has nonzero finitely generated non-idempotent maximal ideal. Both fail the non-finite-generation input.

Lean: `BasicSetup.ofValuationRing`, `isIdempotentElem_maximalIdeal_iff`,
`isIdempotentElem_maximalIdeal_of_not_fg`, `flat_tilde_maximalIdeal`,
`ofValuationRing_eq_ofCompatibleRoots`, and `padicComplex_maximalIdeal_not_fg` (the maximal ideal
of `𝒪_{ℂ_p}` is not finitely generated, so the construction applies to `𝒪_{ℂ_p}`).

### P0.4 firm modules

**Firm modules: a small-hom model of the almost category.** (*firm-module-model*) Let `S = (V, m)`
be a basic setup (`m̃ = m ⊗_V m` flat; flatness is used for the exactness of `m̃ ⊗ −` and the closure
of firm modules under kernels), with `V : Type u` and `ModuleCat.{u} V`, so that `S.Firm : Type (u+1)`
with `Category.{u}`. A `V`-module `M` is firm if the multiplication map `m̃ ⊗_V M → M` is an
isomorphism; firm modules form a full subcategory `S.Firm` of `ModuleCat V`, closed under kernels,
cokernels and finite biproducts (hence abelian, with exact inclusion `j_!`). The functor
`j^* : ModuleCat V → S.Firm`, `M ↦ m̃ ⊗_V M`, is well defined (`m̃ ⊗ m̃ ≅ m̃`), exact, right adjoint to
`j_!`, and has a fully faithful right adjoint `j_* : N ↦ Hom_V(m̃, N)`; its kernel `{M : m̃ ⊗ M = 0}`
is exactly `S.almostZero` ([Bhatt] §4.1, Construction 4.1.5, pp. 18–19). By Mathlib's
`Abelian.isLocalization_isoModSerre_kernel_of_leftAdjoint`, `j^*` is a localization functor for the
almost isomorphisms `S.almostZero.isoModSerre`, which yields the instance
`MorphismProperty.HasLocalization.{u} S.almostZero.isoModSerre` whose localized category has
Hom-types in the universe `u` of the `V`-modules; the almost category of P0.5 is defined through it
([Bhatt] §4.1, Proposition 4.1.7(2) and proof, p. 21; [GR] §2.2, Proposition 2.2.21 and Corollary
2.2.22, p. 15). The categories `S.Firm` and `Almost.Module S` are equivalent; firm modules are the
modules of the form `M_!` (P0.6). Define `BasicSetup.IsFirm` (`Function.Bijective` of
`TensorProduct.lift (multiplication) : S.tilde ⊗[V] M → M`), `BasicSetup.Firm` (the full
subcategory on firm modules), `BasicSetup.firm` (the functor `ModuleCat V ⥤ S.Firm`,
`M ↦ S.tilde ⊗[V] M`), `BasicSetup.Firm.ι` (the inclusion, `j_!`) and `BasicSetup.closure`
(`N ↦ Hom_V(S.tilde, N)`, `j_*`); prove `BasicSetup.firmInclusionAdj` (`Firm.ι ⊣ S.firm`),
`BasicSetup.firmAdj` (`S.firm ⊣ S.closure`, with `S.closure` fully faithful), the kernel identity
`S.firm.kernel = S.almostZero`, `BasicSetup.hasLocalization` (the `HasLocalization.{u}` instance
with localized category `S.Firm`), firmness of `S.tilde` and
`BasicSetup.Firm.abelian` (`Abelian S.Firm`, kernels and cokernels computed in `ModuleCat V`).
*Needs:* P0.1; Mathlib `CategoryTheory.Abelian.isLocalization_isoModSerre_kernel_of_leftAdjoint`,
`CategoryTheory.MorphismProperty.HasLocalization`, `CategoryTheory.Functor.kernel`,
`ModuleCat.monoidalClosedHomEquiv`.

Two supporting results. `m̃ ⊗ m ≅ m̃` and `m̃ ⊗ m̃ ≅ m̃` (*tilde-ideal-idempotent*): for a basic setup
`S`, the map `m̃ ⊗_V m → m̃`, `x ⊗ y ⊗ z ↦ x ⊗ yz`, is an isomorphism, so the multiplication
`m̃ ⊗_V m̃ → m̃` is an isomorphism (`m̃` is an idempotent `V`-module); for every `k ≥ 2` the symmetric
group `S_k` acts trivially on `⊗^k_V m`, in particular `x ⊗ y = y ⊗ x` in `m̃` ([GR] §2.1, Claim
2.1.10 and its proof, p. 9); for the classical setup this is the multiplication `V ⊗ V ⊗ V ≅ V ⊗ V`,
and for a root setup with `ϖ` a nonzerodivisor it is `m ⊗ m ≅ m` (Mathlib `TensorProduct.comm`).
Almost zero modules form a Serre class (*almost-zero-serre-class*): the property `S.almostZero` of
`V`-modules is a Serre class in the abelian category `ModuleCat V` (Mathlib
`ObjectProperty.IsSerreClass`), containing `0` and closed under submodules, quotients and
extensions, and it is a tensor ideal: `M` almost zero implies `M ⊗_V N`, `Hom_V(M, N)` and
`Hom_V(N, M)` almost zero for all `N`; only `m·m = m` is used ([Sch12] §4, Lemma 4.2 with proof,
p. 18; [GR] §2.2, (2.2.2), p. 11; [Berkeley] Lecture 7, §7.4, after Example 7.4.2, p. 52; [AWS] §2.9,
Definition 2.9.4, p. 73). For `V = 𝒪_{ℂ_p}`, `V/m² = V/m = k`, consistent with idempotence, whereas
for `ℤ_p` the module `ℤ/p²` is an extension of almost zero modules not killed by `(p)`; Mathlib's
`SerreClassLocalization` API applies to `S.almostZero` (Mathlib
`CategoryTheory.ObjectProperty.IsSerreClass`, `CategoryTheory.ObjectProperty.IsClosedUnderExtensions`).

**Checks.**

- `firm_test_tilde` (computation) — `S.tilde` is firm and
  `S.firm.obj (ModuleCat.of V V) ≅ S.tilde`.
- Non-example: if `S.ideal ≠ ⊤` then `ModuleCat.of V V` is
  not firm (the image of `m̃ ⊗ V → V` is `m`); and the tempting condition `m·M = M` does not
  characterise firm modules: for `R = 𝔽_p[t^{1/p^∞}]`, `I = (t^{1/p^∞})` and `M` the maximal ideal of
  `R/t`, `IM = M` but `I ⊗ M → M` is not injective ([Bhatt] Warning 4.1.6).
- `firm_test_classical` (degenerate) — for `S = classical V` every `V`-module is firm and
  `S.firm ⋙ BasicSetup.Firm.ι S ≅ 𝟭 (ModuleCat V)`; the composite
  `S.firm ⋙ S.closure` is also naturally isomorphic to the identity.
- Compatibility: `S.firm` satisfies the hypotheses of
  Mathlib's `Abelian.isLocalization_isoModSerre_kernel_of_leftAdjoint`; hence `S.Firm` is equivalent
  to Mathlib's constructed localization `S.almostZero.isoModSerre.Localization` (via
  `Localization.uniq`).

- `firm_test_tilde`, `firm_test_classical`, `firm_test_proper_ideal`, `firm_test_zero_setup` —
  the tensor ideal is firm, every classical module is firm, the ring fails to be firm for a proper
  setup ideal, and in the zero setup firmness means the module is zero. These check `IsFirm` and
  the full-subcategory object property together.
- `firmFunctor_test_classical`, `firmFunctor_test_zero_setup`, `firmFunctor_test_residue` —
  after inclusion, `firm` recovers a classical module, sends every module to zero in the zero
  setup, and kills `V/m`. The composite with `closure` has the same three values.
- `tildeAct_test_two_terms` — on `(ℤ, ℤ)`, `(2 ⊗ 3) ⊗ 5 + (7 ⊗ 11) ⊗ 13` acts as `1031`;
  `tildeAct_test_zero` gives the zero map for `m = 0`, and `tildeAct_test_unit` computes
  `(1 ⊗ 1) ⊗ 1 ↦ 1` over `(ℤ, ℤ)`. These fix the scalar-action convention.
- `firmAdj_test_zero`, `firmAdj_test_classical_identity`, `firmAdj_test_two_terms` — the
  tensor–Hom adjunction sends zero to zero, sends the identity of the classical firmification
  of `ℤ` to a nonzero map, and sends `2f + 3f` to five times the image of `f`.

- `tildeEquiv_test_zero`, `tildeEquiv_test_unit`, `tildeEquiv_test_two_terms` —
  the map `m̃ ⊗ m → m̃`, `(x ⊗ y) ⊗ z ↦ x ⊗ yz`, is zero on the zero setup,
  sends `(1 ⊗ 1) ⊗ 1` to `1 ⊗ 1` classically, and after multiplication sends
  `(2 ⊗ 3) ⊗ 5 + (7 ⊗ 11) ⊗ 13` to `1031`.

Lean: `BasicSetup.almostZero_isSerreClass` (*almost-zero-serre-class*) and
`BasicSetup.isAlmostIso_iff_isoModSerre` (the almost isomorphisms are the morphisms
`S.almostZero.isoModSerre` that the Serre quotient inverts). The firm-module model is specified in
this README only.

### P0.5 almost modules

The almost modules themselves come next: the Serre quotient, almost elements and `M_!`, the
monoidal structure, almost algebras and `B_!!`, the finiteness, flatness and projectivity notions,
duals and traces, and the rank decomposition.

**Almost modules over a basic setup: the Serre quotient `V^a-Mod`.**
(*almost-modules-over-perfectoid-base*) For a basic setup `S = (V, m)` (`m·m = m` and `m̃` flat; no
valuation and no perfectoid field), the category of almost `V`-modules `V^a-Mod := Almost.Module S`
is the localization of `ModuleCat V` at the almost isomorphisms `S.almostZero.isoModSerre` (maps with
almost zero kernel and cokernel), i.e. the Serre quotient of `V`-modules by the almost zero modules,
realised with Mathlib's Serre-class localization: the carrier is `MorphismProperty.Localization'`
for the `HasLocalization.{u}` instance of P0.4, so Hom-types are `V`-modules in the universe of
`V`-modules ([Sch12] §4, Definition 4.3, p. 18; [GR] §2.2, (2.2.2), p. 11). The localization functor
`M ↦ M^a` (`Almost.toAlmost S`) is exact and essentially surjective; `M^a ≅ 0` iff `M` is almost
zero; `f^a` is an isomorphism iff `f` is an almost isomorphism; `V^a-Mod` is abelian (Mathlib
`SerreClassLocalization.abelian`); a functor `G` out of `V^a-Mod` to an abelian category is exact
iff `G ∘ (−)^a` is, and an exact functor out of `ModuleCat V` factors (uniquely up to unique
isomorphism) through `(−)^a` iff it kills almost zero modules. For a perfectoid field `K` and
`S = (K°, K°°)` this is Scholze's `K°a-mod`; for a perfectoid Tate ring `R`, layer P1 instantiates
`S` with `(R°, R°°)` and `(R⁺, root ideal)` ([ECD] §3, Definition 3.21, p. 19; [KL15] §5.5,
Definition 5.5.8, p. 132, for uniform Banach algebras with enough topologically nilpotent units). All
almost-vanishing statements in consumers ("`H^i(X, O⁺)` is almost zero", "`R⁺/ϖ` is almost the
equalizer") are statements in this category. Define `Almost.Module` (`S.almostZero.isoModSerre.Localization'`
for the firm-model instance, a category with Hom-types in `Type u`), `Almost.toAlmost` (the
localization functor `ModuleCat V ⥤ Almost.Module S`) and `Almost.Module.homModule` (the `V`-module
structure on Hom-types, via *almost-hom-formula*, with no nonzero almost zero elements); prove that
`(toAlmost S).IsLocalization S.almostZero.isoModSerre`, `Almost.Module.instAbelian`, that
`toAlmost S` preserves finite limits and finite colimits, that
`IsZero ((toAlmost S).obj M) ↔ S.IsAlmostZero M`, that
`IsIso ((toAlmost S).map f) ↔ S.IsAlmostIso f.hom` (whose module-level half is the Lean
`BasicSetup.isAlmostIso_iff_isoModSerre`), that `toAlmost S` is essentially surjective (every almost
module is some `M^a`, with `M = N_*`), that `G` is exact iff `toAlmost S ⋙ G` is exact, and
`Almost.Module.classicalEquiv` (for `S = classical V`, `ModuleCat V ≌ Almost.Module S`).
For an exact `F : ModuleCat V ⥤ D` with `D` abelian, use Mathlib's
`ObjectProperty.isoModSerre_isInvertedBy_iff` to express killing almost zero modules as
inverting `S.almostZero.isoModSerre`. Then
`SerreClassLocalization.essImage_whiskeringLeft`, applied to `Almost.toAlmost S`, supplies
an exact `G : Almost.Module S ⥤ D` with `F ≅ toAlmost S ⋙ G`;
`SerreClassLocalization.fullyFaithfulWhiskeringLeft` gives the unique comparison
isomorphism compatible with these factorizations. *Needs:* *almost-zero-serre-class* (P0.4); P0.4; Mathlib
`CategoryTheory.ObjectProperty.isoModSerre`, `CategoryTheory.MorphismProperty.Localization'`,
`SerreClassLocalization.abelian`, `SerreClassLocalization.isZero_obj_iff`,
`SerreClassLocalization.isIso_map_iff`, `SerreClassLocalization.preservesFiniteLimits`,
`SerreClassLocalization.preservesFiniteColimits`, `SerreClassLocalization.exactFunctor_comp_iff`,
`SerreClassLocalization.essImage_whiskeringLeft`.

**Checks.**

- `Almost.Module.test_residue_field` (computation) — for `S = ofValuationRing 𝒪_{ℂ_p}`:
  `(toAlmost S).obj (ModuleCat.of V (IsLocalRing.ResidueField V))` is a zero object, while
  `(toAlmost S).obj (ModuleCat.of V (V ⧸ Ideal.span {p}))` is not.
- `Almost.Module.almostModule_test_classical` (compatibility) — for `S = classical V`, `toAlmost S` is an
  equivalence of categories `ModuleCat V ≌ Almost.Module S` (every almost isomorphism is an
  isomorphism).
- `Almost.Module.almostModule_test_zero_setup` (degenerate) — for the setup `(V, ⊥)` every object of
  `Almost.Module S` is a zero object.
- `Almost.Module.test_not_projective` (non-example) — for `S = (K°, K°°)`, `K` the completion of
  `ℚ_p(p^{1/p^∞})`: `V^a` is not a projective object of `Almost.Module S` (`Hom(V^a, −)` does not
  preserve the epimorphism `K°a → (K°/ϖ)^a` on Hom-sets, [Sch12] Remark 4.8); so "projective" in the
  categorical sense is not the notion used for almost modules.
- `Almost.Module.test_hom_formula` (characterisation) — for `V`-modules `M, N`:
  `(toAlmost S).obj M ⟶ (toAlmost S).obj N` is `V`-linearly isomorphic to `S.tilde ⊗[V] M →ₗ[V] N`;
  with `M = V` and `N = K°/ϖ` over `(K°, K°°)` this is `Hom(K°°, K°/ϖ)`.
  The canonical map from `K°/ϖ` need not be surjective ([Sch12] Remark 4.8).

- `almostModule_test_classical` — almostifying and then taking almost elements recovers every classical module.
- `almostModule_test_zero_setup` — every object of the quotient is zero for `m = 0`.
- `almostModule_test_residue` — the almostification of `V/m` is zero for every setup.
- `almostModule_test_classical_nonzero` — the almostification of `ℤ` in the classical setup is nonzero.

The category `Almost.Module S` is specified in this README only; the Lean file carries its input,
the Serre class `BasicSetup.almostZero_isSerreClass` of P0.4, together with
`BasicSetup.isAlmostIso_iff_isoModSerre`.

### P0.6 almost elements and the left adjoint

**Almost elements `M_*` and the left adjoint `M_!` of almostification.** (*almost-hom-and-adjoints*)
For a basic setup `S = (V, m)` (`m̃` flat is used for the exactness of `(−)_!`): (i) the functor of
almost elements `(−)_* : Almost.Module S → ModuleCat V`, `M_* := Hom_{V^a}(V^a, M)` with its
`V`-module structure, is right adjoint to `M ↦ M^a`, with counit `(M_*)^a → M` an isomorphism (so
`(−)_*` is fully faithful) and `(N^a)_* ≅ Hom_V(m̃, N)` naturally ([GR] §2.2, Proposition
2.2.13(ii)-(iii), p. 14); (ii) `(−)_! : Almost.Module S → ModuleCat V`, `M_! := m̃ ⊗_V M_*`, is left
adjoint to `M ↦ M^a` with unit `M → (M_!)^a` an isomorphism (so `(−)_!` is fully faithful), and it
is exact because `m̃` is flat ([GR] §2.2, Proposition 2.2.21, p. 15); (iii) for every `V`-module `N`
the unit `N → (N^a)_*` and the natural map `(N^a)_! → N` are almost isomorphisms, and `M_! → M_*` is
an almost isomorphism ([GR] §2.2, (2.2.9), p. 13); (iv) `(−)^a` preserves all limits and colimits,
`(−)_*` preserves limits and injective objects (not infinite direct sums), `(−)_!` preserves
colimits and tensor products; `Almost.Module S` is complete, cocomplete, satisfies AB5, has the
generator `V^a` and enough injectives ([GR] §2.2, Corollary 2.2.15, p. 14); (v) preimages of almost
elements exist only up to `m`: for an epimorphism `φ : M_1 → M_2`, `x ∈ (M_2)_*` and `ε ∈ m` there
is `y ∈ (M_1)_*` with `φ_*(y) = ε·x` ([GR] §2.2, Remark 2.2.14(ii), p. 14; [Sch12] §4, Proposition
4.6, p. 19, for `(K°, K°°)`). Define `Almost.Module.elements` (`M ↦ M_* = (V^a ⟶ M)`),
`Almost.Module.shriek` (`M ↦ M_! = S.tilde ⊗[V] M.elements`) and `Almost.Module.shriekToElements`
(the natural transformation `shriek ⟶ elements`, an almost isomorphism objectwise); prove
`Almost.Module.adjElements` (`toAlmost S ⊣ elements S`), `Almost.Module.elements_counit_isIso`
(`(M.elements)^a ≅ M`, natural; `elements` is fully faithful), `Almost.Module.elementsToAlmostIso`
(`((toAlmost S).obj N).elements ≅ ModuleCat.of V (S.tilde →ₗ[V] N)`, natural in `N`),
`Almost.Module.adjShriek` (`shriek S ⊣ toAlmost S`), `Almost.Module.shriek_unit_isIso`
(`M ≅ (M.shriek)^a`), `Almost.Module.shriek_preservesFiniteLimits` (`shriek` is exact),
the statement that the unit `N → ((toAlmost S).obj N).elements` is an almost isomorphism,
`Almost.Module.enoughInjectives` (`elements` and `toAlmost` preserve injectives),
`Almost.Module.hasLimits` (all limits and colimits, AB5, preserved by `toAlmost`) and
`Almost.Module.exists_elements_lift_smul` (for an epimorphism `φ`, `x : M₂.elements` and
`ε ∈ S.ideal`, `∃ y, φ.elements y = ε • x`). *Needs:* P0.5; Mathlib `CategoryTheory.Adjunction`,
`ModuleCat.monoidalClosedHomEquiv`.

For every basic setup `S` and `M : Almost.Module S`, the firm representative is
identified with `m̃ ⊗ M_*` by `Almost.Module.shriekIsoTensor`. Its inverse is the
tensor–Hom evaluation map: `Almost.Module.shriekIsoTensor_inv_apply` states
`t ⊗ f ↦ f(t)` for `t : S.tilde` and `f : M_*`. This fixes the comparison on
every pure tensor. `Almost.Module.shriekToElements_apply` then states
`shriekToElements (f(t)) = tildeMul(t) • f`; in particular, for `t = x ⊗ y`
with `x, y ∈ m`, the scalar is `xy`. These are the firm-model formulas for
[GR] §2.2, (2.2.9), p. 13, and (2.2.20), Proposition 2.2.21, p. 15.

Supporting result: `Hom(M^a, N^a) = Hom_V(m̃ ⊗ M, N)` (*almost-hom-formula*). For a basic setup `S`
and `V`-modules `M, N` there is a natural `V`-linear bijection
`Hom_{Almost.Module S}(M^a, N^a) ≅ Hom_V(m̃ ⊗_V M, N)`; in particular Hom-types of `Almost.Module S`
are `V`-modules, functorially in both variables, and contain no nonzero almost zero elements (if
`m·f = 0` then `f = 0`, stated as `Almost.Module.hom_eq_zero_of_ideal_smul` for
any `f : M ⟶ N` with `∀ ε ∈ S.ideal, ε • f = 0`); with `M = V`, `Hom(V^a, N^a) ≅ Hom_V(m̃, N)` ([GR] §2.2, (2.2.4) and the
text after it, p. 12; [Sch12] §4, Proposition 4.4, p. 18). For `S = (K°, K°°)`, `M = K°`,
`N = K°/ϖ` this is `Hom(K°a, (K°/ϖ)^a) = Hom_{K°}(K°°, K°/ϖ)` ([Sch12] Proposition 4.4 and Remark
4.8), proved with Mathlib `ModuleCat.monoidalClosedHomEquiv`.

`Almost.Module.homLinearEquiv_apply` fixes this bijection for every basic setup `S`
and ordinary `V`-modules `M, N`: almostifying the linear representative of `f`
gives `toAlmost.map (S.tildeAct M) ≫ f`, where `tildeAct` sends
`(x ⊗ y) ⊗ z` to `(xy) • z`. This equation characterises the representative.
`Almost.Module.homLinearEquiv_natural` states compatibility with precomposition
by `a : M′ → M` and postcomposition by `b : N → N′`: the representative becomes
`b ∘ homLinearEquiv(f) ∘ (id ⊗ a)`. Both formulas use [GR] (2.2.4), p. 12.

**Checks.**

The three discriminating Checks for `homLinearEquiv` are `homLinearEquiv_zero_setup`,
`homLinearEquiv_two_terms` and `homLinearEquiv_residue_nonexample`, using [GR] (2.2.4),
p. 12. When `S.ideal = ⊥`, every almost map has zero representative and its tensor
source is zero. Over the classical setup on `ℤ`, the displayed two-term tensor has
image `1031`. When `S.ideal ≠ ⊤`, the ordinary identity of `V/S.ideal` is nonzero,
but the representative of its almost identity is zero: this comparison does not
recover ordinary Hom without almostification.

- `homLinearEquiv_zero` (supplementary) — the zero almost map has zero linear representative.
- `homLinearEquiv_identity` — the identity of `M^a` has representative `S.tildeAct M`.
- `homLinearEquiv_two_terms` — over the classical setup on `ℤ`, the representative
  of the almostified identity sends `(2 ⊗ 3) ⊗ 5 + (7 ⊗ 11) ⊗ 13` to `1031`.

- `Almost.Module.elements_test_valuation` (computation) — for `S = ofValuationRing K°` and
  `N = K°°`: `((toAlmost S).obj N).elements ≅ K°` (every `x ∈ K` with `εx ∈ K°°` for all `ε ∈ K°°`
  lies in `K°`).
- `Almost.Module.elements_test_not_identity` (non-example) — for `S = ofValuationRing K°` (`K` the
  completion of `ℚ_p(p^{1/p^∞})`) the unit `K°/ϖ → ((K°/ϖ)^a)_* = Hom(K°°, K°/ϖ)` is injective but
  not surjective ([Sch12] Remark 4.8): almost elements of `N^a` are not elements of `N`. Computed
  witness: set `ϖ = p` and choose compatible roots in `K`. Write
  `K°° = ⋃_n p^{1/p^n}K°`. The compatible residues
  `c_n = Σ_{j<n} p^{1−1/p^j} mod p^{1−1/p^n}` define an almost element.
  If it came from `c ∈ K°`, choose `b ∈ ℚ_p(p^{1/p^N})` with `v_p(c−b) > 1`.
  For `n > N`, compatibility forces `v_p(b−c_n) = 1−1/p^n`, since the next term has this
  valuation. But `b−c_n ∈ ℚ_p(p^{1/p^{n−1}})` has valuation in `p^{−(n−1)}ℤ`, a contradiction.
  This uses the indicated completion; nonconvergence of the displayed sum alone would not suffice.
- `Almost.Module.almostModule_test_classical` (degenerate) — for `S = classical V`:
  `M.elements ≅ M.shriek` and both are inverse to `toAlmost`.
- `Almost.Module.shriek_test_tilde` (compatibility) — `(V^a).shriek ≅ S.tilde` and
  `(N^a).shriek ≅ S.tilde ⊗[V] N`, compatible with Mathlib's tensor product of modules; the map
  `shriek → elements` is induced by `S.tilde → V`.

- `adjunction_test_zero` — both adjunctions send the zero map to the zero map.
- `adjunction_test_classical_identity` — the almostification/elements adjunction sends the identity of the almostified classical `ℤ` to a nonzero map.
- `adjunction_test_two_terms` — its Hom equivalence sends `2f + 3f` to five times the image of `f`, fixing its linearity convention.
- `shriekIsoTensor_zero` — inverse evaluation sends the zero tensor to zero.
- `shriekIsoTensor_two_terms` — inverse evaluation sends `t ⊗ f + u ⊗ g` to `f(t) + g(u)`.
- `shriekIsoTensor_not_negated` — if `f(t) ≠ −f(t)`, the inverse comparison cannot send `t ⊗ f` to `−f(t)`.
- `shriekToElements_zero` — the comparison sends zero to zero.
- `shriekToElements_two_terms` — it sends `f(t) + g(u)` to `tildeMul(t) • f + tildeMul(u) • g`.
- `shriekToElements_classical_unit` — for `(ℚ, ℚ)`, evaluation at `1 ⊗ 1` is sent back to `f`, fixing the scalar as `+1`.
- `shriekToElements_classical_two_terms` — for `(ℚ, ℚ)`, the image of `f(2 ⊗ 3) + f(5 ⊗ 7)` is `41 • f`.

### P0.7 the closed symmetric monoidal structure

**The closed symmetric monoidal structure on almost modules.** (*almost-tensor-and-internal-hom*)
For a basic setup `S = (V, m)`, `Almost.Module S` carries the symmetric monoidal structure localized
from `ModuleCat V` (Mathlib `LocalizedMonoidal` for the monoidal class of almost isomorphisms,
*almost-isomorphisms-are-monoidal*), with unit `V^a` (`≅ m^a`): the localization functor `M ↦ M^a`
is symmetric monoidal, `M^a ⊗ N^a ≅ (M ⊗_V N)^a`, and `M ⊗ N ≅ (M_* ⊗_V N_*)^a ≅ (M_! ⊗_V N_!)^a`
([Sch12] §4, Proposition 4.5, p. 19; [GR] §2.2, (2.2.11), p. 13). The tensor product is right exact
in each variable. The internal hom is `alHom(M, N) := (Hom_{V^a}(M, N))^a`, the almostification of
the `V`-module of *almost-hom-formula*; there is a natural isomorphism
`Hom(L ⊗ M, N) ≅ Hom(L, alHom(M, N))`, so `Almost.Module S` is closed, and
`alHom(M^a, N^a) ≅ (Hom_V(m̃ ⊗ M, N))^a` ([GR] §2.2, (2.2.10)-(2.2.11), p. 13). The functor `(−)_!`
preserves nonempty tensor products, `(M ⊗ N)_! ≅ M_! ⊗_V N_!`, but not the unit (see the next
paragraph), and `(−)_*` is lax symmetric monoidal ([GR] §2.4, Proposition 2.4.35, p. 29). Define `Almost.Module.alHom`
(`alHom M N := (toAlmost S).obj (ModuleCat.of V (M ⟶ N))`); prove that
`S.almostZero.isoModSerre.IsMonoidal`, `CategoryTheory.Localization.Monoidal.instMonoidalCategoryLocalizedMonoidal` (the localized monoidal
structure; `Almost.Module S` is defined as the `LocalizedMonoidal` type synonym),
`CategoryTheory.Localization.Monoidal.instSymmetricCategoryLocalizedMonoidal`, `Almost.toAlmostMonoidal` (`(toAlmost S).Monoidal`, with
`μ : M^a ⊗ N^a ≅ (M ⊗ N)^a` and `ε : 𝟙_ ≅ V^a`), `Almost.Module.tensorIsoElements`
(`M ⊗ N ≅ (toAlmost S).obj (M.elements ⊗ N.elements)`), `Almost.Module.monoidalClosed`
(`ihom M = alHom M −`), `Almost.Module.alHom_toAlmost`
(`alHom (M^a) (N^a) ≅ (toAlmost S).obj (ModuleCat.of V (S.tilde ⊗[V] M →ₗ[V] N))`),
`Almost.Module.shriekTensorIso` (`(M ⊗ N).shriek ≅ M.shriek ⊗ N.shriek`, the tensor comparison of
the oplax structure below),
`Almost.Module.elementsLaxMonoidal` and `Almost.Module.tensor_preservesFiniteColimits` (`tensorLeft M`
and `tensorRight M` preserve finite colimits). *Needs:* *almost-hom-formula* (P0.6); P0.6; Mathlib
`CategoryTheory.LocalizedMonoidal`, `CategoryTheory.Localization.Monoidal.toMonoidalCategory`,
`ModuleCat.monoidalCategory`, `ModuleCat.monoidalClosedHomEquiv`, `CategoryTheory.Functor.LaxMonoidal`.

**The left adjoint `(−)_!` is not unital.** Since `N → (N^a)_*` is an almost isomorphism (P0.6) and
`m̃ ⊗ −` inverts almost isomorphisms (P0.1), `(N^a)_! ≅ m̃ ⊗_V N`; in particular `(V^a)_! ≅ m̃`, not
`V` ([GR] §2.2, (2.2.20)–(2.2.24) and Proposition 2.2.21, p. 15). As the left adjoint of the strong
monoidal functor `(−)^a`, the functor `(−)_! : Almost.Module S → ModuleCat V` is oplax monoidal: its
tensor comparison `(M ⊗ N)_! → M_! ⊗_V N_!` is an isomorphism, because `m̃ ⊗ m̃ ≅ m̃`
(*tilde-ideal-idempotent*, P0.4), and its unit comparison is `(V^a)_! ≅ m̃ → V`, the multiplication
`tildeMul`, which is an isomorphism exactly when `m = V`. No strong monoidal functor
`Almost.Module S → ModuleCat V` is asserted. The strong monoidal statement has a different target:
`(−)_!` is a strong monoidal equivalence onto the category `S.Firm` of firm modules of P0.4, with
tensor product `⊗_V` and unit `m̃` ([Bhatt] §4.1, Construction 4.1.5 and Proposition 4.1.7, pp.
18–21). The missing unit is why P0.9 needs the unitisation `B_!!`: for an almost algebra `B`, `B_!`
is only a non-unital algebra.

Supporting result: almost isomorphisms are stable under tensor products
(*almost-isomorphisms-are-monoidal*). For a basic setup `S`, the class `S.almostZero.isoModSerre`
of almost isomorphisms is multiplicative, and for every `V`-module `X` and almost isomorphism `f`
the maps `X ⊗ f` and `f ⊗ X` are almost isomorphisms; that is, it is a monoidal morphism property
for Mathlib's monoidal structure on `ModuleCat V` (`MorphismProperty.IsMonoidal`); equivalently,
almost zero modules form a tensor ideal ([GR] §2.2, (2.2.5), p. 12; [Bhatt] §4.1, Proposition
4.1.7(3), p. 20). For `f : m ↪ V` and `X = V/m`, `X ⊗ f : m/m² = 0 → V/m` has almost zero cokernel
`V/m`, as predicted (Mathlib `CategoryTheory.MorphismProperty.IsMonoidal`, `ModuleCat.monoidalCategory`).

**Checks.**

- `Almost.Module.tensor_test_residue` (computation) — for `S = ofValuationRing K°`:
  `(toAlmost S).obj k ⊗ X ≅ 0` for every `X`, and
  `(toAlmost S).obj K°° ⊗ (toAlmost S).obj K°° ≅ 𝟙_ (Almost.Module S)`.
- `Almost.Module.tensor_test_unit` (degenerate) —
  `𝟙_ (Almost.Module S) ≅ (toAlmost S).obj (ModuleCat.of V V) ≅ (toAlmost S).obj (ModuleCat.of V S.ideal)`.
- `Almost.Module.tensor_test_mathlib` (compatibility) — `toAlmost S` is a symmetric monoidal functor
  out of Mathlib's `ModuleCat.monoidalCategory`; for `S = classical V` the equivalence
  `Almost.Module.classicalEquiv` is monoidal.
- `Almost.Module.tensor_test_unit_ideal` (non-example) — for a proper ideal `m ≠ V`, in `ModuleCat V` the ideal `m` is not a
  tensor unit (`m ⊗_V V/m = m/m² = 0 ≠ V/m`), whereas `m^a ≅ 𝟙_` in `Almost.Module S`; the unit
  isomorphism only exists after almostification.

- `alHom_test_classical_rank_one` — the almost elements of `alHom(V^a, V^a)` are linearly equivalent to `V` in the classical setup for every commutative ring `V`.
- `alHom_test_zero_source` — internal Hom out of a zero object is zero.
- `alHom_test_residue_target` — internal Hom into `(V/m)^a` is zero.
- `tensor_test_classical_two_terms` — classically, the tensor product of two free modules of rank two is free of rank four.
- `tensor_test_residue` — `(V/m)^a ⊗ M` is zero.
- `tensor_test_unit_ideal` — `m^a` is isomorphic to the tensor unit, including the zero setup.
- Zero setup (negative control for unitality): for `(V, 0)` on a nonzero ring such as `ℤ`, `m̃ = 0`,
  so `(V^a)_! = 0 ≇ V` and the unit comparison `m̃ → V` is not surjective; every module is almost
  zero for this setup.
- Classical setup (positive control): for `m = V`, `m̃ → V` is bijective and `(−)_!` is strong
  monoidal into `ModuleCat V`.

Lean: `BasicSetup.zero` and `BasicSetup.tildeMul`; the two proved examples for the zero setup
show that every module is almost zero and that `tildeMul` is not surjective over a nontrivial ring.
The monoidal structure on `Almost.Module S` is specified in this README only.

### P0.8 almost algebras

**Almost algebras (commutative monoids in `V^a-Mod`) and their modules.** (*almost-algebras*) For a
basic setup `S = (V, m)`, an almost `V`-algebra is a commutative unital monoid object in the
symmetric monoidal category `Almost.Module S`: `Almost.Algebra S := CommMon (Almost.Module S)`
(Mathlib), with monoid morphisms ([GR] §2.2, (2.2.5), p. 12). For an almost algebra `A`,
`A_* = Hom(V^a, A)` is a commutative `V`-algebra (`a·b := μ ∘ (a ⊗ b) ∘ ν⁻¹`); `R ↦ R^a` from
commutative `V`-algebras to almost algebras is left adjoint to `A ↦ A_*`, with counit
`A ≅ (A_*)^a`, so `(−)_*` is fully faithful and `Hom_{Alg}(A, B) ≅ Hom_{V-alg}(A_*, B_*)`; every
almost algebra is `R^a` for `R = A_*` ([Sch12] §4, after Proposition 4.6, p. 19; [Bhatt] §4.2,
Remark 4.2.8, p. 25). An `A`-module is a module object over `A` (Mathlib `CategoryTheory.Mod`);
`A`-modules form an abelian, complete and cocomplete category `A-Mod`, closed symmetric monoidal
with `M ⊗_A N := (M_* ⊗_{A_*} N_*)^a`, and `A-Mod ≃ Almost.Module (S.baseChange A_*)`
by *change-of-basic-setup* (P0.16). An `A`-algebra is a morphism `A → B`. Convention:
module-theoretic notions for `A`-modules (flat, almost projective, almost finitely generated and
presented) are those of P0.11 and P0.10, evaluated through this equivalence on the `A_*`-module
`M_*`. Limits are computed on almost elements, `(lim D)_* ≅ lim D_*`. Colimits are almostified
last: `colim D ≅ (colim D_*)^a`, so free modules `((A_*)^{(I)})^a`, quotients and pushouts are `(−)^a`
of the corresponding constructions on almost elements ([GR] §2.2, Corollary 2.2.15 and 2.2.18, p.
14). The functor `(−)_*` preserves neither colimits nor epimorphisms: an almost element of the
target of an epimorphism lifts only after multiplication by an arbitrary `ε ∈ m` ([GR] §2.2, Remarks
2.2.14(ii) and 2.2.28(i), pp. 14–16), so no exactness or colimit-preservation statement is made for
`(−)_*`. Define `Almost.Algebra` (`CommMon (Almost.Module S)`), `Almost.Algebra.toAlmost`
(`CommAlgCat V ⥤ Almost.Algebra S`, `R ↦ R^a`), `Almost.Algebra.elements` (`A ↦ A_*`),
`Almost.Algebra.ModuleCat` (`A-Mod := CategoryTheory.Mod (Almost.Module S) A.X`),
`Almost.Algebra.free` (the free `A`-module `A^{(I)} := ((A_*)^{(I)})^a`, left adjoint to
`M ↦ (M_* as a set)`, [GR] 2.2.18), `Almost.Algebra.quotient` (for an ideal `I ↪ A`, the almost
algebra `A/I` with `A → A/I` an epimorphism), `Almost.Algebra.prod` (finite products with
`(A₁ × A₂)_* = A₁_* × A₂_*` and `A-Mod` decomposing accordingly) and `Almost.Algebra.tensor`
(pushouts `B ⊗_A C ≅ ((B_* ⊗_{A_*} C_*))^a`); prove `Almost.Algebra.adj` (`toAlmost S ⊣ elements S`),
`Almost.Algebra.counitIso` (`(A.elements)^a ≅ A`; `elements` fully faithful),
`Almost.Algebra.homEquiv` (`(A ⟶ B) ≃ (A.elements →ₐ[V] B.elements)`),
`Almost.Algebra.moduleEquiv` (`A-Mod ≌ Almost.Module (S.baseChange A.elements)`) and
`Almost.Algebra.monoidalCategory` (the closed symmetric monoidal structure on `A-Mod`). *Needs:*
P0.7 for the categories and algebra adjunction; Mathlib `CategoryTheory.CommMon`, `CategoryTheory.Mod`,
`CategoryTheory.MonObj`, `CategoryTheory.IsCommMonObj`. The declaration `moduleEquiv`, its
action and forgetful comparisons, and the closed monoidal structure use
*change-of-basic-setup* (P0.16), whose module categories and algebra adjunction are constructed here.

For every basic setup `S` and almost algebra `A`,
`Almost.Algebra.moduleEquiv_elements_action` identifies the almost elements of
`moduleEquiv.functor.obj M` naturally with `Hom(V^a, M.X)` as additive groups,
with the `A_*` action given by
`e_M(a • m) = γ_M ∘ (elementsHomEquiv(a) ⊗ e_M(m)) ∘ λ⁻¹`.
Here `γ_M : A.X ⊗ M.X → M.X` is the internal module action; naturality is the
equation `e_N(F(f)(m)) = f.hom ∘ e_M(m)` for every `A`-module morphism `f`.
These equations give an `A_*`-linear natural comparison with the almost-element
module constructed from the internal action ([GR] (2.2.8), p. 13; Remark 2.2.12
and Proposition 2.2.13, p. 14).
`Almost.Algebra.moduleEquiv_functor_forget` gives a natural isomorphism from
`moduleEquiv.functor`, followed by almost elements over `A_*`, restriction of
scalars along `V → A_*`, and almostification over `S`, to `Mod.forget A.X`.
Thus the comparison recovers the underlying almost module on objects and on
morphisms, compatibly with Mathlib's restriction of scalars ([GR] Remark 2.2.12
and Proposition 2.2.13, p. 14).

**Checks.**

For `moduleEquiv`, the following Checks use independently specified module objects,
with the comparison of [GR] Remark 2.2.12 and Proposition 2.2.13, p. 14:

- `moduleEquiv_zero_setup` — if `S.ideal = ⊥`, the image of every `A`-module is zero
  and `A_*` is the zero ring.
- `moduleEquiv_regular_dualNumbers` — for the classical setup on `ℚ` and
  `A = (ℚ[ε]/ε²)^a`, apply the equivalence to Mathlib's `Mod.regular A.X`.
  Taking almost elements and restricting scalars to `ℚ` gives dimension `2`.
- `moduleEquiv_nonunit` — for the classical setup on `ℤ` and `A = ℤ^a`, the
  transported multiplication-by-two endomorphism of `Mod.regular A.X` is not an
  isomorphism. Almostification here does not invert the nonunit `2`.
- `moduleEquiv_evaluation_one` — for the classical setup on `ℚ`, restrict the
  regular module of `ℚ^a` along the almostification of evaluation
  `ℚ[T] → ℚ`, `T ↦ 1`. On the almost elements of its image under `moduleEquiv`,
  the scalar coming from `T` through the algebra adjunction unit acts as the
  identity. There is a nonzero element on which it does not act by negation,
  distinguishing this module from evaluation at `T = −1`.

- `Almost.Algebra.test_mod_varpi` (computation) — for `S = ofValuationRing K°`, with
  `K = ℚ_p(p^{1/p^∞})^` and `ϖ = p`: `A := (toAlmost S).obj (K° ⧸ ϖ)` is an almost algebra and
  `A.elements ≅ Hom_{K°}(K°°, K°/ϖ) ⊋ K°/ϖ`.
- `Almost.Algebra.algebra_test_classical` (degenerate) — for `S = classical V`, `Almost.Algebra.toAlmost` is
  an equivalence `CommAlgCat V ≌ Almost.Algebra S`.
- `Almost.Algebra.test_modules` (compatibility) — for a `V`-algebra `R`: modules over `(toAlmost R)`
  in `Almost.Module S` are equivalent to `Almost.Module (S.baseChange R)`, compatibly with Mathlib's
  restriction of scalars `ModuleCat R ⥤ ModuleCat V`.
- `Almost.Algebra.test_hom_elements` (characterisation) — for `A = (K°/p)^a` with
  `K = ℚ_p(p^{1/p^∞})^` and `ϖ = p`, `A.elements ⊋ K°/ϖ`, and a morphism `A ⟶ B` of almost algebras is an algebra map
  `A.elements →ₐ[V] B.elements` of almost elements, with restriction along `K°/ϖ → A.elements`. For a target `B.elements`, the adjunction
  also identifies it with a map out of `K°/ϖ`; the larger ring of almost elements does not
  invalidate this latter description.

- `algebra_test_classical` — almostifying an ordinary algebra and taking its almost elements recovers it in the classical setup.
- `algebra_test_zero_setup` — for `m = 0`, every almost algebra and its algebra of almost elements are zero.
- `algebra_test_residue` — both `(V/m)^a` and its almost elements are zero.
- `algebra_test_nonzero` — `ℤ^a` over the classical setup is a nonzero algebra.
- Colimits are not computed on almost elements: for multiplication by `p` on `(K°)^a`, with `K` the
  completion of `ℚ_p(p^{1/p^∞})`, the cokernel computed on almost elements is `K°/p`, the almost
  elements of the cokernel are `((K°/p)^a)_*`, and the comparison `K°/p → ((K°/p)^a)_*` is not
  surjective (the witness of P0.6 with `ϖ = p`).

The formulas for `free`, `prod`, and `tensor` are respectively almostification of
finitely supported functions, the product of the algebras of almost elements, and their tensor
over `V`. The map `homEquiv` is `elements.map`; the hom of `counitIso` is the adjunction counit.

**Checks for these operations.**

- `Almost.Algebra.counitIso_zero`, `Almost.Algebra.homEquiv_zero`, `Almost.Algebra.prod_zero`: for the zero setup all almost elements and the product vanish.
- `Almost.Algebra.counitIso_classical`, `Almost.Algebra.homEquiv_classical`, `Almost.Algebra.prod_classical`: the classical setup recovers ordinary algebra maps and products.
- `Almost.Algebra.counitIso_two`, `Almost.Algebra.homEquiv_two`, `Almost.Algebra.prod_two`: the product of two copies of the classical rational algebra has distinct endomorphisms and an invertible counit.
- `Almost.Algebra.free_empty`: the free module on the empty set is zero.
- `Almost.Algebra.free_one`: one generator gives the regular module.
- `Almost.Algebra.free_two`: two generators give two copies of the regular module.
- `Almost.Algebra.tensor_unit`: tensoring with the base algebra returns the algebra.
- `Almost.Algebra.tensor_zero`: tensoring with the zero algebra gives zero.
- `Almost.Algebra.tensor_two`: tensoring with two copies of the base gives the product of two copies.

### P0.9 the left adjoint `B_!!` on algebras

**The left adjoint `B ↦ B_!!` of almostification on algebras.** (*almost-algebra-lower-shriek-shriek*)
For a basic setup `S = (V, m)` and an almost `V`-algebra `B`, `B_! = m̃ ⊗_V B_*` is a non-unital
`V`-algebra: it inherits a multiplication from `B` through `m̃ ⊗ m̃ ≅ m̃`, but no unit, since
`(V^a)_! ≅ m̃` and the unit comparison `m̃ → V` of the oplax monoidal functor `(−)_!` is not an
isomorphism unless `m = V` (P0.7). Its unitisation is `B_!! := (V ⊕ B_!)/I`, where `I` is the `V`-submodule spanned by the elements
`(xy, −x ⊗ y ⊗ 1)` for `x, y ∈ m`, is a unital commutative `V`-algebra with `B ≅ (B_!!)^a`, and
`B ↦ B_!!` is left adjoint to `R ↦ R^a`. The span is already the ideal of the unitalisation
`V ⊕ B_!` that Gabber–Ramero quotient by: in `B_! = m̃ ⊗_V B_*` one has
`(x ⊗ y ⊗ 1)·(x' ⊗ y' ⊗ b) = x ⊗ yx'y' ⊗ b = (xy)·(x' ⊗ y' ⊗ b)`, so
`(xy, −x ⊗ y ⊗ 1)·(v, b) = v·(xy, −x ⊗ y ⊗ 1)`, and the two descriptions of `I` agree. The
adjunction is between `A_!!`-algebras and `A`-algebras, for any almost algebra `A` with `B` an
`A`-algebra ([GR] §2.2, Proposition 2.2.27, p. 15). `B` is exact if
`0 → m̃ → V ⊕ B_! → B_!! → 0` is exact; every almost algebra is exact when `m̃ → m` is an isomorphism
(e.g. `m` flat), and `V^a × B` is always exact ([GR] §2.2, Definition 2.2.25 and Remark 2.2.26,
p. 15). `B ↦ B_!!` commutes with all colimits, in particular with tensor products ([GR] §2.2, Remark
2.2.28(i), p. 16); the natural map `B_!! → B_*` is an almost isomorphism ([Bhatt] §4.2, Remark
4.2.8, p. 25); `B ↦ B_!!` preserves faithful flatness of morphisms but not flatness,
see *lower-shriek-shriek-faithful-flatness* (P0.25). Define `Almost.Algebra.shriekShriek`
(`Almost.Algebra S ⥤ CommAlgCat V`, `B ↦ B_!!`), `Almost.Algebra.IsExact` and
`Almost.Algebra.shriekShriekToElements` (the natural map `B_!! → B_*`); prove
`Almost.Algebra.shriekShriekAdj` (over `A`: `B ↦ B_!!` is left adjoint to `R ↦ R^a`),
`Almost.Algebra.shriekShriekUnitIso` (`B ≅ (B.shriekShriek)^a`), `Almost.Algebra.isExact_of_bijective`
(if `S.tilde → S.ideal` is bijective, every almost algebra is exact), `Almost.Algebra.isExact_prod`
(`V^a × B` is exact), `Almost.Algebra.shriekShriek_preservesColimits` (with `shriekShriekTensorIso : (B ⊗_A C)_!! ≅ B_!! ⊗_{A_!!} C_!!`). *Needs:* P0.8's categories and algebra adjunction.
The comparison `Almost.Algebra.shriekShriek_faithfullyFlat_iff` (`φ` faithfully flat
`↔ φ_!!` faithfully flat) is *lower-shriek-shriek-faithful-flatness* (P0.25), using this
unitization and P0.11's flatness theory.

**Checks.**

- `Almost.Algebra.shriekShriek_test_unit` (computation) —
  `(Almost.Algebra.toAlmost S (CommAlgCat.of V V)).shriekShriek ≅ V` as `V`-algebras: `(V^a)_! = m̃`
  and `(V ⊕ m̃)/I ≅ V` via `(v, x ⊗ y) ↦ v + xy`.
- `Almost.Algebra.shriekShriek_test_not_flat` (non-example) — for `S = ofValuationRing V` with
  residue field `k`: the projection `V × V → V` is flat but its image under `(−)^a` then `(−)_!!` is
  `V ×_k V → V`, which is not flat ([GR] Remark 3.1.3(ii)); so flatness of almost algebra maps cannot
  be defined through `(−)_!!`.
- `Almost.Algebra.shriekShriek_test_classical` (degenerate) — for `S = classical V`:
  `B.shriekShriek ≅ B.elements` for every almost algebra `B` (`I` contains `(1, −1 ⊗ 1 ⊗ 1)`, and
  `(v, b) ↦ v·1 + b` is the isomorphism).
- `Almost.Algebra.shriekShriek_test_zero_setup` (degenerate) — for the zero setup `(V, ⊥)`:
  `B_! = 0`, `I = 0` and `B_!! = V` for the unique almost algebra `B = 0`; `(B_!!)^a = V^a = 0 = B`
  and `B_!! = V → B_* = 0` is an almost isomorphism, as every `V`-module is almost zero.
- `Almost.Algebra.shriekShriek_test_elements` (compatibility) — the map
  `B.shriekShriek → B.elements` is an almost isomorphism of `V`-algebras, and `(toAlmost S).map` of
  it is the identity of `B` up to the unit isomorphisms.

- `shriekRelation_test_two_terms` — over any classical setup (in particular `(ℤ, ℤ)`), adding the relations for `(2,3)` and `(5,7)` gives
  `(41, −((2 ⊗ 3) ⊗ 1 + (5 ⊗ 7) ⊗ 1))` in the actual unitalization.
  Thus the relation identifies a positive scalar product with the corresponding positive tensor in the quotient.
  The classical, zero-setup, and unit-algebra cases also satisfy `IsExact`.
  The three direct quotient computations `shriekShriek_test_classical`, `shriekShriek_test_zero_setup`, and
  `shriekShriek_test_unit` mirror the classical, zero-setup, and unit-algebra Checks above.

### P0.10 almost finite generation and presentation

**Almost finitely generated, uniformly almost finitely generated and almost finitely presented
modules.** (*almost-finitely-generated-and-presented*) Let `S = (V, m)` be a basic setup, `R` a
`V`-algebra and `N` an `R`-module, `M = N^a` over `A = R^a`. `N^a` is almost finitely generated if
for every finitely generated ideal `m₀ ⊆ m` there are `n` and an `R`-linear `φ : Rⁿ → N` with
`m₀·coker φ = 0`; it is almost finitely presented if for every such `m₀` there is a complex
`R^k →ψ Rⁿ →φ N` with `m₀·coker φ = 0` and `m₀·ker φ ⊆ im ψ`
([GR] Proposition 2.3.10(i), (ii)(c), p. 18). Both conditions depend only on `N^a`.
For an arbitrary almost algebra `A`, apply these two formulas to `M_*` over `A_*` (P0.8).

The invariant uniform notion is `Almost.Module.UniformlyAlmostFG M n`: the isomorphism class of
`M` belongs to `U_n(A)`, the closure of the classes of `A`-modules generated by at most `n`
elements ([GR] Definition 2.3.8(iii), p. 17). The uniform structure is that of [GR]
Definition 2.3.1(ii), p. 16: for each finitely generated `m₀ ⊆ m`, a basic entourage consists of
pairs admitting a common source whose two maps have kernels and cokernels annihilated by `m₀`.
Thus the specified bound is an invariant of the almost module for every basic setup.

Separately, the ordinary-module predicate `BasicSetup.UniformlyAlmostFG N n` requires an
ordinary `φ : Rⁿ → N` with `m₀·coker φ = 0` for every finitely generated `m₀ ⊆ m`.
This implies membership of `N^a` in `U_n(A)` for every setup. Under condition (A),
`BasicSetup.IsFilteredPrincipal`, it is equivalent to that membership and is preserved by
almost isomorphisms with the same bound `n` ([GR] Remark 2.3.9, p. 18).
Without (A), this stronger predicate can depend on the ordinary representative, and evaluating it
on `M_*` does not characterize all representatives of `M`. The non-uniform predicates under (A)
also agree with Scholze's `ε`-formulation: for every `ε ∈ m` there is a finitely generated
(presented) `N_ε → N` with kernel and cokernel killed by `ε`
(*almost-finite-generation-epsilon-criterion*, P0.11; [Sch12] Definition/Proposition 4.7(iii), p. 19).
The Lean forms use `V`-modules over `S`; for `R`-modules use `S.baseChange R`.
In almost finite presentation, `k`, `n`, `ψ` and `φ` may depend on `m₀`, with no uniform kernel
condition ([GR] Proposition 2.3.10(ii)(c)). Define `BasicSetup.AlmostFG`
(`∀ m₀ ≤ S.ideal, m₀.FG → ∃ n (φ : (Fin n → V) →ₗ[V] N), ∀ ε ∈ m₀, ∀ x, ε • x ∈ range φ`),
`BasicSetup.UniformlyAlmostFG` (the ordinary-module predicate with `n : ℕ` fixed) and `BasicSetup.AlmostFP`
(`∀ m₀` f.g., `∃` complex `V^k —ψ→ Vⁿ —φ→ N` with `φ ∘ ψ = 0`, `m₀ • coker φ = 0` and
`m₀ • ker φ ≤ range ψ`); prove `BasicSetup.AlmostFP.almostFG` (almost finitely presented implies
almost finitely generated), invariance of `AlmostFG` and `AlmostFP` under almost
isomorphisms, `BasicSetup.UniformlyAlmostFG.iff_of_isAlmostIso` (under (A), an almost
isomorphism preserves the ordinary-module predicate with the same `n`),
`Almost.Module.almostFG_of_finite` (`Module.Finite R N → AlmostFG`;
`Module.FinitePresentation R N → AlmostFP`), `BasicSetup.AlmostFG.extension` and
`BasicSetup.AlmostFP.extension` (in a short exact sequence of ordinary `V`-modules,
almost f.g., respectively almost f.p., endpoints imply the same property for the middle term;
for `M` almost f.p., a quotient is almost f.p. iff the kernel is almost f.g.,
[GR] Lemma 2.3.17(i)–(ii), p. 20), `BasicSetup.AlmostFG.baseChange` (base change along
`A → B` preserves `AlmostFG`, `AlmostFP` and uniform bounds), `Almost.Module.almostFP_iff_alHom_colim`
(`AlmostFP M ↔ alHom_A(M, −)` commutes with filtered colimits, [GR] Proposition 2.3.15) and
`Almost.Module.almostFG_iff_epsilon` (under (A): `AlmostFG ↔ ∀ ε ∈ m, ∃` f.g. `N_ε → N` with kernel
and cokernel killed by `ε`). *Needs:* P0.8; Mathlib `Submodule.FG`, `Module.Finite`,
`Module.FinitePresentation`.

**Checks.**

- `Almost.Module.AlmostFG_test_quadratic` (computation) — for `K` the completion of
  `ℚ_p(p^{1/p^∞})`, `p ≠ 2`, and `L = K(√p)` (quadratic over `K`, since `√p` has value `1/2`,
  outside the value group `ℤ[1/p]` of `K`; at `p = 2`, `√2 = 2^{1/2}` lies in `K` and `L = K`, so this
  `L` is used for odd `p` only, here and in P0.11–P0.17, where it is called the odd-`p` `L` of
  P0.10): with `R_n = K° ⊕ p^{1/2p^n}K° ⊆ L°`, the cokernel of
  `R_n → L°` is killed by `p^{1/p^{n−1}}`, so `L°a` is `UniformlyAlmostFG` with bound `2` and
  `AlmostFP` ([Bhatt] Example 4.2.7).
- `Almost.Module.finiteness_test_almost_zero` (degenerate) — every almost zero `R`-module `N` gives
  `N^a ≅ 0`, which is `UniformlyAlmostFG` with bound `0` and `AlmostFP`.
- `Almost.Module.finiteness_test_classical` (compatibility) — for `S = classical V`:
  `AlmostFG (N^a) ↔ Module.Finite R N` and `AlmostFP (N^a) ↔ Module.FinitePresentation R N`.
- Boundary of representative independence: let `D` be a characteristic-`p` Dedekind domain
  with a maximal ideal `P` of infinite ideal-class order, `V = D^perf` and `m = √(PV)`.
  This is a basic setup: `m` is idempotent and is the filtered union of the extensions of the
  invertible ideals `P^{1/p^r}`; hence `m` and `m ⊗_V m` are flat ([Stacks] Tag 0AUW).
  The inclusion `m → V` is an almost isomorphism, so both almost modules have invariant bound `1`.
  The ordinary-map predicate has bound `1` for `V`, but not for `m`: for `m₀ = PV`, a map with
  image `zV` would require `(PV)m ⊆ zV`. Then `z` is a unit away from `P`, and a nonzero
  `f = z^{p^r} ∈ D` has `(f) = P^d`, `d > 0`, a contradiction.
  One instance is `p = 5`, `k = algebraicClosure(𝔽₅(u))`,
  `D = k[x,y]/(y² − x³ + x)`, `P = (x − u, y − v)` with `v² = u³ − u`.
  The point `(u,v)` has infinite order on the constant elliptic curve: torsion points have
  coordinates algebraic over `𝔽₅`, whereas `u` is transcendental. The ideal-class identification is
  [Milne, *Elliptic Curves*, I §4, Aside 4.9 and Proposition 4.10, p. 32](https://www.jmilne.org/math/Books/EC2.pdf).
- `Almost.Module.almostFG_test_infinite_sum_valuation` (non-example) — for `S = ofValuationRing K°`:
  `N = ⊕_{n∈ℕ} K°` is not almost finitely generated: for a fixed `0 ≠ ϖ ∈ K°°` and `m₀ = (ϖ)`, no map `(K°)^n → N` has
  cokernel killed by `ϖ` (the image misses all but finitely many summands).

- `finiteness_test_classical` — for every commutative `V`, the classical predicates are
  `Module.Finite V N` and `Module.FinitePresentation V N`.
- `finiteness_test_almost_zero` — an almost zero module satisfies all three predicates, with uniform bound zero.
- `finiteness_test_free` — `Vⁿ` satisfies all three, with uniform bound `n`, including `n = 0, 1, 2`.
- `finiteness_test_infinite_sum` — the countable direct sum of copies of `ℚ` fails all three predicates in the classical setup, for every finite uniform bound.
- `uniform_test_rank_bound`, `uniform_test_zero_bound` — over a field, the bound for `k^r` is `n` exactly when `r ≤ n`; for any setup the bound zero means almost zero.

- Over any domain, a nonzero element of the setup ideal prevents the countable free module from being almost finitely generated: any finite family misses a basis coordinate on which that element still acts nontrivially.
- Over `(ℤ, ℤ)`, `ℤ` is almost finitely generated and presented, whereas `ℕ →₀ ℤ` is neither.
  The same infinite free module satisfies both predicates for the zero setup.

Lean: `BasicSetup.AlmostFG`, `BasicSetup.UniformlyAlmostFG`, `BasicSetup.AlmostFP`,
`BasicSetup.AlmostFP.almostFG`, `BasicSetup.UniformlyAlmostFG.iff_of_isAlmostIso`, and
`BasicSetup.isAlmostFG_classical_iff` and `BasicSetup.isAlmostFP_classical_iff` (classical
almost finite generation and presentation are ordinary finite generation and presentation). The finite-rank, infinite-rank and zero-setup Checks are specified in prose.

### P0.11 flatness and almost projectivity

**Flat, faithfully flat, almost projective and almost finite projective almost modules.**
(*almost-flat-projective-finiteness*) Let `S = (V, m)` be a basic setup, `A` an almost `V`-algebra
and `M` an `A`-module (`A-Mod` as in P0.8). `M` is flat if `M ⊗_A − : A-Mod → A-Mod` is exact,
faithfully flat if moreover `M ⊗_A N ≅ 0` implies `N ≅ 0`, almost projective if
`alHom_A(M, −) : A-Mod → A-Mod` is exact, and almost finite projective if it is almost projective
and almost finitely generated (P0.10); "uniformly almost finite projective" adds a uniform bound
([GR] §2.4, Definition 2.4.4, p. 23). Convention: categorical projectivity (`Hom(M, −)`
exact) is never used — `Almost.Module S` has no nonzero projective objects when `V` is local and
`m ≠ V` ([GR] §2.4, Example 2.4.5, p. 23; [Sch12] §4, Remark 4.8, p. 20). For a `V`-algebra `R` and
an `R`-module `N`: `N` flat (faithfully flat, projective) implies `N^a` flat (faithfully flat,
almost projective) over `R^a` ([GR] §2.4, (2.4.8), p. 24). Characterisations: the Tor and Ext
criteria *almost-flatness-tor-criterion*, the `ε`-factorisation
*almost-projective-epsilon-factorisation* (P0.11) and finite projective = flat + almost finitely
presented, *almost-finite-projective-iff* (P0.11). Define `Almost.Module.Flat` (`M ⊗_A −` preserves
finite limits), `Almost.Module.FaithfullyFlat` (flat and `M ⊗_A N ≅ 0 → N ≅ 0`),
`Almost.Module.AlmostProjective` (`alHom_A(M, −)` preserves epimorphisms) and
`Almost.Module.AlmostFiniteProjective` (`AlmostProjective ∧ AlmostFG`; uniformly with bound `n`);
prove `Almost.Module.flat_toAlmost` (`Module.Flat R N → Flat ((toAlmost _).obj N)`; likewise
`FaithfullyFlat`, and `Projective → AlmostProjective`), `Almost.Module.AlmostProjective.flat`,
`Almost.Module.Flat.tensor` (`Flat` and `AlmostProjective` are stable under `⊗_A` and under base
change `B ⊗_A −`), `Almost.Module.Flat.trans` (if `B` is a flat (almost projective, almost f.g.,
almost f.p.) `A`-algebra and `M` such a `B`-module, `M` is such an `A`-module, [GR] Lemma 2.4.7),
and `Almost.Module.flat_iff_tor` (`Flat (N^a) ↔ ∀ X i, 0 < i → IsAlmostZero (Tor_i^R(N, X))`);
the characterisations `Almost.Module.almostProjective_iff_factor` (for `M` almost f.g.:
`AlmostProjective M ↔ ∀ ε ∈ m, ∃ n u v, v ∘ u = ε • 𝟙` through `A^n`) and
`Almost.Module.almostFiniteProjective_iff` (`AlmostFiniteProjective M ↔ Flat M ∧ AlmostFP M`) are
proved with the supporting results *almost-projective-epsilon-factorisation* (P0.11) and
*almost-finite-projective-iff* (P0.11). *Needs:* P0.8; P0.10; Mathlib `Module.Flat`,
`Module.FaithfullyFlat`, `Module.Projective`, `CategoryTheory.Functor.PreservesEpimorphisms`.

Two supporting results. Tor and Ext criteria for almost flatness and almost projectivity
(*almost-flatness-tor-criterion*): for a basic setup `S`, a `V`-algebra `R` and `R`-modules `M, N`,
(i) there are natural isomorphisms `Tor_i^{R^a}(M^a, N^a) ≅ Tor_i^R(M, N)^a` and
`alExt^i_{R^a}(M^a, N^a) ≅ Ext^i_R(M, N)^a`, where `Tor^{R^a}` and `alExt` are the derived functors
of `⊗` and `alHom` on `R^a-Mod` (computed by flat resp. injective resolutions; Mathlib
`CategoryTheory.Abelian.Ext`); (ii) `N^a` is flat over `R^a` iff `Tor_i^R(N, X)` is almost zero for
all `R`-modules `X` and all `i > 0`; (iii) `N^a` is almost projective iff `Ext^i_R(N, X)` is almost
zero for all `X` and `i > 0`; (iv) `N^a` is faithfully flat iff it is flat and `N ⊗_R X` almost zero
implies `X` almost zero; (v) an almost projective module is flat ([GR] §2.4, (2.4.10), p. 24; [GR]
§2.4, Remark 2.4.12(i), p. 24; [Sch12] §4, Definition/Proposition 4.7(i), p. 19). Over `(K°, K°°)`, `M^a` is flat iff `Tor_1(M, K°/ϖ) = M[ϖ]` is almost zero, iff
`M_*` has no `ϖ`-torsion ([Sch12] Lemma 5.3(i)). The `ε`-criterion for almost finite generation and
presentation under condition (A) (*almost-finite-generation-epsilon-criterion*): for a basic setup
satisfying (A) (root and valuation setups), a `V`-algebra `R` and an `R`-module `N`, `N^a` is almost
finitely generated (resp. presented) iff for every `ε ∈ m` there are a finitely generated (resp.
presented) `R`-module `N_ε` and `N_ε → N` with kernel and cokernel killed by `ε`, and `N^a` is
uniformly almost finitely generated with bound `n` iff for every `ε ∈ m` there is `Rⁿ → N` with
cokernel killed by `ε`. For general basic setups, the non-uniform generation and presentation
criteria use finitely generated subideals `m₀ ⊆ m` as in P0.10; the fixed-rank ordinary-map
criterion implies the invariant uniform bound, with the converse asserted under (A)
([GR] §2.3, Remark 2.3.9, p. 18;
[Sch12] §4, Definition/Proposition 4.7(iii), p. 19).

**Checks.**

- `Almost.Module.Flat_test_valuation` (computation) — for `S = ofValuationRing K°` and `0 ≠ ϖ ∈ K°°`:
  `(toAlmost S).obj K°` is `Flat` and `(toAlmost S).obj (K° ⧸ ϖ)` is not `Flat` (its almost elements
  have `ϖ`-torsion).
- `Almost.Module.projectivity_test_classical` (compatibility) — for `S = classical V` and a `V`-module `N`:
  `Flat (N^a) ↔ Module.Flat V N`, `FaithfullyFlat (N^a) ↔ Module.FaithfullyFlat V N`, and
  `AlmostProjective (N^a) ↔ Module.Projective V N`.
- `Almost.Module.projectivity_test_zero` (degenerate) — the zero module is `Flat` and
  `AlmostProjective` but not `FaithfullyFlat` (unless `A ≅ 0`).
- `Almost.Module.almostProjective_test_unit_epsilon` (degenerate) — if some `ε ∈ m` is a unit then
  `m = V` and the factorisation `v ∘ u = ε • 𝟙` through `Aⁿ` of `almostProjective_iff_factor` is an
  honest splitting, so `AlmostProjective` is `Module.Projective` and the `ε`-criterion
  `N_ε → N` of *almost-finite-generation-epsilon-criterion* is an isomorphism from a finitely generated module; both
  reduce to the classical statements, as `Almost.Module.projectivity_test_classical` requires.

Supporting result: Flat + almost finitely presented ⟺ almost projective + almost finitely
generated (*almost-finite-projective-iff*, `Almost.Module.almostFiniteProjective_iff`): for a basic
setup `S`, an almost `V`-algebra `A` and an `A`-module `M`, (i) if `M` is almost projective and
almost finitely generated, `M` is almost finitely presented (and flat); (ii) if `M` is flat and
almost finitely presented, `M` is almost projective;
hence Scholze's "finite projective" (flat and almost finitely presented) coincides with almost
finite projective; (iii) if `A → B` is a monomorphism of almost algebras and `M` is an almost
finitely generated flat `A`-module with `B ⊗_A M` almost finitely presented over `B`, then `M` is
almost finitely presented ([GR] §2.4, Proposition 2.4.18, p. 26; [Sch12] §4, Proposition 4.10 and
the text after it, p. 20). Classically this says finitely presented flat modules are projective and
finitely generated projective modules are finitely presented. 

Almost projectivity via
`ε`-factorisations through free modules (*almost-projective-epsilon-factorisation*,
`Almost.Module.almostProjective_iff_factor`): for an almost algebra `A` and an `A`-module `M`,
(i) `M` is almost projective iff for every `ε ∈ m` there are a set `I` and `A`-linear maps
`M →u A^{(I)} →v M` with `v ∘ u = ε·1_M`; (ii) if `M` is almost finitely
generated, `M` is almost projective iff for every `ε ∈ m` there are `n(ε) ∈ ℕ` and
`M →u Aⁿ →v M` with `v ∘ u = ε·1_M`, and, under condition (A), `M` is uniformly almost finite projective with bound `n`
iff `n` can be taken independent of `ε`, Kedlaya–Liu's definition ([GR] §2.4, Lemma 2.4.15, p. 25, and the proof of Lemma 2.4.29, p. 28;
[KL15] §5.5, Definition 5.5.8, p. 132). For `P = L°a` over `K°a` (`L = K(√p)`, the odd-`p` `L` of
P0.10) the maps
`L° → (K°)² → L°` with composite `p^{1/p^{n−1}}` ([Bhatt] Example 4.2.7) exhibit `P` as uniformly
almost finite projective with bound `2`.

- `projectivity_test_classical` — flatness, faithful flatness and almost projectivity become the three ordinary module predicates in the classical setup.
- `projectivity_test_zero` — the zero object is flat, almost projective, almost finite projective, and uniformly so with bound zero; it is faithfully flat exactly when the tensor unit is zero.
- `projectivity_test_free` — the almostification of `Vⁿ` is almost finite projective with uniform bound `n`, including `n = 0, 1, 2`.
- `projectivity_test_torsion` — classically, `ℤ/2` is neither flat nor almost projective.
- `faithful_test_unit` — the tensor unit is faithfully flat in every setup.
- `uniformProjectivity_test_rank` — over a field in the classical setup, a free module of rank `r` has uniform projective bound `n` exactly when `r ≤ n`.

`Almost.Module.ordinaryTor` and `ordinaryExt` use Mathlib's derived tensor and
internal-Hom functors on ordinary modules. `tor` and `alExt` almostify their values on
almost elements; `tor_toAlmost` and `alExt_toAlmost` identify the result for any honest
representatives ([GR] (2.4.10)).

- `derived_test_degree_zero` — degree zero recovers almost tensor and internal Hom.
- `derived_test_free` — positive Tor and almost Ext vanish from every finite free almost module, including rank zero.
- `derived_test_torsion` — `Tor₁^ℤ(ℤ/2, ℤ/2) = ℤ/2` and `Ext¹_ℤ(ℤ/2, ℤ) = ℤ/2`;
  their almost versions are nonzero in the classical setup.

Lean: `BasicSetup.IsAlmostFiniteProjective`, the `ε`-factorisation of
*almost-projective-epsilon-factorisation*(ii) for a `V`-module over `S`;
`BasicSetup.IsAlmostFiniteProjective.almostFG`; and
`BasicSetup.isAlmostFiniteProjective_classical_iff`, identifying the classical `ε`-form with
`Module.Projective ∧ Module.Finite`.

### P0.12 duals and traces

**Duals, traces of endomorphisms and the trace form of an almost finite projective algebra.**
(*almost-dual-and-trace*) Provides the duality trace whose internal almost-module realization is
used by the algebra-trace and trace-form consumers. Supplied by PivotalSpherical §Layer 2 / Left
and right trace and the common spherical `trace`.
Convention gap: this roadmap proves the required almost duality, instantiates the supplier's
standard symmetric pivotal/spherical structure (§Layer 7), and identifies the internal map
`End_A(P)^a → A` with its tensor-unit-valued trace on endomorphisms.

Let `S` be a basic setup and `A` an almost `V`-algebra. For an `A`-module `M`, the dual is
`M* := alHom_A(M, A)` with evaluation `ev_{M/A} : M ⊗_A M* → A`. For `P` almost finite projective,
`ω_{P/A} : P ⊗_A P* → End_A(P)^a` is an isomorphism by
*tensor-hom-comparison-almost-projective*. On the full subcategory of almost finite projective
`A`-modules, prove closure under duals and tensor products and both triangle identities for the
evaluation and the coevaluation obtained from `ω_{P/A}⁻¹` applied to the identity; these give the
rigid symmetric category needed by the supplier. Retain `Almost.Module.trace` only as the local
internal realization `tr_{P/A} := ev_{P/A} ∘ ω_{P/A}⁻¹ : End_A(P)^a → A`. Its comparison with the
supplied trace is an equality of internal morphisms, natural on maps `L → End_A(P)^a` for every
test `A`-module `L`, using the corresponding contraction `L ⊗_A P → P`; at `L = A` it recovers
PivotalSpherical §Layer 2's trace through `alHom_A(A, A) ≅ A`.
The carrier's docstring is: “Internal almost-module realization of PivotalSpherical §Layer 2's
trace for `Almost.Algebra.trace` and trace-form consumers. The tensor–Hom/evaluation and scalar
identifications are proved here.”

Provides cross-object cyclicity for the almost endomorphism trace through the duality comparison.
Supplied by PivotalSpherical §Layer 2 / Cyclicity (`leftTrace_comm` and the common spherical trace).
Convention gap: both source objects must carry
the proved almost duality, and this roadmap translates composition between `ψ∘φ` and Lean's
left-to-right `f ≫ g` before transporting scalar traces through the internal comparison.

Provides tensor multiplicativity for the almost endomorphism trace. Supplied by PivotalSpherical
§Layer 2 / The left trace is monoidal (`leftTrace_tensor` and the common spherical trace).
Convention gap: this roadmap transports the
supplier's tensor-unit endomorphisms and unitor formula to multiplication in `A`, preserving the
chosen tensor/dual identifications.

For an `A`-algebra `B` which is almost finite projective as an `A`-module,
`μ : B → End_A(B)^a` (`b ↦`
multiplication by `b`, defined on almost elements; [Sch12] §4, Remark 4.15, p. 20) gives the almost
trace `Tr_{B/A} := tr_{B/A} ∘ μ : B → A` using the internal realization of PivotalSpherical
§Layer 2 ([GR] §4.1, (4.1.7), p. 74), the trace form
`t_{B/A} := Tr_{B/A} ∘ μ_{B/A} : B ⊗_A B → A` and `τ_{B/A} : B → B*`; `t_{B/A}` is perfect, by
definition, iff `τ_{B/A}` is an isomorphism ([GR] §4.1, (4.1.12), p. 75). Properties:
additivity on short exact sequences of almost finite projective modules;
`Tr_{C⊗_A B/C} = 1_C ⊗ Tr_{B/A}`; `Tr_{C/A} = Tr_{B/A} ∘ Tr_{C/B}` ([GR]
§4.1, Proposition 4.1.8(ii)-(iii), p. 74); `τ` is `B`-linear and compatible with base change.
Mathlib's `LinearMap.trace` (defined through a finite basis, and `0` when the module is not finite
free) is not used for almost finite projective modules; it agrees with `tr` on finite free modules
after almostification via AdicSpacesPartIII §R3.8 / *projective-module-trace*,
`LinearMap.projectiveTrace_eq_trace` and the internal comparison above.
Define `Almost.Module.dual` (`M*` with its evaluation), `Almost.Algebra.trace` (`Tr_{B/A} : B ⟶ A`),
`Almost.Algebra.traceForm` (`t_{B/A}` and `τ_{B/A}`) and `Almost.Algebra.IsPerfectTraceForm`
(`τ_{B/A}` is an isomorphism); prove `Almost.Module.dualTensorHomIso` (`ω_{P/A} : P ⊗_A P* ≅ End_A(P)^a` for `P` almost finite projective), `Almost.Module.trace_additive`, `Almost.Module.isReflexive`
(`P ≅ P**`), `Almost.Algebra.trace_baseChange`, `Almost.Algebra.trace_trans`,
`Almost.Algebra.trace_of_iso` (simp: if `A → B` is an isomorphism, `Tr_{B/A}` is its inverse) and
`Almost.Algebra.traceFormDual_baseChange` (`η ∘ (1_C ⊗ τ_{B/A}) = τ_{C⊗B/C}`, where
`η : C ⊗_A B* → (C ⊗_A B)*` is the dual base-change comparison, [GR] Lemma 4.1.13).
*Needs:* P0.7; P0.8; P0.11; PivotalSpherical §Layer 2 (trace, cyclicity and monoidality) and §Layer 7
(standard symmetric pivotal/spherical structure); AdicSpacesPartIII §R3.8 /
*projective-module-trace*; Mathlib `LinearMap.trace`, `Algebra.trace`, `Algebra.traceForm`.

Supporting result: `E ⊗ alHom(F, N) → alHom(F, E ⊗ N)` is an isomorphism for almost finite
projective `E` or `F` (*tensor-hom-comparison-almost-projective*). For a basic setup `S`, an almost
algebra `A` and `A`-modules `E, F, N`, the natural map `E ⊗_A alHom_A(F, N) → alHom_A(F, E ⊗_A N)`
is (i) an isomorphism if `E` or `F` is almost finite projective, if `E` is flat and `F` almost
finitely presented, if `F` is almost projective and `E` almost finitely presented, or if `E` is
almost projective and `F` almost finitely generated; (ii) a monomorphism if `E` is flat and `F`
almost finitely generated, or if `E` is almost projective; (iii) an epimorphism if `F` is almost
projective and `E` almost finitely generated ([GR] §2.4, Lemma 2.4.29, p. 27). Consequently, for `P`
almost finite projective, `ω_{P/A}` is an isomorphism ([GR] §2.4, Lemma 2.4.29(i)(b), p. 27), `P` is
reflexive ([GR] §2.4, Lemma 2.4.31(ii), p. 28), and
`B ⊗_A alHom_A(P, N) ≅ alHom_B(B ⊗_A P, B ⊗_A N)` for every `A`-algebra `B`; for `P = (Rⁿ)^a` free,
`ω_{P/A}` is the usual `Rⁿ ⊗ (Rⁿ)^∨ ≅ End(Rⁿ)` after almostification.

**Checks.**

- `Almost.Module.trace_test_free_endomorphism` (compatibility) — for `P = (toAlmost _).obj (Fin n → R)`:
  `Almost.Module.trace P ((toAlmost _).map f) = (toAlmost _).map (LinearMap.trace R _ f)`, and for a
  free finite `R`-algebra `R'`, `Almost.Algebra.trace = (Algebra.trace R R')^a`.
- `Almost.Algebra.trace_test_quadratic` (computation) — for `K` the completion of
  `ℚ_p(p^{1/p^∞})`, `p ≠ 2`, `L = K(√p)` (the odd-`p` `L` of P0.10, for which `Tr(1) = 2` is a
  unit): `Tr_{L°a/K°a}(1) = 2` and `IsPerfectTraceForm (L°a/K°a)`,
  while the honest trace form on `R_n = K° ⊕ p^{1/2p^n}K°` has discriminant `4p^{1/p^n}` ([Bhatt]
  Examples 4.2.7 and 4.3.2, pp. 24 and 26).
- `Almost.Algebra.trace_test_quadratic_order` — for the monic quadratic order
  `R[X]/(X²-a)`, the trace matrix on `1, X` has determinant `4a`; taking `a = p^{1/pⁿ}`
  gives the discriminant of the orders in the preceding Check.
- `Almost.Module.trace_test_almost_zero` (degenerate) — for `P = 0`, `tr_{P/A} = 0`; for `B = 0`,
  `Tr_{B/A} = 0` and `t_{B/A}` is perfect (`τ : 0 → alHom(0, A) = 0`).
- `Almost.Algebra.trace_test_self_product_dual_numbers` (computation) — for `B = A`:
  `Tr_{A/A} = 𝟙_A` and `t(b ⊗ b') = bb'` is perfect (`τ = 𝟙`); for `B = A × A`:
  `Tr((b₁, b₂)) = b₁ + b₂`, `t((b₁, b₂) ⊗ (c₁, c₂)) = b₁c₁ + b₂c₂`, perfect; for `B = A[ε]/(ε²)`:
  `Tr(1) = 2`, `Tr(ε) = 0`, `t` has matrix `[[2, 0], [0, 0]]` on the basis `1, ε` and `τ(ε) = 0`,
  so for `A ≠ 0`, `t` is not perfect even when `2` is a unit, and `B` is not unramified (P0.15).
  For `A = 0` both `B` and its trace pairing are zero, and the pairing is perfect.
- `Almost.Module.trace_test_not_linearMap_trace` (non-example) — for `P = L°a` over `K°a`, with `L`
  the odd-`p` `L` of P0.10 (`L°` is
  not a free `K°`-module), `(LinearMap.trace K° L°)^a = 0 ≠ tr_{P/A}(1) = 2`: the almost trace cannot
  be defined through Mathlib's `LinearMap.trace` of a representative.

- `dual_test_zero`, `dual_test_unit`, `dual_test_rank_two` — the duals of zero, the unit, and a free module of rank two are respectively zero, the unit, and a free module of rank two.
- `trace_test_almost_zero`, `trace_test_almost_unit`, `trace_test_almost_two` — the actual almost trace gives `0`, `1`, and `2` on the identities of those three objects.
- `trace_test_free_endomorphism` — for every endomorphism of `Vⁿ`, including ranks zero, one, and two, the trace after almostification is the scalar endomorphism of the tensor unit given by Mathlib's ordinary finite-free trace.
- `trace_test_char_two` — the identity of a classical rank-two vector space in characteristic two has trace zero.
- `tensorHom_test_zero`, `tensorHom_test_unit`, `tensorHom_test_two` — the specified tensor–Hom comparison is invertible for zero source, unit tensor factor, and free rank-two source respectively.

- `trace_test_product`, `trace_test_dual_numbers`, `trace_test_unramified` — the ordinary library computations
  give product trace matrix `diag(1,1)`, dual-number trace matrix `diag(2,0)`, and failure of formal
  unramifiedness for dual numbers over a nonzero ring. These are comparisons for the almost trace, not its definition.

- `algebraTrace_test_zero` — the relative algebra trace and its bilinear form are zero for a zero target, and the resulting map from zero to its dual is invertible.
- `algebraTrace_test_product` — the trace form of the almostification of `V × V` is perfect; its trace is nonzero unless the tensor unit is zero.
- `algebraTrace_test_dual_numbers` — over the classical setup on `ℚ`, the dual-number trace form is nonzero but not perfect. This distinguishes a nondegenerate pairing from a merely nonzero map.

`Almost.Algebra.traceElements` is the trace on almost-element rings. Its characterizing
identity `traceElements_spec` identifies its almostification with `trace` using the
monoidal unit comparison. The base-change and dual-trace identities are evaluated on
pure tensors; the transitivity and isomorphism identities are evaluated on almost elements.

**Checks.**

- `traceElements_identity` — the trace for the identity algebra is the identity function.
- `traceElements_zero` — the trace from a zero almost algebra is zero.
- `traceElements_product` — the product algebra sends `(a,b)` to `a+b`, including characteristic two.

`Almost.kummerField p r` is the closure in `ℂ_p` of the subfield generated by the
range of `r`, where `r₀ = p` and `rₙ₊₁^p = rₙ`. Its equation is `kummerField_eq`.
This choice specifies one compatible root sequence. The quadratic examples use
`L = K(√p)` inside `ℂ_p`; `kummerIntegersMap` is the inclusion on valuation rings,
characterized by its unchanged image in `ℂ_p`.

**Checks.**

- `kummerField_zero` — the zero sequence generates the closed copy of `ℚ_p`.
- `kummerField_roots` — every chosen root belongs to `K` and has norm whose `pⁿ`-th power is `p⁻¹`.
- `kummerField_quadratic` — for odd `p`, a square root of `p` is outside `K`.
- `kummerIntegersMap_identity` — equal subfields give the identity map.
- `kummerIntegersMap_composition` — two inclusions compose to the direct inclusion.
- `kummerIntegersMap_prime` — the inclusion sends the integral scalar `p` to `p`.
- `kummerIntegersMap_quadratic_not_surjective` — for odd `p`, the inclusion into
  the quadratic extension misses the integral square root of `p`.

### P0.13 rank decomposition

**Rank decomposition of uniformly almost finite projective modules.**
(*rank-decomposition-of-finite-projective*) Let `S` be a basic setup satisfying condition (B)
(`k`-th powers generate `m`; automatic from flatness of `m̃` by [GR] Proposition 2.1.7(ii), p. 9; (B) is used for
the uniform continuity of `Λ^r`, [GR] Lemma 2.3.7(vi), and for Proposition 4.3.4), `A` an almost
`V`-algebra and `P` a uniformly almost finite projective `A`-module. Then `P` has finite rank:
`Λ^r_A P = 0` for some `r` ([GR] §4.3, Remark 4.3.10(i), p. 85). For `P` almost finite projective with
`Λ^r_A P = 0` there is a unique decomposition `A ≅ A_0 × A_1 × ⋯ × A_{r−1}` such that
`P_i := P ⊗_A A_i` has constant rank `i` over `A_i`: `Λ^{i+1}_{A_i} P_i = 0` and `Λ^i_{A_i} P_i` is
an invertible `A_i`-module (`Λ^i P_i ⊗ (Λ^i P_i)* ≅ A_i`); the decomposition is compatible with base change
`A → A'` ([GR] §4.3, Proposition 4.3.27, p. 88, with the standing finiteness hypothesis on p. 83;
[GR] §4.3, Definition 4.3.9(iv), p. 85; [Sch12] §4,
Theorem 4.11, p. 20, over `(K°, K°°)`, citing the book numbering). For `L°a` over `K°a`
(`L = K(√p)`, the odd-`p` `L` of P0.10): `r = 3` and `A = A_2` (constant rank `2`); for `A = K°a × K°a` and `P = K°a × 0`,
`A_1` is the first factor and `A_0` the second. *Needs:* P0.12; Mathlib `Module.Invertible`,
`ExteriorAlgebra.exteriorPower`.

`Almost.Module.rankDecomposition_baseChange` states compatibility for every almost-algebra
map `f : A → A'`, every almost finite projective module `P` over `S.baseChange A_*`, and every
`D : RankDecomposition P r`. It produces a rank decomposition of
`(Almost.Module.baseChange S f).obj P` and a comparison between the new tensor unit and the
scalar extension of the old one. Each new projector is the corresponding transported projector
conjugated by that comparison. No flatness or faithfulness of `f` is required; a factor may become
zero. This is the naturality of [GR] Proposition 4.3.27, p. 88.

**Checks.**

- `Almost.Module.rankFactors_test_zero` (degenerate) — `P = 0` has `Λ^1 P = 0`, so `r = 1` and the
  decomposition is `A = A_0`; a module of constant rank `0` is zero, since `Λ^1 P = P`; for a
  uniformly almost finite projective `P` with invariant bound `n`, `Λ^{n+1} P = 0`:
  `Λ^{n+1}` is uniformly continuous under (B) and vanishes on modules generated by at most `n`
  elements, hence on their closure `U_n(A)` ([GR] Lemma 2.3.7(vi), p. 17,
  and Remark 4.3.10(i), p. 85).

- `exterior_test_degree_zero` — `Λ⁰ M` is the tensor unit, including `M = 0`.
- `exterior_test_degree_one` — `Λ¹ M ≅ M`; constant rank zero therefore forces `M = 0`.
- `exterior_test_rank_two` — for `(V²)^a`, `Λ³ = 0` and `Λ² ≅ V^a`.
- `Almost.Module.rankFactors_test_one` — a rank-one free module has `r = 2`, with `A₁ = A` and `A₀ = 0`.
- `Almost.Module.rankFactors_test_two` — a rank-two free module has `r = 3`, with `A₂ = A` and `A₀ = A₁ = 0`.
  `RankDecomposition` expresses the factors by orthogonal idempotent endomorphisms of the tensor unit,
  whose sum is the identity. The factor modules are their images. On factor `i`, `Λ^{i+1} P` vanishes
  and the evaluation of `Λ^i P` against its dual is invertible. Thus the carrier records constant rank,
  not merely a list of ring factors. `rankDecomposition` constructs it and `rankDecomposition_unique`
  fixes its projectors uniquely; `RankDecomposition.unitIso` is the base decomposition.
- `rankFactors_test_zero`, `rankFactors_test_one`, `rankFactors_test_two` — for free ranks `0, 1, 2`,
  only the matching projector is the identity and the other projectors are zero.
- `rankFactors_test_empty` — an empty rank decomposition forces the base unit to be zero.
- `Almost.Module.rankDecomposition_infinite_support` (negative control) — let `k` be a field,
  `V = ∏_{i∈ℕ} k` with the classical setup, and `I = ⊕_{i∈ℕ} V δ_i ⊂ V`, where `δ_i` is the
  coordinate idempotent. For `P = I^a`, projectivity follows from the direct sum of the projective
  summands `V δ_i`, and `Λ²_V P = 0`: distinct summands have zero tensor product, and each summand
  is cyclic. Yet `P` is not almost finitely generated, since any finite set of generators has
  support in a common finite subset of `ℕ`. Its annihilator is zero, so a rank decomposition with
  `r = 2` would force the rank-zero projector to be zero and the rank-one projector to be the
  identity. But the image of evaluation `P ⊗ P* → V^a` is `I^a`, which misses `1`; evaluation is
  not invertible and `RankDecomposition S P 2` is empty. Thus vanishing of an exterior power does
  not replace the almost finite generation hypothesis in `rankDecomposition`.

### P0.14 flat, unramified, étale and finite étale morphisms

Almost finite étale algebras are built next: the morphism classes, the trace-form criterion,
inversion of an element of `m`, the comparison with Mathlib's finite étale algebras, almost Kähler
differentials and the cotangent complex of a ring map.

**Flat, unramified, étale and finite étale morphisms of almost algebras.**
(*almost-finite-projective-and-finite-etale-algebras*) Let `S` be a basic setup and `φ : A → B` a
morphism of almost `V`-algebras; view `B` as a `B ⊗_A B`-algebra through the multiplication
`μ_{B/A} : B ⊗_A B → B`, with kernel `I_{B/A}`; the module notions are those of P0.11 and P0.10,
applied to `B` as an `A`-module and as a `B ⊗_A B`-module. `φ` is flat (faithfully flat, almost
projective) if `B` is a flat (faithfully flat, almost projective) `A`-module; (uniformly) almost
finite if `B` is a (uniformly) almost finitely generated `A`-module; weakly unramified (unramified)
if `B` is a flat (almost projective) `B ⊗_A B`-module; weakly étale (étale) if flat and weakly
unramified (unramified) ([GR] §3.1, Definition 3.1.1(iii)-(iv), p. 39). `φ` is finite étale if it is
étale and `B` is an almost finitely presented `A`-module ([Sch12] §4, Definition 4.13, p. 20;
Gabber–Ramero's `A-Ét_afp`, [GR] §3.2, (3.2.27), p. 46; Kedlaya–Liu's and Bhatt's "almost finite
étale", [Bhatt] §4.3, Definition 4.3.1, p. 25); then `B` is almost finite projective over `A`
by *almost-finite-projective-iff* (P0.11). `A_fet` denotes the full subcategory of `A`-algebras that
are finite étale over `A`. Equivalent descriptions: unramified iff there is a diagonal idempotent
`e ∈ (B ⊗_A B)_*` with `μ(e) = 1` and `I_{B/A,*}·e = 0` (*unramified-idempotent-characterisation* in P0.15,
Scholze's definition); for `B` almost finite projective over `A`, étale iff the trace form
`t_{B/A}` is perfect (P0.15). Convention: "finite étale" always means étale and almost finitely
presented; in the classical setup it is Mathlib's finite étale (P0.17), while Gabber–Ramero's
"étale" without finiteness is more general than EGA étale ([GR] §3.1, Remark 3.1.3(i), p. 39).
Define `Almost.Algebra.IsFlat` (flat, faithfully flat, almost projective `φ`),
`Almost.Algebra.IsUnramified` (with `IsWeaklyUnramified`), `Almost.Algebra.IsEtale`
(`IsFlat ∧ IsUnramified`, with `IsWeaklyEtale`), `Almost.Algebra.IsFiniteEtale`
(`IsEtale ∧ AlmostFP B` over `A`) and `Almost.Algebra.FiniteEtale` (`A_fet`, the full subcategory
of `Under A`); prove `Almost.Algebra.IsEtale.baseChange` and `Almost.Algebra.IsEtale.comp`
(stability of flat, unramified, étale, weakly étale and finite étale morphisms under base change
and composition, *etale-morphisms-stability*),
`Almost.Algebra.IsFiniteEtale.almostFiniteProjective` (by *almost-finite-projective-iff*, P0.11) and
`Almost.Algebra.isFiniteEtale_toAlmost` (Mathlib finite étale `R → R'` gives finite étale
`R^a → R'^a`); the characterisations `Almost.Algebra.isUnramified_iff_exists_idempotent` and
`Almost.Algebra.isEtale_iff_isPerfectTraceForm` are P0.15's. *Needs:* P0.11; P0.10; P0.8;
*almost-finite-projective-iff* (P0.11);
Mathlib `Algebra.Etale`, `Algebra.FormallyUnramified`, `Module.Finite`.

**Checks.**

- `Almost.Algebra.IsFiniteEtale_test_quadratic` (computation) — for `K` the completion of
  `ℚ_p(p^{1/p^∞})`, `p ≠ 2`, `L = K(√p)` (the odd-`p` `L` of P0.10):
  `IsFiniteEtale ((toAlmost _).map (algebraMap K° L°))` over
  `S = ofValuationRing K°` ([Bhatt] Example 4.3.2).
- `Almost.Algebra.IsFiniteEtale_test_idempotent_not_honest` (non-example) — in the same example the
  diagonal idempotent `e = (1/(2√p ⊗ 1))(1 ⊗ √p + √p ⊗ 1)` is an almost element of `L°a ⊗ L°a` but
  does not lie in `L° ⊗_{K°} L°` (only `p^{1/p^n}e` does, for every `n`): a definition requiring an
  honest idempotent of `B_* ⊗_{A_*} B_*` would reject `L°a/K°a`.
- `Almost.Algebra.finiteEtale_test_classical` (compatibility) — for `S = classical V` and a
  `V`-algebra map `R → R'`: `IsFiniteEtale ((toAlmost S).map f) ↔ Algebra.Etale R R' ∧ Module.Finite R R'` (Mathlib).
- `Almost.Algebra.IsFiniteEtale_test_trivial` (degenerate) — `𝟙 A`, `A → 0` and the diagonal
  `A → A × A` are finite étale.

- `algebraPredicates_test_identity` — every identity map is faithfully flat, almost projective, almost finite, weakly étale, and finite étale.
- `algebraPredicates_test_zero` — a map to the zero algebra has all these properties except faithful flatness, which holds exactly when the source is zero.
- `algebraPredicates_test_product` — the diagonal `V^a → (V × V)^a` has all these properties.
- `algebraPredicates_test_dual_numbers` — the classical map `ℚ → ℚ[ε]/ε²` is faithfully flat, finite free, and almost projective, but neither weakly unramified nor unramified nor finite étale.

- `finiteEtale_test_classical` — for an arbitrary tower `V → R → T` of commutative rings, the almostified map in the classical setup is finite étale exactly when Mathlib's `Algebra.Etale R T` and `Module.Finite R T` hold.

### P0.15 the trace-form criterion

**Almost finite projective algebras are étale iff the trace form is perfect.**
(*trace-form-criterion*) Let `S` be a basic setup and `φ : A → B` a morphism of almost algebras with
`B` almost finite projective as an `A`-module. Then `φ` is étale iff the trace form
`t_{B/A} : B ⊗_A B → A` is a perfect pairing, i.e. `τ_{B/A} : B → B* = alHom_A(B, A)` is an
isomorphism (`Almost.Algebra.isEtale_iff_isPerfectTraceForm`; [GR] §4.1, Theorem 4.1.14, p. 75;
[Sch12] §4, Theorem 4.16, p. 21; [KL15] §5.5, Definition 5.5.8, p. 132, where the criterion is the
definition). In that case
`(1_B ⊗ τ_{B/A})(e_{B/A}) = ζ_B`, where `e_{B/A}` is the diagonal idempotent and
`ζ_B = ω_{B/A}⁻¹(1_B)` ([GR] §4.1, Remark 4.1.17, p. 76). In particular, for `B` almost finitely
presented, `φ` is finite étale iff `B` is flat over `A` and `t_{B/A}` is perfect. For `L°a/K°a`
(`K = ℚ_p(p^{1/p^∞})^`, `p ≠ 2`, `L = K(√p)`) the trace form is perfect on the almost level (the discriminants `4p^{1/p^n}` of the
approximations `R_n` tend to a unit up to `m`); in the classical setup with `B` finite free this is
the criterion "finite free étale iff the discriminant is a unit". The trace uses P0.12's internal
realization of PivotalSpherical §Layer 2. *Needs:* P0.12; P0.14.

`Almost.Algebra.diagonalIdempotent_trace` states the identity on the relative module
`P = Almost.Algebra.relativeModule S φ`. The element `e : A → P ⊗_A P` is specified by
`μ(e) = 1` and `(b ⊗ 1)e = (1 ⊗ b)e`, expressed as equality of morphisms
`P → P ⊗_A P`; these conditions characterize the diagonal idempotent.
For `P` almost finite projective, the conclusion is
`e ≫ (1 ⊗ traceFormDual) = ζ_B`, with `ζ_B` explicitly the internal-Hom representative of
`1_P` followed by `dualTensorHomIso.inv` ([GR] Remark 4.1.17, p. 76).
`Almost.Algebra.isFiniteEtale_iff_flat_perfectTraceForm` assumes almost finite presentation of
`P` and states finite étaleness iff there is a flatness proof for which the trace pairing is
perfect. Flatness and presentation supply the almost finite projectivity needed to define that
trace ([GR] Proposition 2.4.18, p. 26, and Theorem 4.1.14, p. 75).

**Checks.**

- `Almost.Algebra.traceForm_test_product` (computation) — `B = A × A` over `A`: the diagonal
  idempotent is `e = (1,0) ⊗ (1,0) + (0,1) ⊗ (0,1)`, `μ(e) = 1`, and `t_{B/A}` has matrix `[[1, 0], [0, 1]]`, so both sides of the criterion hold.
- `Almost.Algebra.traceForm_test_dual_numbers` (non-example) — for a nonzero almost algebra `A`,
  `B = A[ε]/(ε²)` is finite free,
  `t_{B/A}` has matrix `[[2, 0], [0, 0]]` and is not perfect; correspondingly `I_{B/A}/I_{B/A}² = Ω_{B/A} = (B/(2ε))·dε ≠ 0`, so no idempotent `e` with `μ(e) = 1` and `I_{B/A}·e = 0` exists: both
  sides of the criterion fail. Over `ℚ`, `Ω` has dimension `1`; over `𝔽₂` it has dimension `2`.
- `Almost.Algebra.traceForm_test_product_diagonal` — in the split product algebra, the displayed
  tensor is idempotent, its multiplication is one, and it annihilates the diagonal ideal.
- `Almost.Algebra.traceForm_test_dual_numbers_quotient` — the differential module is
  `B/(2ε)`, with `dε` sent to one; the formula includes characteristic two.
- `Almost.Algebra.traceForm_test_dual_numbers_differentials` — after almostification,
  the module of differentials of `R[ε]/(ε²)` remains nonzero whenever `R^a` is nonzero.
- `Almost.Algebra.algebraTrace_test_zero` (degenerate) — `B = 0`: `t_{B/A}` is perfect on `0 ⊗ 0` and
  `A → 0` is étale, as `0 = A/(1)` is an idempotent quotient.

Unramified morphisms are characterised by a diagonal idempotent
(*unramified-idempotent-characterisation*, `Almost.Algebra.isUnramified_iff_exists_idempotent`):
for a basic setup `S` and `φ : A → B`, `φ` is unramified (`B` almost projective over
`B ⊗_A B`) iff there is an almost element
`e_{B/A} ∈ (B ⊗_A B)_*` with `e² = e`, `μ_{B/A}(e) = 1` and `x·e = 0` for every `x ∈ (I_{B/A})_*`;
`e` is unique, and `B ⊗_A B ≅ I_{B/A} × B` as `B ⊗_A B`-algebras (`I_{B/A}` with unit `1 − e`). More
generally, for an ideal `I` of an almost algebra `A`, `A/I` is almost projective over `A` iff `I` is
generated by an idempotent of `A_*`, which is then unique ([GR] §3.1, Proposition 3.1.4, p. 40; [GR]
§3.1, Remark 3.1.8, p. 40; [Sch12] §4, Definition 4.12(i), p. 20; [Bhatt] §4.3, Definition 4.3.1(2),
p. 26). In the classical setup with `B` finite étale over `A`, `ker μ` is a finitely generated
idempotent ideal generated by `1 − e` (Mathlib `Ideal.isIdempotentElem_iff_of_fg`, `IsIdempotentElem`).

`Almost.Algebra.diagonalIdempotent_unique` equates any two almost elements satisfying the
three displayed conditions. `Almost.Algebra.diagonalIdempotent_split` gives the product
isomorphism on almost elements: with `D = B ⊗_A B`, it is the additive and multiplicative
bijection `D_* → (diagonalIdeal S φ) × B_*`, sending `x` to `((1 − e)x, μ(x))`.
Its first factor has inherited multiplication and unit `1 − e`; its `D_*`-action is multiplication,
and the action on `B_*` is through `μ`. Thus almostification gives the asserted isomorphism of
`D`-algebras ([GR] Proposition 3.1.4, Remark 3.1.8 and Corollary 3.1.9, p. 40).

### P0.16 inverting an element of `m`

**Inverting an element of `m`: almost modules become honest modules over `V[1/ϖ]`.**
(*almost-inversion-and-completion*) Let `S = (V, m)` be a basic setup and `ϖ ∈ m` (so `ϖ` kills
almost zero modules; `ϖ` need not be a nonzerodivisor). The localisation `M ↦ M[1/ϖ] = V[1/ϖ] ⊗_V M`
is exact and kills almost zero modules (`ϖM = 0`), so it factors, uniquely up to unique
isomorphism, through an exact functor `Almost.invert ϖ : Almost.Module S → ModuleCat V[1/ϖ]` with
`invert(M^a) ≅ M[1/ϖ]` and `invert(M) ≅ M_*[1/ϖ] ≅ M_![1/ϖ]` ([GR] §5.2, proof of Corollary 5.2.15,
p. 110; [Bhatt] §4.1, Exercise 4.1.9, p. 22). Consequences: almost isomorphisms become isomorphisms
and almost exact complexes become exact after inverting `ϖ`; the setup `S.baseChange V[1/ϖ]` is
classical (`m·V[1/ϖ] = V[1/ϖ]`), so `Almost.Module` over it is `ModuleCat V[1/ϖ]`; `invert` is
symmetric monoidal and sends almost algebras to `V[1/ϖ]`-algebras ([Berkeley] Lecture 8, proof of
Proposition 8.2.8, p. 60: almost vanishing of `H^i(X, O⁺)` gives vanishing of
`H^i(X, O) = H^i(X, O⁺)[1/ϖ]`). The `ϖ`-adic completion side of this passage — completeness of `M`
versus `M_*`, and exactness of complete torsion-free complexes that are almost exact modulo `ϖ`,
with the `lim¹` term kept — is *almost-completeness-via-almost-elements* and
*almost-exactness-passes-to-completion* (P0.28). Define `Almost.invert` (the exact functor
`Almost.Module S ⥤ ModuleCat (Localization.Away ϖ)` for `ϖ ∈ S.ideal`); prove
`Almost.toAlmostCompInvertIso` (`toAlmost S ⋙ invert ϖ ≅` localisation at the powers of `ϖ`),
that `invert ϖ` preserves finite limits, `Almost.invertElementsIso`
(`(invert ϖ).obj M ≅ LocalizedModule (powers ϖ) M.elements`), that localising at the powers of `ϖ`
turns almost isomorphisms into isomorphisms and almost zero modules into zero modules,
`Almost.BasicSetup.baseChange_away_eq_classical`
(`S.baseChange (Localization.Away ϖ) = classical (Localization.Away ϖ)`), `Almost.invertMonoidal`
(with `Almost.Algebra.invert` on almost algebras) and that a complex of `V`-modules with almost
zero cohomology becomes exact after inverting `ϖ`. *Needs:* P0.5 for inversion;
P0.8's `Almost.Algebra`, `Almost.Algebra.ModuleCat` and algebra adjunction for
*change-of-basic-setup* (i); Mathlib
`SerreClassLocalization.essImage_whiskeringLeft`, `LocalizedModule`, `Localization.Away`.

On algebras, `Almost.Algebra.invert` sends `A` to `A_*[1/ε]`; on canonical numerators its map is induced by `elements.map`.

**Checks.**

- `Almost.Algebra.invert_zero`: inverting zero gives the zero algebra.
- `Almost.Algebra.invert_unit`: inverting a unit preserves the algebra of almost elements.
- `Almost.Algebra.invert_torsion`: an algebra killed by a power of the inverted scalar becomes zero.

Two supporting results. Base change of a basic setup along a ring map `V → W`
(*basic-setup-base-change*): for a basic setup `S = (V, m)` and any ring homomorphism `V → W`,
`S_W := (W, mW)` is a basic setup: `(mW)·(mW) = mW`, and the natural map `W ⊗_V m̃ → (mW) ⊗_W (mW)`
is an isomorphism, so `(mW)~` is `W`-flat; flatness of `m` alone does not pass to `mW`, which is why
the setup requires `m̃`, not `m`, to be flat ([GR] §2.1, Remark 2.1.4(ii), p. 8). For `V → V[1/ϖ]`
with `ϖ ∈ m`, `S_W` is the classical setup of `V[1/ϖ]`; for `V → V/I`, `S_{V/I} = (V/I, (m + I)/I)`,
and the base change of a root setup is the root setup of the images of the roots (Mathlib
`Module.Flat.baseChange`, `Ideal.map`). Change of basic setup (*change-of-basic-setup*): with
`S_W = (W, mW)` as above, (i) restriction of scalars gives a functor from `Almost.Module S_W` to
modules over the almost `V`-algebra `W^a` in `Almost.Module S`, and it is an equivalence with
quasi-inverse `M ↦ (M_*)^a`; it commutes with `(−)_*` and `(−)_!`, and the analogous statement holds
for almost algebras; (ii) if `V → W` is an almost isomorphism of `V`-modules (kernel and cokernel
killed by `m`), then `W^a ≅ V^a` as almost `V`-algebras and restriction of scalars
`Almost.Module S_W → Almost.Module S` is an equivalence ([GR] §2.2, Remark 2.2.12, p. 14; [Berkeley]
Lecture 7, §7.4, after Definition 7.4.3, p. 53). The case `R⁺ ⊆ R°` (`R°/R⁺` almost zero, [ECD]
Example 3.22(ii)) is the comparison `R°a-mod ≃ R⁺a-mod` stated in [ECD] after Definition 3.23,
proved in layer P1 from this lemma; for `W = V/I`, (i) identifies almost `V/I`-modules with almost
`V`-modules killed by `I` (Mathlib `ModuleCat.restrictScalars`).

**Checks.**

- Computation: for `S = ofValuationRing K°` and `ϖ ∈ K°°` nonzero:
  `(invert ϖ).obj ((toAlmost S).obj (K° ⧸ ϖ)) ≅ 0` and `(invert ϖ).obj ((toAlmost S).obj K°) ≅ K`.
- Degenerate: `0 ∈ S.ideal` always, and `invert 0` lands in
  modules over the zero ring `V[1/0] = 0`: every object goes to `0`, every statement "after
  inverting `0`" is vacuous; the localisation still detects the part of a module on which `ϖ` is invertible,
  even when the module also has `ϖ`-torsion.
- Degenerate: if `ϖ` is a unit of `V` (then `S.ideal = ⊤` as
  `ϖ ∈ S.ideal`), `invert ϖ ≅ Almost.Module.elements` composed with the identification
  `V[1/ϖ] = V`.
- Non-example: `invert ϖ` is not faithful: for
  `S = ofValuationRing K°` and `0 ≠ ϖ ∈ K°°`, the nonzero almost module `(K°/ϖ)^a` is sent to `0` ([Bhatt] Exercise
  4.1.9); almost statements are strictly finer than statements after inverting `ϖ`.
- Compatibility: `toAlmost S ⋙ invert ϖ ≅` the Mathlib localisation
  functor `M ↦ LocalizedModule (Submonoid.powers ϖ) M`, as functors
  `ModuleCat V ⥤ ModuleCat (Localization.Away ϖ)`.

- `invert_functor_test_zero` — the actual inversion functor at zero sends every almost module to zero.
- `invert_functor_test_quotient` — inversion at `ε ∈ m` kills `(V/(ε))^a`.
- `invert_functor_test_ring` — it sends `V^a` to the free rank-one module over `V[1/ε]`.

Lean: `BasicSetup.baseChange` with `isIdempotentElem_map` and `flat_tilde_map`
(*basic-setup-base-change*), and `BasicSetup.baseChange_away_eq_classical`. The functor
`Almost.invert` and *change-of-basic-setup* are specified in this README only.

### P0.17 comparison with Mathlib's finite étale algebras

**Comparison of finite étale almost algebras with Mathlib's finite étale algebras.**
(*finite-etale-localizes*) Let `S = (V, m)` be a basic setup and `R → R'` a map of commutative
`V`-algebras. (i) If `R'` is finite étale over `R` in Mathlib's sense (`Algebra.Etale R R'` and
`Module.Finite R R'`), then `R^a → R'^a` is finite étale in the almost sense ([Bhatt] §4.3,
Construction 4.3.3, p. 26: a finite étale algebra is a direct summand of a free module via the
diagonal idempotent and the trace). (ii) In the classical setup (`m = V`) the converse holds, so
`A_fet` for `A = R^a` is equivalent to Mathlib's category of finite étale `R`-algebras. (iii) For
`ϖ ∈ m`, if `R'^a` is finite étale over `R^a` then `R'[1/ϖ]` is finite étale over `R[1/ϖ]` in
Mathlib's sense (inverting `ϖ` turns almost statements into honest ones, P0.16). (iv) Gabber–Ramero's
étale morphisms in the classical setup are exactly the absolutely flat (weakly étale) maps whose
diagonal is almost projective; they need not be of finite presentation, and a classical étale map in
EGA's sense is étale in their sense ([GR] §3.4, (3.4.44), p. 64, which states the comparison in the
classical limit `m = V`; [GR] §3.1, Remark 3.1.3(i), p. 39; for a general basic setup, (i) follows because a finite étale `R'` is finite projective
over `R` and carries the diagonal idempotent, so `R'^a` is almost finite projective over `R^a` by
[GR] (2.4.8) and unramified by *unramified-idempotent-characterisation*; (iii) because finite étale
morphisms are stable under the base change `R^a → R[1/ϖ]^a` (*etale-morphisms-stability*) and
`S.baseChange V[1/ϖ]` is classical, so by *change-of-basic-setup* the base-changed morphism is the
honest `R[1/ϖ] → R'[1/ϖ]` in the classical setup, where (ii) applies). For `K` a perfectoid field
and `L/K` finite separable,
`L°a` is finite étale over `K°a` while `L°` need not be finite étale over `K°` in Mathlib's sense (it
is not finitely presented in general; for instance the odd-`p` `L°` of P0.10 need not be finitely
presented over `K°`), so (i) is strictly weaker than the almost statement; for
`R → R' = R × R` the diagonal idempotent is `(1,0) ⊗ (1,0) + (0,1) ⊗ (0,1)`. *Needs:* P0.14; P0.11;
P0.16; Mathlib `Algebra.Etale`, `Module.Finite`, `Ideal.isIdempotentElem_iff_of_fg`,
`KaehlerDifferential`.

### P0.18 almost Kähler differentials

**Almost Kähler differentials `Ω_{B/A}` and almost derivations.** (*almost-differentials*) Let `S` be
a basic setup and `A → B` a morphism of almost algebras, `I_{B/A} = ker(μ_{B/A} : B ⊗_A B → B)`.
The module of almost differentials is the `B`-module `Ω_{B/A} := I_{B/A}/I_{B/A}²`, with the
`A`-derivation `δ : B → Ω_{B/A}`, `b ↦ 1 ⊗ b − b ⊗ 1` on almost elements ([GR] §2.5, Definition
2.5.22(ii), p. 33). For a `B`-module `M`, an `A`-derivation `B → M` is an `A`-linear `∂` with
`∂(b₁b₂) = b₁∂(b₂) + b₂∂(b₁)` on almost elements; they form a `V`-module `Der_A(B, M)`, and
`f ↦ f ∘ δ` is an isomorphism `Hom_B(Ω_{B/A}, M) ≅ Der_A(B, M)` ([GR] §2.5, (2.5.23), p. 33).
Properties: `Ω_{C⊗_A B/C} ≅ C ⊗_A Ω_{B/A}`; for `A → B → C` an exact sequence
`C ⊗_B Ω_{B/A} → Ω_{C/A} → Ω_{C/B} → 0`; for `C = B/I` an exact sequence
`I/I² → C ⊗_B Ω_{B/A} → Ω_{C/A} → 0` ([GR] §2.5, Proposition 2.5.25(i)-(ii), p. 33); `Ω` commutes
with colimits; `(Ω_{B/A})_! ≅ Ω_{B!!/A!!}` ([GR] §2.5, Lemma 2.5.28, p. 34); for `V`-algebras
`R → R'`, `Ω_{R'^a/R^a} ≅ (Ω_{R'/R})^a` (Mathlib `KaehlerDifferential`). Define
`Almost.Algebra.kaehlerDifferential` (`Ω_{B/A}` as a `B`-module),
`Almost.Algebra.kaehlerDifferentialD` (the universal derivation `δ`) and `Almost.Algebra.Derivation`
(`A`-derivations `B ⟶ M`, a `V`-module); prove `Almost.Algebra.kaehlerDifferential.liftEquiv`
(`(Ω_{B/A} ⟶ M) ≃ₗ[V] Derivation A B M`), `Almost.Algebra.kaehlerDifferential.baseChangeIso`,
`Almost.Algebra.kaehlerDifferential.exact_seq` (the two exact sequences),
`Almost.Algebra.kaehlerDifferential.shriekIso` (`(Ω_{B/A})_! ≅ Ω_{B!!/A!!}`),
`Almost.Algebra.kaehlerDifferential_toAlmost` (`Ω_{R'^a/R^a} ≅ (KaehlerDifferential R R')^a`) and
`Almost.Algebra.kaehlerDifferential_eq_zero_of_isUnramified` (`φ` unramified `⇒ Ω_{B/A} = 0`, since
the diagonal idempotent `e` satisfies `1 − e ∈ I_{B/A}` and `x = x(1 − e)` for every almost
element `x ∈ I_{B/A}`, giving `I_{B/A} = I_{B/A}²`; [GR] Proposition 3.1.4, p. 40, and
Definition 2.5.22(ii), p. 33). *Needs:* P0.8; Mathlib `KaehlerDifferential`, `Derivation`.

**Checks.**

- `Almost.Algebra.kaehler_test_polynomial` (computation) — for `A` an almost algebra and
  `B = (A_*[X])^a`: `Ω_{B/A} ≅ B`, generated by `δX`.
- `Almost.Algebra.kaehler_almost_test_identity`, `Almost.Algebra.kaehler_almost_test_quotient` (degenerate) — `Ω_{A/A} = 0` and, for an ideal `I`,
  `Ω_{(A/I)/A} = 0`.
- `Almost.Algebra.kaehler_test_mathlib` (compatibility) — for `V`-algebras `R → R'`:
  `Ω_{R'^a/R^a} ≅ (toAlmost _).obj (KaehlerDifferential R R')` as `R'^a`-modules, compatible with the
  derivations.
- `Almost.Algebra.kaehler_test_not_elements` (comparison of representatives) — the
  unital differential construction uses `B_*` or `B_!!`; `B_!` is a non-unital algebra.
  The comparison `kaehlerDifferential.shriekIso` identifies `Ω_{B!!/A!!}` with
  `(Ω_{B/A})_!` ([GR] Lemma 2.5.28, p. 34), retaining the scalar extension from `V`.

- `differentials_test_sign` — over `ℤ[X]`, the convention `δb = 1 ⊗ b − b ⊗ 1`
  agrees with Mathlib: under its polynomial differential equivalence, `δ(X² + X) = 2X + 1`.
  Reversing the convention would give `−2X − 1`.

`Almost.Algebra.kaehlerDifferential` is an almost module over `B_*`, for the base-changed setup.
`kaehlerDifferentialD` is the ordinary universal derivation followed by the unit map to almost elements.

- `kaehler_almost_test_identity` — both the differential module and derivation of the identity are zero.
- `kaehler_almost_test_quotient` — both vanish for `A → A/I`, including the zero and unit ideals.
- `kaehler_almost_test_polynomial` — for classical `ℚ → ℚ[X]`, the module and derivation are nonzero.
- `kaehler_almost_test_leibniz` — on almost elements, `d(xy) = x·dy + y·dx`.

- `differentials_test_char_two` — ordinary differentials of dual numbers have dimension one over `ℚ`
  and dimension two in characteristic two.

**Checks for `Almost.Algebra.Derivation`.** `Derivation_identity` gives only the zero derivation for an identity map; `Derivation_zero` does so for zero coefficients; `Derivation_polynomial` gives a nonzero derivation for the classical rational polynomial algebra, detected by its value on `X`. These use [GR] Proposition 2.5.25.

The ordinary module `Almost.Module.homModule` has scalar multiplication
`(a • f)(x) = a • f(x)`. Its Checks are `homModule_zero` (zero source),
`homModule_classical_one` (endomorphisms of the rank-one classical module), and
`homModule_classical_two` (the two-by-two matrix module).

### P0.19 the cotangent complex of a ring map, consumed

**The cotangent complex of a ring map and its deformation theory, consumed.**
(*cotangent-complex-of-ring-map*) Ownership: the ordinary cotangent complex (simplicial
resolutions, [Stacks] Tag 08PL) and its square-zero obstruction class belong to
DerivedModuleAlgebra §§0.4 and 0.25, consistently with SchemeAndStackFoundations §4.6; Mathlib's
`Algebra.Extension.cotangentComplex` and `Algebra.H1Cotangent` cover only the two lowest degrees.
This roadmap consumes that carrier and owns only the almost construction of P0.20. What follows is
the contract that P0.20–P0.24 need of the consumed carrier, with its Checks; the names
`CotangentComplex` and its API below stand for the corresponding declarations of that roadmap. For a map of commutative rings `R → S`, the cotangent complex
`L_{S/R} ∈ D(S)` is the complex of Kähler differentials of a simplicial resolution of `S` by
polynomial `R`-algebras, tensored termwise with `S` along the augmentation
([Stacks] Tag [08PL](https://stacks.math.columbia.edu/tag/08PL), Definitions 92.3.1–92.3.2), together
with: (a) `H_0(L_{S/R}) ≅ Ω_{S/R}`, compatibly with Mathlib's `KaehlerDifferential`, and `H_1`
compatible with the truncation `Algebra.Extension.Cotangent` and with `Algebra.H1Cotangent`
(Mathlib has the degree `≤ 1` truncation only, no simplicial `L_{S/R}`); (b) the transitivity
triangle `S ⊗_T^L L_{T/R} → L_{S/R} → L_{S/T}` for `R → T → S` ([Stacks] Tag 08QX), whose
`H_1`/`H_0` part is Mathlib's Jacobi–Zariski sequence `Algebra.Generators.Cotangent.exact`; (c) base change
`L_{S'/R'} ≃ S' ⊗_S^L L_{S/R}` for a Tor-independent square ([Stacks] Tag 08QQ); (d) commutation
with filtered colimits of rings ([Stacks] Tag 08S9); (e) `Ext^0_S(L_{S/R}, M) ≅ Der_R(S, M)` and
`Ext^1_S(L_{S/R}, M) ≅ Exal_R(S, M)`, the group of square-zero `R`-extensions of `S` by `M`
([Stacks] Tag [08S3](https://stacks.math.columbia.edu/tag/08S3), Lemma 91.2.3,
and Tag [08RB](https://stacks.math.columbia.edu/tag/08RB) for the truncation comparison); (f) for a flat `R`-algebra `S` and a square-zero extension
`R' → R` with kernel `J`, the obstruction to a flat deformation of `S` over `R'` lies in
`Ext^2_S(L_{S/R}, S ⊗_R J)`, deformations form a torsor under `Ext^1` and automorphisms are `Ext^0`
([Stacks] Tag 08SP), and for a square-zero extension `T′ → T` of `R`-algebras with kernel `J`,
the obstruction to lifting an `R`-map `S → T` to `T′` lies in `Ext^1_S(L_{S/R}, J)`,
where `J` is an `S`-module through `S → T`, with lifts a torsor under `Ext^0`
([Stacks] Tag [08S3](https://stacks.math.columbia.edu/tag/08S3), Lemma 91.2.1); (g) for a map of simplicial rings `R_• → S_•`, the diagonal complex `L^Δ` with the
spectral sequence `H_j(L_{S_i/R_i}) ⇒ H_{i+j}(L^Δ)`, and `L_{S/R} ≃ L^Δ` for simplicial resolutions
([Illusie] Chapter II, 1.2.6.2). For the free simplicial resolution `S_• → S`
and `R_• = S_• ×_S R` used to compare cotangent complexes of almost-isomorphic algebras,
[GR] Proposition 8.1.7, proof, printed pp. 220–221, gives precisely
`E¹_{i,j} = H_j(L_{S_i/R_i}) ⇒ H_{i+j}(L^Δ)` in homological degrees.
The consumed carrier provides `CotangentComplex` with `CotangentComplex.homologyZeroIso`
(`H_0(L_{S/R}) ≅ Ω_{S/R}`), `CotangentComplex.transitivityTriangle`,
`CotangentComplex.baseChange` under the Tor-independence in (c), `CotangentComplex.filteredColimit`
for (d), `CotangentComplex.extZero` and `CotangentComplex.extOne` for (e),
`CotangentComplex.flatDeformation` and `CotangentComplex.liftMorphism` with the hypotheses in (f),
and `CotangentComplex.diagonalComparison` for (g). Grading is homological: `H_i(L)` is `H^{−i}(L)` in
the cohomological indexing of [Stacks]; `M[n]` places `M` in homological degree `n`, so `M[1]` places
`M` in homological degree one. The obstruction groups of P0.21–P0.23 are `Ext` groups of the full
complex, never of a two-term truncation. Tensor products of complexes in the derived category
mean derived tensor products. *Needs:* DerivedModuleAlgebra §§0.1–0.2, 0.4–0.11, 0.13 and 0.25;
Mathlib
`KaehlerDifferential`, `Algebra.Extension.Cotangent`, `Algebra.H1Cotangent`,
`Algebra.Generators.Cotangent.exact`, `CategoryTheory.SimplicialObject`.

**Checks.**

- `cotangentComplex_polynomial` (computation) — for `S = R[X_i]` the complex is `Ω_{S/R}` in degree
  `0` (free on the `dX_i`).
- `cotangentComplex_perfect_charP` (computation) — for a perfect `𝔽_p`-algebra `S`,
  `L_{S/𝔽_p} ≃ 0` (Frobenius is an isomorphism, so the relative Frobenius criterion applies).
- `cotangentComplex_not_concentrated_in_degree_zero` (non-example) — for `S = R/(f)` with `f` a
  nonzerodivisor and `(f) ≠ R`, `L_{S/R} ≃ (f)/(f²)[1] ≠ 0` ([Stacks] Tag 08SJ); for instance
  `ℤ → ℤ/2` has `L ≃ ℤ/2[1]`: a definition returning only `Ω_{S/R} = 0` is wrong.
- `cotangentComplex_identity_and_quotient` (degenerate) — `L_{R/R} ≃ 0` (the resolution is
  constant); for a surjection `R → R/I`, `H_0 = Ω_{(R/I)/R} = 0` and `H_1 = I/I²` (Mathlib
  `Algebra.H1Cotangent`), so `H_1(L_{(R/I)/R}) = 0` iff `I = I²`; `L_{(R/I)/R} ≃ 0` needs in
  addition `H_i = 0` for `i ≥ 2` (automatic when `I` is generated by an idempotent); for `I = R`
  the target is the zero ring and `L_{0/R} = 0`. For an arbitrary ideal `I`, only `H_0 = 0` and
  `H_1 = I/I²` hold in general; the formula `(f)/(f²)[1]` needs `f` to be a nonzerodivisor.
- `cotangentComplex_dual_numbers` (negative control for the nonzerodivisor hypothesis) — for
  `R = k[ε]/(ε²)` and the zero-divisor `f = ε`, the hypersurface presentation `R = k[x]/(x²)` gives
  `k ⊗ᴸ_R L_{R/k} ≃ [k —0→ k] ≃ k[0] ⊕ k[1]`, and the transitivity triangle of `k → R → k` gives
  `L_{k/R} ≃ k[1] ⊕ k[2]`, whose `H_2 = k` is missing from `(ε)/(ε²)[1] = k[1]`.

- `cotangent_test_unit_ideal` — for `I = V`, both `H₁(L_{(V/I)/V})` and `Ω_{(V/I)/V}` are zero,
  which is the negative control for the proper-ideal hypothesis above.

- `cotangent_model_test_identity` — the full derived object for `R → R` and every homology module of its model are zero.
- `cotangent_model_test_polynomial` — for `R → R[X]`, positive homology vanishes and `H₀ ≅ R[X]`.
- `cotangent_model_test_regular_quotient` — for `ℤ → ℤ/2`, `H₀ = 0` but `H₁ ≅ ℤ/2 ≠ 0`, excluding the differential-only construction.
- `cotangent_model_test_zero_target` — the full derived object for `R → R/(1)` is zero.
- `cotangent_model_test_two_faces` — the degree-one differential on `x + y` is
  `d₀x − d₁x + d₀y − d₁y`; the degreewise faces have no built-in sign.
- `cotangent_model_test_shift` — homological degrees `0, 1, 2` are placed in cohomological degrees `0, −1, −2`.
- For the standard resolution of `ℤ → ℤ/2`, `P₀ = ℤ[ℤ/2]` is additively torsion-free.
  Every additive map `ℤ/2 → P₀` is zero, so no inverse in simplicial abelian groups can
  induce the identity on `H₀ = ℤ/2`. The set-level contraction supplies no additive contraction.

`CotangentComplex.PolynomialResolution` is the carrier
`Algebra.CotangentComplex.PolynomialResolution` of DerivedModuleAlgebra §0.1: an
augmented simplicial commutative `R`-algebra with polynomial presentations in each degree
and augmentation to `T` a trivial Kan fibration of underlying simplicial sets. The supplied
standard resolution has an extra degeneracy after forgetting to sets. Its augmentation is
therefore a homotopy equivalence of simplicial sets, induces a quasi-isomorphism on the
associated complexes of abelian groups, and is degreewise surjective; the last two properties
imply that it is a trivial Kan fibration of underlying simplicial sets
([Stacks, Lemma 92.4.2, first proof](https://stacks.math.columbia.edu/tag/08PT),
[Lemma 14.31.9](https://stacks.math.columbia.edu/tag/08P2) and
[Lemma 14.31.8](https://stacks.math.columbia.edu/tag/08P1)). The supplier's
`Algebra.CotangentComplex.PolynomialResolution.toQuasiIso` supplies the quasi-isomorphism,
and `Algebra.CotangentComplex.PolynomialResolution.comparison` uses the trivial-Kan-fibration
criterion of [Stacks, Lemma 92.4.1](https://stacks.math.columbia.edu/tag/08PS).
Its `differentials` have terms
`T ⊗_{P_n} Ω_{P_n/R}`; `termIso` and `face_tmul` pin this identification and its face maps.
`complex` is Mathlib's alternating face complex. `model` uses a chosen free resolution,
and `object` extends it to nonpositive cohomological degrees before applying `DerivedCategory.Q`.
`resolutionIso` states independence of the resolution; `homologyZeroIso` and `homologyOneIso`
compare with Mathlib's differentials and first cotangent homology.

- `cotangent_test_quotient` — for `R → R/I`, Mathlib's `H1Cotangent` is zero exactly when `I² = I`;
  this is a test of first homology, not vanishing of the full complex.

DerivedModuleAlgebra §0.5 supplies `CotangentComplex.SquareZeroExtension`, using Mathlib's algebra extension,
with an injective additive kernel map whose image is the kernel, whose scalar action
is induced by the quotient, and whose products vanish. `extensionSetoid` identifies
extensions by algebra isomorphisms preserving the quotient and the kernel map;
`Exal` is its quotient. `CotangentComplex.extensionClass` assigns the specified
extension its class, and `CotangentComplex.extOne` asserts that this particular
map is bijective and sends precisely the split extensions to zero (DerivedModuleAlgebra §0.6).
`CotangentComplex.extensionClass_mk` identifies it with
`CotangentComplex.morphismObstruction` for the identity map. For an arbitrary
`R`-algebra map `f : T → B`, `morphismObstruction` is the class of the pullback
of the specified extension of `B`; `CotangentComplex.liftMorphism` tests the
vanishing of this class. Its normalization is given by
`CotangentComplex.morphismObstruction_cocycle`: lift the degree-zero polynomial
resolution to the extension, subtract its two lifted faces in the order
`d₀ − d₁`, and use the given kernel identification. The resulting degree-one
cocycle represents the class for every such lift. This is the simplicial form
of [Stacks, Remark 91.2.4](https://stacks.math.columbia.edu/tag/08S3), with the
full-complex comparison of [Stacks, Lemma 92.11.3](https://stacks.math.columbia.edu/tag/08RB).

**Checks.** For each of `morphismObstruction` and `extensionClass`, `_zero`
uses the zero coefficient module, `_split` uses an actual algebra lift or
section, and `_nonsplit` uses `ℤ/9 → ℤ/3` and gives a nonzero class.

**Checks.** For each of `SquareZeroExtension`, `extensionSetoid` and `Exal`, the
`_zero` Check has zero kernel and a bijective quotient; `_dual` uses the nonzero
square-zero element of rational dual numbers, whose extension class is zero;
`_nonsplit` uses `ℤ/4 → ℤ/2`, which has no ring section and a nonzero class.
These are the split and nonsplit cases of [Stacks] Tag 08S3.

DerivedModuleAlgebra §§0.4, 0.9 and 0.11 supply the following functorial models.
The functor `CotangentComplex.arrowFunctor` retains every homological degree and the
maps induced by commutative squares of rings. `arrowResolution` is the standard
polynomial resolution, `resolutionRings` forgets its base-algebra structure, and
`resolutionRingMap` commutes with the augmentation and base scalars.
`arrowModelMap_tmul` fixes its map on generators: `t ⊗ dx` goes to
`b(t) ⊗ d(a(x))`. Thus `filteredColimit` compares the canonical diagram of
complexes, with both source and target rings varying, as in [Stacks] Tag 08S9.
`diagonalModel` is the total complex of the alternating chains of this simplicial
complex. `diagonalComparison` has differential bidegree `(-r,r-1)` and an
exhaustive finite filtration on each homology group; its resolution comparison
uses extra degeneracies on underlying sets ([Illusie] II 1.2.6.2).

**Checks.** `resolutionRingMap_identity`, `resolutionRingMap_composition`, and
`resolutionRingMap_zero_target` fix functoriality and the zero-target augmentation.
For `arrowResolution`, `resolutionRings`, `arrowModel`, `arrowModelMap`, and
`arrowFunctor`, the `_identity`, `_polynomial`, and `_regular_quotient` Checks
respectively give zero homology, zero positive homology, and nonzero first
homology for `ℤ → ℤ/2`. `diagonalModel_constant`, `diagonalModel_identity`, and
`diagonalModel_regular_quotient` distinguish totalization from taking only
ordinary differentials.

The chain maps `CotangentComplex.transitivityFirst` and
`CotangentComplex.transitivitySecond` of DerivedModuleAlgebra §0.8 are induced by the squares
`(A → B) → (A → C)` and `(A → C) → (B → C)`, respectively.
`CotangentComplex.transitivityFirst_tmul` sends `c ⊗ x` to `c` times the
image of `x` under `arrowModelMap`; `CotangentComplex.transitivitySecond_apply`
is the same functorial map with the target ring fixed. Their images under
`DerivedCategory.Q` are the first two arrows in `transitivityTriangle`;
only the connecting arrow is existentially quantified. This fixes the maps
in the transitivity sequence ([Stacks] Tag 08QX; [GR] Theorem 2.5.32, p. 35).

**Checks.** `transitivityFirst_identity` is invertible when the second ring map
is the identity; `transitivityFirst_zero_source` vanishes in the derived
category when the first ring map is the identity;
`transitivityFirst_polynomial` is nonzero for `ℚ → ℚ[X] → ℚ[X]`.
`transitivitySecond_identity` is the identity when `A = B`;
`transitivitySecond_zero_target` vanishes when `B = C`;
`transitivitySecond_regular_quotient` is nonzero on the degree-one cotangent
homology of `ℤ → ℤ → ℤ/3`.

### P0.20 the almost cotangent complex

**The almost cotangent complex `L_{B/A}`.** (*almost-cotangent-complex*) Let `S` be a basic setup
(`m̃` flat is used in [GR] Proposition 2.5.34 and Theorem 2.5.35) and `A → B` a morphism of almost
algebras. Put `Ã := V^a × A` and `B̃ := V^a × B` (exact almost algebras). The almost cotangent
complex is `L_{B/A} := B_!! ⊗_{B̃_!!} L_{B̃_!!/Ã_!!}`, an object of the derived category of
`B_!!`-modules, where `L_{−/−}` is the cotangent complex of the ring map `Ã_!! → B̃_!!` (simplicial
polynomial resolution, the consumed carrier of P0.19, whose `Ext` groups are those of the full
complex); this almost construction is the one this roadmap owns. Its almostification `L^a_{B/A} ∈ D(B-Mod)` is the object used in
deformation theory ([GR] §2.5, Definition 2.5.20, p. 32; [Sch12] §5, before Remark 5.15, p. 25).
Properties: `H_0(L^a_{B/A}) ≅ Ω_{B/A}`; `Ext^i_{B!!}(L_{B/A}, M_!) ≅ Ext^i_B(L^a_{B/A}, M)` ([GR]
(3.2.8)); `Ext⁰ ≅ Der_A(B, M)` ([GR] 2.5.29) and `Ext¹ ≅ Exal_A(B, M)` (P0.21); the transitivity
triangle *cotangent-transitivity* (P0.22); Tor-independent base change
*cotangent-flat-base-change*; `L_{colim B_λ/colim A_λ} ≅ colim L_{B_λ/A_λ}` for filtered systems
([GR] §2.5, Proposition 2.5.33, p. 35); vanishing for weakly étale maps
*cotangent-vanishing-weakly-etale* (P0.24); and for `V`-algebras `R → S`,
`L^a_{S^a/R^a} ≅ (L_{S/R})^a` by *almost-cotangent-comparison-classical*, so the almost cotangent
complex depends only on `R^a → S^a` ([Sch12] §5). Define `Almost.cotangentComplex` (`L_{B/A}` in
`DerivedCategory (ModuleCat B_!!)`) and `Almost.almostCotangentComplex` (`L^a_{B/A}` in
`DerivedCategory (B-Mod)`); prove the identification `H_0(L^a_{B/A}) ≅ Ω_{B/A}`, the comparison
`Ext^i_{B!!}(L_{B/A}, M_!) ≅ Ext^i_B(L^a_{B/A}, M)`, the identifications `Ext⁰ ≅ Der_A(B, M)` and
`Ext¹ ≅ Exal_A(B, M)`, the transitivity statement (the distinguished triangle
`C ⊗ L_{B/A} → L_{C/A} → L_{C/B} →`), Tor-independent base change, the colimit statement, the
comparison `L^a_{S^a/R^a} ≅ (L_{S/R})^a` with the almostified classical complex, and vanishing for
weakly étale maps (weakly étale, more generally Tor-trivial diagonal, `⇒ L_{B/A} ≃ 0`). *Needs:* P0.9; P0.18; P0.8; P0.19; Mathlib
`Algebra.Extension.cotangentComplex`, `Algebra.H1Cotangent`, `DerivedCategory`.

The named additive map `Almost.cotangentDerivationComparison` sends a degree-zero
class to its derivation. The identification `Ext⁰ ≅ Der_A(B, M)` above asserts bijectivity
of this map. For every basic setup, algebra map and coefficient module, its evaluation
formula applies `H⁰` to the class, transports along the identification
`H_0(L^a_{B/A}) ≅ Ω_{B/A}`, and evaluates on `Algebra.kaehlerDifferentialD`. It
commutes with every coefficient-module map, and a comparison statement
identifies its value with the underlying V-linear
`cotangentDerivation` of P0.22. These are the universal-derivation comparison of
[GR] (2.5.24), p. 33, and Proposition 2.5.29, p. 34.

`Almost.cotangentExtComparison` is the named additive map from ordinary Ext
with lower-shriek coefficients to almost Ext, in each integer degree.
The comparison `Ext^i_{B!!}(L_{B/A}, M_!) ≅ Ext^i_B(L^a_{B/A}, M)` above asserts its bijectivity.
Its formula is fixed as follows: apply Mathlib's
`mapDerivedCategory` to scalar extension from `B_!!` to `B_*` followed by
almostification, use the comparison with extension by zero in every
nonpositive degree, and use the scalar-extension counit and the inverse
lower-shriek unit at the coefficient module. The canonical shift and
single-object comparisons determine the degree convention.
It is compatible with coefficient maps. Sources: [GR] Proposition 2.5.42, p. 36, and (3.2.8), p. 41.

**Checks.** `cotangentDerivationComparison_zero` sends zero to zero;
`cotangentDerivationComparison_identity` vanishes for the identity algebra map;
`cotangentDerivationComparison_polynomial` realizes `d/dX` over `ℚ[X]` and
sends every polynomial `p` to `p′` on almost elements. Thus its values at
`1`, `X`, and `X²` are `0`, `1`, and `2X`, respectively.
`cotangentExtComparison_zero` sends zero to zero;
`cotangentExtComparison_identity` is zero in every degree for an identity
algebra map; `cotangentExtComparison_nonzero` preserves every nonzero class.

Two supporting results. The almost cotangent complex of `R^a → S^a` is the almostified classical one
(*almost-cotangent-comparison-classical*): for a basic setup `S` and a map `f : R → S` of
`V`-algebras, (i) if `f^a : R^a → S^a` is an isomorphism then `(L_{S/R})^a ≃ 0` in `D(S^a-Mod)`;
(ii) the natural map `L^a_{S^a/R^a} → (L_{S/R})^a` is an isomorphism in `D(S^a-Mod)`; hence the
almost cotangent complex of a morphism coming from `V`-algebras is computed by the cotangent complex
of P0.19 and depends only on `R^a → S^a` ([GR] §8.1, Proposition 8.1.7(i)-(ii), pp. 219–220; [Sch12]
§5, before Remark 5.15, p. 25). For `R = V` and `S = V × k` (`k = V/m`), `f^a` is an isomorphism and
`L_{S/V} ≅ 0 × L_{k/V}` has homology consisting of `k`-modules, hence almost zero, as (i) predicts.
Tor-independent base change (*cotangent-flat-base-change*): for `A → B` and `A → A'` morphisms of
almost algebras, `B' := B ⊗_A A'`, with `B ⊗^L_A A' → B'` an isomorphism (`Tor^A_i(B, A') = 0` for
`i > 0`; e.g. `B` or `A'` flat over `A`), the maps `B'_!! ⊗_{B!!} L_{B/A} → L_{B'/A'}` and
`(B'_!! ⊗_{B!!} L_{B/A}) ⊕ (B'_!! ⊗_{A'!!} L_{A'/A}) → L_{B'/A}` are quasi-isomorphisms ([GR] §2.5,
Theorem 2.5.35, p. 35, and Proposition 2.5.34, p. 35; cited by [Sch12] §5, proof of Theorem 5.10,
p. 26, as Theorem 2.5.36 in the book numbering). In that proof, for `A_n` flat over `(K°/ϖ^n)^a`
the square `(K°/ϖ^n)^a → (K°/ϖ^{n−1})^a`, `A_n → A_{n−1}` is Tor-independent, so
`L_{A_{n−1}/(K°/ϖ^{n−1})^a} ≅ A_{n−1} ⊗ L_{A_n/(K°/ϖ^n)^a}`.

**Checks.**

- Computation: for `B = (A_*[X])^a`:
  `H_0(L^a_{B/A}) ≅ B·δX` and `H_i(L^a_{B/A}) = 0` for `i > 0`.
- Degenerate: `L_{A/A} ≃ 0` for every almost algebra `A`
  (the identity of `Ã_!!`).
- Compatibility: for `S = classical V` and `V`-algebras
  `R → R'`: `L^a_{R'^a/R^a} ≅ L_{R'/R}` (the cotangent complex of P0.19), and `H_1` agrees with
  Mathlib's `Algebra.H1Cotangent R R'`.
- Non-example: defining `L_{B/A}` as the cotangent
  complex of `A_* → B_*` would be wrong: for `A_* → B_*` not of the form `Ã_!! → B̃_!!` the result is
  only almost quasi-isomorphic after almostification ([GR] Proposition 8.1.7), and for `B_*` the unit
  and the non-exactness of `B_!` intervene; the definition must pass through the exact algebras
  `V^a × A`, `V^a × B`.

The concrete `cotangentComplex_test_not_naive` uses `m = 0` over a nonzero
ring `V`: the unalmostified coefficient ring `B_!!` is nonzero (indeed `V`),
whereas `B_*` is zero. There is no ring isomorphism between them. This tests
the coefficient-ring distinction; both cotangent complexes of an identity map
still vanish.

The chain model `Almost.cotangentModel` applies ordinary scalar extension to the polynomial-resolution model for `Ã_!! → B̃_!!`, along `B̃_!! → B_!!`. `almostCotangentModel` then extends to `B_*` and almostifies in the setup `S.baseChange B_*`. Both derived objects use cohomological degrees `0, −1, −2, …`. The comparison `L^a_{S^a/R^a} ≅ (L_{S/R})^a` concerns the almostified complex; it does not identify the unalmostified complexes over `B_!!` and `B_*`.

- `exactification_test_classical` — in the classical setup the almost elements of the exactification of `R^a` are `V × R`.
- `exactification_test_zero` — exactifying a zero almost algebra gives the unit algebra, whose lower shriek-shriek is `V`.
- `exactification_test_unit` — exactifying the unit algebra gives `(V × V)^a`.
- `almostCotangent_test_identity` — both derived complexes vanish for an identity map.
- `almostCotangent_test_zero_setup` — both vanish for `m = 0`: the lower shriek-shriek algebras are all `V`, and their maps are identities.
- `almostCotangent_test_polynomial` — for every honest `V`-algebra `R`, the almostified polynomial map `R → R[X]` has degree-zero homology equal to the tensor unit over its target and no positive homology.
- `almostCotangent_test_classical_quotient` — for `ℤ → ℤ/2` in the classical setup, almost cotangent homology in degree zero is zero and degree one is nonzero.

The functor `cotangentBaseFunctor` retains the `V`-module structure of
`cotangentBaseModel`. Its `cotangentBaseMap` is specified on scalar-extended
polynomial differential generators. The colimit statement says that the
canonical map from the filtered colimit of these complexes is a
quasi-isomorphism ([GR] Proposition 2.5.33).

**Checks.** For `cotangentBaseModel` and `cotangentBaseFunctor`, `_identity`
has zero homology, `_classical` agrees with the ordinary cotangent complex, and
`_regular_quotient` detects nonzero first homology of `ℤ → ℤ/2`.
`cotangentBaseMap_identity`, `cotangentBaseMap_composition`, and
`cotangentBaseMap_zero` check identities, compositions, and that the identity of
the nonzero regular-quotient model is not the zero map.

The chain maps `Almost.cotangentTransitivityFirst` and
`Almost.cotangentTransitivitySecond` use these same two squares after
exactification and lower shriek-shriek. Their formulas on pure tensors are
those of `CotangentComplex.transitivityFirst_tmul` and
`CotangentComplex.transitivitySecond_apply` in P0.19, expressed through
`cotangentBaseMap`. The almost transitivity triangle fixes these first
two arrows and asserts existence only of the third ([GR] Theorem 2.5.32, p. 35).

**Checks.** `cotangentTransitivityFirst_identity` is invertible,
`cotangentTransitivityFirst_zero_source` is zero after passing to the derived
category, and `cotangentTransitivityFirst_polynomial` is nonzero for the
classical rational polynomial algebra. `cotangentTransitivitySecond_identity`
is invertible, `cotangentTransitivitySecond_zero_target` is zero in the derived
category, and `cotangentTransitivitySecond_regular_quotient` detects the
nonzero cotangent homology of the classical map `ℤ → ℤ/3`.

### P0.21 square-zero extensions and `Ext¹`

**Square-zero extensions of almost algebras are classified by `Ext¹` of the almost cotangent
complex.** (*almost-exal-ext-comparison*) Let `S` be a basic setup, `A → B` a morphism of almost
algebras and `M` a `B`-module; `A`-extensions of `B` by `M` are short exact sequences
`0 → M → E → B → 0` in `A-Mod` with `E` an `A`-algebra and `M` a square-zero ideal with its given
`B`-module structure ([GR] 2.5.2–2.5.7). (i) Almostification from ordinary `A_!!`-extensions
of `B_!!` by `M_*` to almost `A`-extensions of `B` by `M` is an equivalence of categories,
inducing the group isomorphism `Exal_{A!!}(B_!!, M_*) ≅ Exal_A(B, M)` on
isomorphism classes, functorial in all arguments ([GR] §2.5, Lemma 2.5.17(i)–(ii), p. 32). (ii) There are
natural isomorphisms `Ext⁰_{B!!}(L_{B/A}, M_!) ≅ Der_A(B, M)` ([GR] §2.5, Proposition 2.5.29, p. 34)
and `Exal_A(B, M) ≅ Ext¹_{B!!}(L_{B/A}, M_!) ≅ Ext¹_B(L^a_{B/A}, M)` ([GR] §2.5, Theorem 2.5.30,
p. 34; [GR] §3.2, (3.2.8), p. 41). (iii) For an epimorphism `A → B` with kernel `I`,
`Hom_B(I/I², M) ≅ Exal_A(B, M)` via pushout along `u : I/I² → M` of `0 → I/I² → A/I² → B → 0`. For
the trivial extension `B ⊕ M` the class is `0`; for `A → B = A/I` with `I² = 0` the extension `A`
itself corresponds to `1_I`; in [Sch12] Theorem 5.11 these groups classify deformations of
perfectoid `K°a/ϖ`-algebras.
*Needs:* P0.20; P0.18; P0.9.

`Almost.SquareZeroExtension` records the short exact sequence of underlying
almost modules, the algebra map from the base, and the quotient-compatible kernel
action. `Almost.extensionSetoid` fixes these maps under algebra isomorphisms,
and `Almost.Exal` is its quotient.

`CotangentComplex.extensionCategory` and `Almost.extensionCategory` give these
existing extension carriers their categories: an arrow is an algebra homomorphism
that fixes the base, quotient and specified kernel. In each namespace,
`extensionSetoid_iff_iso` identifies the existing relation with categorical
isomorphism ([GR] 2.5.4–2.5.7, p. 30). For every basic setup `S`, almost algebra
map `f : A → B` and `B`-module `M`, `Almost.ordinaryExtensions S f M` is the ordinary
extension category over `A_!!`, with quotient `B_!!` and kernel `M_*`, whose
`B_!!`-action is restricted along `B_!! → B_*`.
`Almost.extensionToAlmost` applies algebra and module almostification to each
term and each arrow, using `shriekShriekUnitIso` at the base and quotient and
`moduleEquiv_functor_forget` at the kernel. That `Almost.extensionToAlmost` is an
equivalence is the category-level assertion of [GR] 2.5.17(i), p. 32, for this specific functor;
in particular it preserves and recovers automorphisms.

In both namespaces `exalAddCommGroup` equips the existing quotient with Baer
addition. The theorem `CotangentComplex.exal_add_representative` characterizes
its sum by the fibre product modulo `(i_E(m), −i_F(m))`, with
`(i_E(m), i_F(n))` mapping to `i_G(m+n)`.
The almost representative statement gives the same construction on almost modules:
the two kernel inclusions `l,r` into the algebra fibre product have equal images
in the sum, and `0 → M →[l−r] E ×_B F → G → 0` is exact.
Both constructions fix the base and quotient maps ([GR] 2.5.5–2.5.7, p. 30).
An additive equivalence is derived from this equivalence of categories, with the image of every
representative equal to the class of its termwise almostification ([GR] 2.5.17(ii), p. 32); its
underlying class bijection is the isomorphism `Exal_{A!!}(B_!!, M_*) ≅ Exal_A(B, M)` of (i).

**Checks.** `CotangentComplex.extensionCategory_zero` and
`CotangentComplex.exalAddCommGroup_zero` give a unique arrow and zero sum when the
kernel is zero. Their `_classical` checks give only the identity automorphism
and the zero class for extensions of `ℚ` over itself.
`CotangentComplex.extensionCategory_polynomial` uses
`E = ℚ[X] ⊕ εℚ[X]`: the nonidentity automorphism sends `a+εb` to
`a+ε(b+a′)`, hence `X` to `X+ε`, while fixing the kernel and quotient.
`CotangentComplex.exalAddCommGroup_nonsplit` exhibits the nonzero order-two
class for `ℤ/4 → ℤ/2`.
For each of `Almost.ordinaryExtensions`, `Almost.extensionCategory`,
`Almost.extensionToAlmost`, and `Almost.exalAddCommGroup`, `_zero` checks the
zero kernel and `_classical` checks the rational base over itself. The first
three also have `_polynomial`: a nonidentity automorphism remains nonidentity
after almostification, although all extension classes of `ℚ[X]` by itself agree.
The `_nonsplit` Check of `Almost.exalAddCommGroup` gives the nonzero order-two class over `ℤ/2`.

**Checks.** Each carrier has `_zero` (zero kernel forces an invertible projection),
`_classical` (extensions of the rational base over itself split), and `_nonsplit`
(the classical `ℤ/4 → ℤ/2` class) examples. The zero-class characterization in
the identification `Ext¹ ≅ Exal_A(B, M)` of P0.20 uses an algebra section over the specified base,
as in [GR] Theorem 2.5.30.

The identification `Ext¹ ≅ Exal_A(B, M)` of P0.20 uses the named additive equivalence
`Almost.cotangentExtensionClass`, including its specified kernel identification.
It identifies addition with the Baer sum and kernel pushout with
postcomposition, and its almostification compatibility identifies the
class of each termwise almostified ordinary extension with the image of
`CotangentComplex.extensionClass` (the consumed ordinary classifier of P0.19). Its source map is induced by the projection
from exactified rings, with `t ⊗ x ↦ t·image(x)` in every degree; its target
map is the scalar-extension counit followed by the almost-elements counit.
This fixes the class itself, including its sign ([GR] Theorem 2.5.30,
p. 34, Lemma 2.5.17, p. 32, and (3.2.8), p. 41).

For a surjection of ordinary commutative rings `R → T`,
`CotangentComplex.conormalComparison` identifies `Ext¹(L_{T/R}, M)` with
`Hom_T(I/I², M)`, using Mathlib's conormal module.
`CotangentComplex.conormalComparison_extensionClass` specifies its value:
the image of `r mod I²` under the class of `E` is the unique `m` whose
kernel inclusion equals the base image of `r` in `E`.
For every commutative algebra map, `CotangentComplex.extensionClass_add`
asserts compatibility with Baer addition ([GR] Theorem 2.5.30, p. 34).
Together with this almostification compatibility, this is the conormal
normalization of the almost classifier ([GR] Lemma 3.2.1 and (3.2.2), p. 41).

**Checks.** `conormalComparison_zero` is the zero map;
`conormalComparison_identity` vanishes for `R → R`;
`conormalComparison_zmodNine` uses `ℤ/9 → ℤ/3`, with kernel generator sent
to `3 mod 9`. The class takes `3 mod 9` in `(3)/(9)` to `1 mod 3`, while
its negative takes it to `2 mod 3`; it has order three.
`cotangentExtensionClass_zmodNine` carries this extension to the classical
almost setup, where the class differs from its negative, doubling is Baer
addition, and three copies sum to zero.

### P0.22 flat square-zero deformations

Nilpotent deformations of almost algebras come next: flat deformations, lifting of morphisms,
lifting of étale algebras, faithfully flat descent, tight ideals and henselian pairs, and modules
over complete almost algebras.

**Flat square-zero deformations of almost algebras: obstruction, torsor, automorphisms.**
(*almost-deformation-theory*) Let `S` be a basic setup (`m̃` flat), `A` an almost algebra,
`B̃ = (0 → I → B → B₀ → 0)` an `A`-extension (`I² = 0`) and `f₀ : B₀ → C₀` a flat morphism of almost
algebras. Let `ω(B̃, f₀) := ∂(I → C₀ ⊗_{B₀} I) ∈ Ext²_{C₀}(L^a_{C₀/B₀}, C₀ ⊗_{B₀} I)`, `∂` the
boundary of the transitivity sequence for `B → B₀ → C₀`. Then (i) `ω(B̃, f₀) = 0` iff there is a
flat deformation of `C₀` over `B`, i.e. a `B`-extension `C̃ = (0 → C₀ ⊗ I → C → C₀ → 0)` with `C`
flat over `B` and `C ⊗_B B₀ ≅ C₀`; (ii) if `ω = 0`, the isomorphism classes of flat deformations
form a torsor under `Exal_{B₀}(C₀, C₀ ⊗_{B₀} I) ≅ Ext¹_{C₀}(L^a_{C₀/B₀}, C₀ ⊗_{B₀} I)`; (iii) the
automorphism group of a flat deformation (acting trivially on `C₀` and on `C₀ ⊗ I`) is
`Der_{B₀}(C₀, C₀ ⊗_{B₀} I) ≅ Ext⁰_{C₀}(L^a_{C₀/B₀}, C₀ ⊗_{B₀} I)` ([GR] §3.2, Corollary
3.2.11(i)-(iii), p. 42). More generally, for a `C₀`-module `J` and `B₀`-linear `u : I → J`, let
`u_C : C₀ ⊗_{B₀} I → J` be `c ⊗ i ↦ c·u(i)`. The obstruction
`ω(B̃, f₀, u) = (u_C)_* ω(B̃, f₀)` governs extensions `C̃` with kernel `J` ([GR] §3.2, Proposition
3.2.9, p. 41; [Sch12] §5, Theorem 5.11, p. 24, cites this proposition under the same number). If
`L^a_{C₀/B₀} ≃ 0` (e.g. `C₀` étale over `B₀`, or a perfectoid `(K°/ϖ)^a`-algebra) a flat deformation
exists and is unique up to unique isomorphism, which [Sch12] Theorem 5.10 applies over
`(K°/ϖ^n)^a → (K°/ϖ^{n−1})^a`; in the classical setup, for `A = B = B₀ ⊕ I` trivial and `C₀ = B₀[X]`, the trivial
deformation `C₀ ⊕ C₀ ⊗ I` exists and deformations form a torsor under `Ext¹(L^a, −) = 0`. *Needs:*
P0.21; P0.20; P0.18; P0.19; DerivedModuleAlgebra §0.25.

**Checks.**

- Degenerate: for `I = 0` (`B = B₀`): `ω = 0`, the only
  flat deformation is `C₀` itself, the torsor `Ext¹(L^a_{C₀/B₀}, C₀ ⊗ 0) = 0` is a point and the
  automorphism group `Der_{B₀}(C₀, 0) = 0` is trivial, as it must be.
- Computation: in the classical setup, `C₀ = B₀[X]`: `L^a_{C₀/B₀} ≃ C₀·dX` in
  degree `0`, so `Ext² = Ext¹ = 0`: every square-zero `B` admits a flat deformation, unique up to
  isomorphism, with automorphisms `Der_{B₀}(B₀[X], B₀[X] ⊗ I) ≅ B₀[X] ⊗ I` (the value on `X`).
  In a general basic setup, `B₀[X]` is only almost projective as a module over itself:
  this does not imply vanishing of ordinary `Ext¹` or `Ext²`.
- Non-example: flatness of the deformation `C` over `B` is not
  decorative: for `B₀ = k`, `B = k[t]/(t²)`, `C₀ = k`, the `B`-algebra `C = k` (`t ↦ 0`) has
  `C ⊗_B k = k` but is not flat over `B` (`(t) ⊗_B k = k → (t)·k = 0` is the zero map, see
  *almost-local-flatness-criterion*) and its kernel `0` is not `C₀ ⊗_{B₀} I = k`; the flat
  deformation of `C₀ = k` over `B` is `B` itself.

**Check `CotangentComplex.transitivityTriangle_cone_sign`.** Use Mathlib's
`CochainComplex.mappingCone.triangle`: for the zero map `ℚ[0] → ℚ[0]`,
the cone has one copy of `ℚ` in degree `−1` and one in degree `0`.
Its connecting map sends the degree `−1` generator to `−1` in `ℚ[1]`.
Thus the cone triangle uses minus projection onto the shifted source;
this is the convention for interpreting the transitivity boundary in the
obstruction formula above. The supplier is
`CochainComplex.mappingCone.inl_v_triangle_mor₃_f` in
`Mathlib/Algebra/Homology/HomotopyCategory/Pretriangulated.lean`.

Two supporting results. The transitivity triangle for almost cotangent complexes
(*cotangent-transitivity*): for `A → B → C` morphisms of almost algebras there is a natural
distinguished triangle `C_!! ⊗_{B_!!} L_{B/A} → L_{C/A} → L_{C/B} → C_!! ⊗_{B_!!} L_{B/A}[1]` in
`D(C_!!-Mod)`, hence `C ⊗_B L^a_{B/A} → L^a_{C/A} → L^a_{C/B} →` in `D(C-Mod)`; on `H_0` it extends
the exact sequence `C ⊗_B Ω_{B/A} → Ω_{C/A} → Ω_{C/B} → 0` ([GR] §2.5, Theorem 2.5.32, p. 35). For
`A → A[X] → A` (`X ↦ 0`) it gives `L^a_{A/A[X]} ≃ (X)/(X²)[1] ≅ A[1]`, the conormal module in
homological degree `1`. [Sch12]'s proof of Theorem 5.10 instead tensors `0 → A → A_n → A_{n−1} → 0`
with `L^a_{A_n/(K°/ϖ^n)^a}` and uses base change ([GR] §2.5, Theorem 2.5.35, pp. 35–36, which [Sch12]
cites as Theorem 2.5.36) to get
`L^a_{A/(K°/ϖ)^a} → L^a_{A_n/(K°/ϖ^n)^a} → L^a_{A_{n−1}/(K°/ϖ^{n−1})^a} →`. The local flatness criterion for square-zero thickenings
(*almost-local-flatness-criterion*): (i) for a ring `R`, an ideal `J ⊆ R` with `J² = 0`, `R' = R/J`
and an `R`-module `M` with `M' = M/JM`, `M` is `R`-flat iff `M'` is `R'`-flat and
`u(M) : J ⊗_{R'} M' → JM`, `x ⊗ m' ↦ xm`, is an isomorphism ([GR] §3.2, (3.2.19), p. 44); (ii) for
a basic setup `S` and a morphism `f̃ : B̃ → C̃` of `A`-extensions `0 → I → B → B₀ → 0` and
`0 → J → C → C₀ → 0` (`I² = J² = 0`) whose square is cofibred (`C₀ ≅ C ⊗_B B₀`), `B → C` is flat iff
`B₀ → C₀` is flat and `C₀ ⊗_{B₀} I → J` is an isomorphism ([GR] §3.2, Lemma 3.2.6 with proof,
p. 41); the general nilpotent case follows by induction. For `R = k[ε]/(ε²)` and `M = R`, `M' = k`
is flat and `u(M) : (ε) ⊗ k → εR` is bijective, while for `M = k`, `u(M) : (ε) ⊗ k → 0` is not
injective and `k` is not flat over `R` (Mathlib `Module.Flat`, `TensorProduct`).

The square-zero flatness criterion uses the canonical `R/J` actions on `J`,
`N = JM`, and `M/N`: a residue class `[r]` acts by the original `r` action.
The first two actions use `Module.IsTorsionBySet.module`, with annihilation
following from `J² = 0`; the third is Mathlib's quotient-module instance.
Thus the multiplication map is linear for these specified actions in both
the ordinary and almost flatness criteria ([GR] (3.2.19), p. 44, and
Lemma 3.2.6, p. 41).

**Checks.** `flat_squareZero_ideal_action` computes `[r] • a = ra`,
`flat_squareZero_image_action` computes `[r] • n = r • n`, and
`flat_squareZero_quotient_action` computes `[r] • [m] = [r • m]`.
`flat_squareZero_regular` and `flat_squareZero_residue` distinguish the
regular module over rational dual numbers from its nonflat residue module.

`Almost.FlatDeformation` is the upper algebra of the specified pushout square,
with a flat map from the thickened base. `flatDeformationSetoid` identifies exactly
isomorphisms commuting with the base and reduction maps. The obstruction and torsor
are stated by `Almost.flatDeformation`; `flatDeformation_unique` includes unique
lifting of isomorphisms when the cotangent complex vanishes ([GR] Corollary 3.2.11).

**Checks.** For both carriers, `_zero` forces an invertible reduction, `_polynomial`
gives one isomorphism class with the expected polynomial kernel automorphisms in
the classical setup, and `_nonflat` excludes the residue algebra of dual numbers.

The normalization in `Almost.flatDeformation` uses a specified universal
`B₀`-linear map `u : I → J`: for each `C₀`-module `M`, composition with `u`
identifies `Hom_{C₀}(J,M)` with `Hom_{B₀}(I,M)`. Thus `J` is the tensor
kernel `C₀ ⊗_{B₀} I`, with its generator map retained.
Its kernel statement supplies this universal map and identifies its
coefficient module with `Module.baseChange f I` ([GR] (3.2.5), p. 41). For every deformation,
the theorem supplies an extension by this kernel and an isomorphism to the
given upper algebra, commuting with the base, reduction, and kernel maps.
These hypotheses include flatness of `B₀ → C₀`; no flatness of `I` is required.

`Almost.cotangentConnecting` is the connecting arrow for the functorial
polynomial-resolution transitivity triangle, after extension to `C₀_*` and
almostification. Its triangle statement specifies its first
two arrows as the images of `cotangentTransitivityFirst` and
`cotangentTransitivitySecond`. The cone convention is the minus projection
described above ([GR] Theorem 2.5.32, p. 35). Regard the given extension
`E = (I → B → B₀)` as an extension over `B`, with base map `id_B`.
`Almost.kernelExtensionClass E f J u` is the scalar-adjoint class of its
pushout by `u`. Its degree-one conormal map is `u`, using the given
inclusion of `I`, as in [GR] (3.2.2), (3.2.7), p. 41. Define
`Almost.deformationBoundary E f J u` by composing the connecting arrow
with this class shifted by one, followed by the canonical identification
`J[1][1] ≅ J[2]`. `Almost.flatDeformation` tests this specific element for
vanishing ([GR] (3.2.10), Corollary 3.2.11, p. 42).

`Almost.cotangentExtensionClass` is the additive extension classifier:
its polynomial-resolution cocycle uses the given kernel inclusion and the
order `d₀ − d₁`, with the exactification and almostification comparisons
([GR] (2.5.31), p. 34, and (3.2.8), p. 41). `Almost.exalRestriction`
forgets part of the base without changing the upper algebra, reduction,
or kernel. Its injectivity requires the square-zero
extension of the base and identifies the injection in [GR] (3.2.15), p. 43.
The torsor in `Almost.flatDeformation` is specified by
`restriction(class([D] −ᵥ [F])) = [D] − [F]` in `Exal_B(C₀,J)`.
Consequently translation by an `Ext¹` class is Baer addition with that class;
reversing the difference changes its sign.

The automorphisms fixing the base and reduction form a subgroup of
`CategoryTheory.Aut`. Their comparison is a multiplicative equivalence to
`Multiplicative Ext⁰`, so composition corresponds to addition. Its value
is determined by
`h − id = reduction ≫ derivation ≫ kernel inclusion`, using the specified
identification of the deformation with its extension. The additive map
`Almost.cotangentDerivation` evaluates a degree-zero derived morphism on
`H⁰(L) = Ω`. Its value on the universal differential is specified, and its range
statement gives injectivity and the exact Leibniz and base-vanishing conditions on its image
([GR] Proposition 2.5.25, p. 33, (2.5.29), p. 34, and Corollary
3.2.11(iii), p. 42). In particular identity maps to zero and inverses map
to negative derivations.

DerivedModuleAlgebra §0.25 supplies the following normalized ordinary deformation laws,
on the consumed carrier of P0.19. The target `CotangentComplex.flatDeformation` retains the same
flatness and square-zero hypotheses. Its
`CotangentComplex.deformationObstruction` is the composite of
`CotangentComplex.transitivityConnecting` with the shifted
`CotangentComplex.kernelExtensionClass`. The latter sends the conormal
class of `i` to `1 ⊗ i`; it does not discard the given kernel inclusion.
`CotangentComplex.transitivityConnecting_triangle` identifies the first
two arrows with `transitivityFirst` and `transitivitySecond`. The theorem
specifies ordinary extension representatives of each classical almost
flat lift. Restriction of the torsor difference is the difference of their
`CotangentComplex.extensionClass` values. Its automorphism comparison is
again a group isomorphism with value `h − id` under
`CotangentComplex.derivationEvaluation`;
`CotangentComplex.derivationEvaluation_differential` evaluates the map on
`H⁰` at `d x` ([GR] (3.2.7)–(3.2.11), pp. 41–42, in the classical setup).

**Checks.** Each label below names a Check, specified in this README only.

- For `Almost.cotangentExtensionClass`: `_zero`, `_split`, and `_nonsplit` send zero
  and split extensions to zero and distinguish an extension without a section.
- For `Almost.cotangentDerivation`: `_zero`, `_base`, and `_nonzero` compute zero,
  vanishing on the base, and detection of a nonzero degree-zero class.
- For `Almost.exalRestriction`: `_identity`, `_zero`, and `_surjection` retain the
  extension for identity restriction, preserve zero, and preserve nonzero
  classes when the base map is the specified square-zero surjection.
- For `Almost.cotangentConnecting`: `_identity_source`, `_identity_target`, and
  `_polynomial_quotient` give zero for an identity leg and a nonzero arrow
  for `ℚ → ℚ[X] → ℚ`, with `X ↦ 0`.
- For `Almost.kernelExtensionClass`: `_zero`, `_negation`, and `_identity_nonzero`
  compute the zero kernel map, change sign when the map changes sign, and
  detect an invertible map from a nonzero kernel with `f = id`.
- For `Almost.deformationBoundary`: `_zero`, `_split`, and `_negation` give zero
  for the zero kernel map, zero for a split base extension, and the negative
  boundary for the negative kernel map. A split base extension may still
  have a nonzero kernel-extension class over its upper algebra.
- `CotangentComplex.transitivityConnecting_identity_source`,
  `_identity_target`, and `_polynomial_quotient` make the same three
  transitivity computations on the ordinary model.
- `CotangentComplex.kernelExtensionClass_zero`, `_nonzero`, and
  `_faithfully_flat` distinguish the zero kernel, a nonzero kernel with
  `T = R`, and its nonzero image under faithfully flat scalar extension.
- `CotangentComplex.deformationObstruction_zero`, `_split`, and
  `_polynomial` give zero for a zero kernel, a split base extension, and
  the polynomial algebra `ℚ[X]`, respectively.
- `CotangentComplex.derivationEvaluation_zero`, `_base`, and `_nonzero`
  test zero, vanishing on scalars, and injectivity of the comparison.

`CotangentComplex.polynomialLift_sign` computes the automorphism convention
on the flat lift `ℚ[ε]/(ε²)[X]` of `ℚ[X]`. For each `a : ℚ`, translation
`hₐ(X) = X + aε` has a unique `Ext⁰` class satisfying the difference formula;
its derivation takes `X` to `a` and `X²` to `2aX`. For a polynomial `P`
over the dual numbers, the kernel inclusion of that derivation at `P mod ε`
is exactly `hₐ(P) − P`. **Checks** `CotangentComplex.polynomialLift_positive`,
`_inverse`, and `_composition` compute the values `ε`, `−ε`, and the law
`hₐ ∘ h_b = h_{a+b}`. Since `1 ≠ −1` in `ℚ[X]`, the positive translation
cannot be assigned the negative derivation under this comparison. This is
the polynomial computation of [GR] Corollary 3.2.11(iii), p. 42.

### P0.23 lifting morphisms along square-zero extensions

**Lifting morphisms along square-zero extensions of almost algebras.** (*almost-morphism-lifting*)
Let `S` be a basic setup, `A` an almost algebra, `B̃` an `A`-extension `0 → I → B → B₀ → 0`
(`I² = 0`), and `C̃ⁱ = (0 → Jⁱ → Cⁱ → C₀ⁱ → 0)`, `i = 1, 2`, `A`-extensions with morphisms
`f̃ⁱ : B̃ → C̃ⁱ` (kernel maps `uⁱ : I → Jⁱ` and reductions `f₀ⁱ : B₀ → C₀ⁱ`), together with a `B₀`-algebra map `g₀ : C₀¹ → C₀²` and a
`C₀¹`-linear map `v : J¹ → J²` (the target action is through `g₀`) such that `u² = v ∘ u¹` and
`f₀² = g₀ ∘ f₀¹`. Regard `C̃¹` and `C̃²` as extensions over `B`, via `f̃¹` and `f̃²`.
Their class difference `v_* e(C̃¹) − e(C̃²) ∗ g₀` belongs to
`Ext¹_{C₀¹}(L^a_{C₀¹/B}, J²)`. Compatibility on `I` makes its restriction to
`Hom_{B₀}(I, J²)` zero. Its unique preimage under the transitivity injection
`Ext¹_{C₀¹}(L^a_{C₀¹/B₀}, J²) → Ext¹_{C₀¹}(L^a_{C₀¹/B}, J²)` is the
obstruction to a morphism of `A`-extensions `g̃ : C̃¹ → C̃²` over `v` and `g₀` with
`f̃² = g̃ ∘ f̃¹`; when it vanishes, such `g̃` form a torsor under
`Der_{B₀}(C₀¹, J²) ≅ Ext⁰_{C₀²}(C₀² ⊗ L^a_{C₀¹/B₀}, J²)` ([GR] §3.2, Proposition 3.2.16, p. 43). In
particular, if `C` and `C'` are flat over `B` with reductions `C₀`, `C₀'` and `f₀ : C₀ → C₀'`, the
obstruction to lifting `f₀` lies in `Ext¹(L^a_{C₀/B₀}, C₀' ⊗ I)` and lifts form a torsor under
`Hom(L^a_{C₀/B₀}, C₀' ⊗ I)` ([Sch12] §5, Theorem 5.12, p. 25, the almost version); with `L^a ≃ 0`
this gives unique lifting of morphisms, the full faithfulness in [Sch12] Theorem 5.10, and for `C₀`
étale over `B₀` morphisms lift uniquely (P0.24(i)). *Needs:* P0.21; P0.20.

`Almost.exalPush` pushes an extension along a linear map of its identified
kernel, and `Almost.exalPull` pulls it back along a map of quotient algebras.
Their representative statements specify
the resulting classes through maps of extensions that commute with the base,
quotient, and kernel inclusions ([GR] (2.5.5)–(2.5.7), p. 30, and (3.2.12)–(3.2.15),
pp. 42–43). `Almost.morphismObstruction` is the `Ext¹` class of the unique
preimage of `v_*[D₁] − g₀^*[D₂]` under `exalRestriction`.
Its existence statement supplies the unique preimage,
and its difference statement states its equality, with the
source-first order of subtraction. Its signature retains the two specified
kernel maps, their compatibility with the base extension, and `u₂ = v ∘ u₁`.

`Almost.liftMorphism` tests that particular obstruction for zero. Its torsor
is specified by the equality
`h − k = projection₁ ≫ cotangentDerivation(h −ᵥ k) ≫ inclusion₂`.
Thus a derivation acts by adding its composite through the quotient and
kernel; the negative action does not satisfy the contract. This also fixes
the identity-based automorphism comparison used in P0.22. If the relative
cotangent complex vanishes, a lift exists and is unique ([GR] Proposition
3.2.16, p. 43).

**Checks.** For `Almost.exalPush`, `_identity`, `_zero`, and `_negation` test identity,
zero, and negative identity on the kernel. For `Almost.exalPull`, `_identity`,
`_zero`, and `_section` test identity pullback, the zero extension class,
and a pullback with a specified section. For `Almost.morphismObstruction`, `_lift`,
`_zero_coefficient`, and `_unequal_classes` give zero for a supplied compatible
lift, zero for a zero target kernel, and a nonzero obstruction when the two
specified extension classes differ. These Checks are specified in this README only.

### P0.24 étale algebras lift along nilpotent ideals

**Étale almost algebras lift uniquely along nilpotent ideals.**
(*nilpotent-lifting-of-etale-almost-algebras*) Let `S` be a basic setup, `A` an almost algebra and
`I ⊆ A` a nilpotent ideal (induction reduces to `I² = 0`), `A' := A/I`. (i) For `B` weakly étale
over `A`, `C` any `A`-algebra and `J ⊆ C` nilpotent, `Hom_{A-Alg}(B, C) → Hom_{A-Alg}(B, C/J)` is
bijective ([GR] §3.2, Theorem 3.2.18(i), p. 43). (ii) `B ↦ A' ⊗_A B` is an equivalence from weakly
étale `A`-algebras to weakly étale `A'`-algebras; (iii) it restricts to an equivalence of étale
algebras `A-Ét ≃ A'-Ét` ([GR] §3.2, Theorem 3.2.18(iii), p. 43) and (iv) to an equivalence of finite
étale algebras `A_fet ≃ A'_fet` (étale and almost finitely presented; [GR] §3.2, Theorem
3.2.28(ii), p. 46; almost finite presentation lifts by [GR] §3.2, Lemma 3.2.25(iii), p. 45). For
`A = (K°/ϖ^n)^a`, `n ≥ 2`, and `I = ϖ^{n−1}A`, finite étale `(K°/ϖ^{n−1})^a`-algebras lift uniquely to
`(K°/ϖ^n)^a`, the finite-level step of [GR] Theorem 5.3.27; in the classical setup the restriction of (iii) to finitely presented étale algebras
is [SGA1] Exposé I, Corollaries 5.5 and 8.3, printed pp. 6 and 12
([public edition](https://arxiv.org/abs/math/0206203)); [GR] Theorem 3.2.18 supplies the larger category used here.
*Needs:* P0.22; P0.23; P0.14; P0.10.

Supporting result: vanishing of the almost cotangent complex when the diagonal is Tor-trivial
(*cotangent-vanishing-weakly-etale*). For a basic setup `S` and a morphism `R → S'` of almost
algebras with `Tor^R_i(S', S') = 0` and `Tor^{S'⊗_R S'}_i(S', S') = 0` for all `i > 0` (the
`S' ⊗_R S'`-module structure via `μ`), `L_{S'/R} ≃ 0` in `D(S'_!!-Mod)`; in particular
`L_{S'/R} ≃ 0` for every weakly étale (hence every étale, every finite étale) morphism `R → S'`
([GR] §2.5, Theorem 2.5.36, p. 36; used for weakly étale lifting in [GR] §3.2, proof of Theorem
3.2.18, p. 43). Every idempotent quotient `A → A/eA` is weakly étale, so `L_{(A/eA)/A} ≃ 0`; for a
perfect `𝔽_p`-algebra the vanishing `L_{R/𝔽_p} = 0` is of a different nature (Frobenius; [Sch12]
Proposition 5.13, layer P1).

### P0.25 faithfully flat descent

**Faithfully flat descent for almost modules and almost algebras.**
(*almost-faithfully-flat-descent*) Let `S` be a basic setup and `φ : A → B` a faithfully flat
morphism of almost algebras (P0.11). (i) `φ` is of effective descent for modules:
`M ↦ (B ⊗_A M, can)` is an equivalence from `A-Mod` to the category `Desc(B/A)` of `B`-modules `N`
with a `B ⊗_A B`-linear isomorphism `β : p₂*N ≅ p₁*N` satisfying the cocycle condition
`p₁₂*(β) ∘ p₂₃*(β) = p₁₃*(β)`; (ii) the same holds for `A`-algebras; (iii) both statements are
stable under base change `A → A'` (universal effective descent) ([GR] §3.4, introduction and
(3.4.1), p. 55); (iv) under (i) the full subcategories of flat, almost finitely generated, almost
finitely presented and almost finite projective modules correspond, and under (ii) those of weakly
étale, étale and finite étale algebras correspond (finite étale: combine étale with almost finitely
generated projective) ([GR] §3.4, (3.4.1), pp. 55–56); (v) almost
projectivity descends when `B` is almost finitely presented over `A` (then `B` is almost finite
projective over `A`, so `B ⊗_A alHom_A(M, N) ≅ alHom_A(M, B ⊗_A N)` by
*tensor-hom-comparison-almost-projective*, `B ⊗_A I` is `alHom_A(M, −)`-acyclic for `I`
injective, and `B ⊗_A alExt^i_A(M, N) ≅ alExt^i_B(B ⊗_A M, B ⊗_A N) = 0` for `i > 0`, whence
`alExt^i_A(M, N) = 0` by faithful flatness); for a general faithfully flat `φ` it is not asserted
([GR] §3.2, Remark 3.2.26(iii)–(iv), p. 45). For `M` almost finitely generated no hypothesis on `B` is
needed, since almost finite projective = flat + almost finitely presented descends by (iv). The
honest case is part of this target: for `V`-algebras `R → R'` with `R'` faithfully flat, (i) and
(ii) for the classical setup `S = classical V` are Grothendieck's effective descent of modules and
algebras ([GR] §3.4, (3.4.1)). Mathlib supplies the comonadicity of `R' ⊗_R −` (`comonadicExtendScalars`)
and the degree `≤ 1` Amitsur statement for the ring itself (`CommRingCat.isLimitForkPushoutSelfOfFaithfullyFlat`),
and descends injectivity, surjectivity and bijectivity of linear maps
(`Module.FaithfullyFlat.bijective_of_tensorProduct`); the identification of comodules with descent
data, hence the equivalence `R-Mod ≃ Desc(R'/R)`, is proved here, and for a general setup the
almostification of that descent is (i) for `R^a → R'^a`.
DiamondsAndVStacks layer D3 applies (ii) with `(S⁺/ϖ)^a` faithfully flat over `(R⁺/ϖ)^a` to descend
the datum on `(S̃⁺/ϖ)^a` to an almost `(R⁺/ϖ)^a`-algebra. *Needs:* P0.7; P0.10; Mathlib
`comonadicExtendScalars`, `ModuleCat.extendScalars`, `CategoryTheory.Pseudofunctor.DescentData`,
`CommRingCat.isLimitForkPushoutSelfOfFaithfullyFlat`.

Three supporting results. Stability of flat, unramified and
étale morphisms (*etale-morphisms-stability*): for `φ : A → B`, `ψ : B → C`, (i) flatness, almost
projectivity, faithful flatness, almost finiteness, weak unramifiedness, unramifiedness, weak
étaleness and étaleness are stable under base change and (ii) under composition; (iii) if `φ` is
flat and `ψ ∘ φ` faithfully flat then `φ` is faithfully flat; (iv) if `φ` is weakly unramified and
`ψ ∘ φ` flat (weakly étale) then `ψ` is flat (weakly étale); (v) if `φ` is unramified and `ψ ∘ φ`
étale then `ψ` is étale; (vi) `φ` is faithfully flat iff it is a monomorphism and `B/A` is a flat
`A`-module; (vii) if `φ` is almost finite and weakly unramified then `φ` is unramified; (viii) if `ψ`
is faithfully flat and `ψ ∘ φ` flat (weakly unramified) then `φ` is flat (weakly unramified); finite
étale morphisms are stable under base change and composition ([GR] §3.1, Lemma 3.1.2(i) and
(v)-(vii), p. 39). Classically (vi) is "faithfully flat iff injective with flat cokernel"; base change
of `L°a/K°a` along `K°a → (K°/ϖ)^a` gives a finite étale `(L°/ϖ)^a` over `(K°/ϖ)^a`. `B ↦ B_!!`
preserves and reflects faithful flatness, but not flatness (*lower-shriek-shriek-faithful-flatness*):
(i) `φ` is faithfully flat iff `φ` is a monomorphism and `B/A` is a flat `A`-module ([GR] §3.1,
Lemma 3.1.2(vi), p. 39); (ii) `φ` is a monomorphism iff `φ_!! : A_!! → B_!!` is injective, and
`B_!!/A_!! ≅ B_!/A_!` is flat over `A_!!` iff `B/A` is flat over `A` (`m̃` flat is used here through
[GR] Proposition 2.4.35); (iii) hence `φ` is faithfully flat iff `φ_!!` is a faithfully flat ring map
([GR] §3.1, Remark 3.1.3(ii), p. 40); (iv) flatness is not preserved: for `S = ofValuationRing V`
with residue field `k`, the flat projection `V^a × V^a → V^a` becomes `V ×_k V → V` under `(−)_!!`,
which is not flat ([GR] §3.1, Remark 3.1.3(ii), p. 39; [Bhatt] §4.2, Remark 4.2.8, p. 25; Mathlib
`Module.FaithfullyFlat`). The Amitsur complex of an almost faithfully flat map is exact
(*almost-amitsur-complex-exact*): for `φ : A → B` faithfully flat and `M` an `A`-module, the
augmented Amitsur complex `0 → M → M ⊗_A B → M ⊗_A B ⊗_A B → ⋯` (differentials the alternating sums
of the coface maps) is exact in `A-Mod`; in honest terms, for `V`-algebras `R → R'` such that `R'^a`
is faithfully flat over `R^a` (e.g. `Tor_i^R(X, R')` almost zero for `i > 0` and `X ⊗_R R'` almost
zero only if `X` is) and any `R`-module `N`, the cohomology of
`0 → N → N ⊗_R R' → N ⊗_R R' ⊗_R R' → ⋯` is almost zero, so `N^a` is the equalizer of
`(N ⊗ R')^a ⇉ (N ⊗ R' ⊗ R')^a` ([GR] §3.4, (3.4.1), p. 55; [ECD] §8, proof of Proposition 8.8,
p. 43; Mathlib `CommRingCat.isLimitForkPushoutSelfOfFaithfullyFlat`). For `A → A × A` the Amitsur
complex is split exact; DiamondsAndVStacks layer D2 uses the almost exactness of
`R⁺/ϖ → S⁺/ϖ ⇉ S⁺/ϖ ⊗ S⁺/ϖ`.

`Module.baseChange` takes almost elements, extends scalars along the map of
almost-element rings, and almostifies. `Module.descentPseudofunctor` uses these
functors; `Algebra.descentPseudofunctor` uses pushouts of almost algebras. Their
descent data are Mathlib's `Pseudofunctor.DescentData`, including the cocycle equation.
The canonical comparison is an equivalence, also after any base change, by [GR] (3.4.1).

The coherence isomorphisms are fixed as follows. `Module.baseChangeUnit`
sends an almost element through the ordinary scalar-extension unit and the
almost-elements adjunction unit. `Module.baseChangeIdIso_exists` and
`Module.baseChangeCompIso_exists` give unique natural isomorphisms with the
prescribed generator equations; `Module.baseChangeIdIso` and
`Module.baseChangeCompIso` select them. `Module.baseChangeIdIso_unit` sends
the identity-extension generator back to the original element.
`Module.baseChangeCompIso_unit` sends an iterated generator back to its
direct-extension generator. These are the almost versions of Mathlib's
`ModuleCat.extendScalarsId` and `ModuleCat.extendScalarsComp`, and supply
`mapId` and `mapComp` of `Module.descentPseudofunctor`. For
`Algebra.descentPseudofunctor` those fields are exactly `Under.pushoutId`
and `Under.pushoutComp`. Both use `pseudofunctorOfIsLocallyDiscrete` for
2-morphisms; associativity and the two unit equations concern these fixed
isomorphisms. Faithfully flat descent (*almost-faithfully-flat-descent*) therefore applies to the
specified scalar-extension descent data, universally after base change
([GR] (3.4.1), p. 55).

**Checks.** `baseChangeUnit_zero`, `baseChangeUnit_add`, and
`baseChangeUnit_identity_injective` test zero, addition, and injectivity
along an identity base map. For each of `baseChangeIdIso` and
`baseChangeCompIso`, `_generator` fixes the comparison on generators,
`_zero` gives zero, and `_not_negated` excludes the negative comparison
whenever the generator differs from its negative.

**Checks.** The identity, zero and rank-two Checks of almost-module base change give
the identity functor, the zero module, and the free rank-two module. The module
pseudofunctor has the corresponding `_identity`, `_zero`, and `_two` Checks.
The algebra pseudofunctor's `_identity`, `_unit`, and `_two` Checks give the
identity functor, the base algebra, and the two-factor product after scalar extension.

`Algebra.relativeForget` uses the underlying module over the source's almost
elements. Its `_unit`, `_zero`, and `_two` Checks give the tensor unit, the zero
module, and two copies of the unit. `amitsurCosimplicial` is the Čech conerve in
almost algebras over the base, followed by this forgetful functor and tensoring
with the coefficient module. `amitsurComplex` takes its alternating cofaces.

**Checks.** Both Amitsur constructions have `_identity` (cohomology is the
coefficient module in degree zero only), `_zero` (zero coefficients), and `_two`
(the split cover `A → A × A`). In the last case the first differential is
`δ₀ − δ₁` and is nonzero for a nonzero base; on two scalar values this is
`(a,b) ↦ 1 ⊗ (a,b) − (a,b) ⊗ 1`, fixing the sign in [GR] (3.4.1).

### P0.26 tight ideals and henselian pairs

**Tight ideals, the Jacobson radical and henselian pairs of almost algebras.** (*tight-ideal*) Let
`S = (V, m)` be a basic setup and `A` an almost `V`-algebra; ideals of `A` are subobjects `I ↪ A`
stable under multiplication, with powers `Iⁿ` the images of `I^{⊗n}`. An ideal `I ⊆ A` is tight if
there are a finitely generated ideal `m₀ ⊆ m` and `n ∈ ℕ` with `Iⁿ ⊆ m₀·A` ([GR] §5.1, Definition
5.1.5, p. 103); for `ϖ ∈ m` the ideal `ϖA` is tight. The Jacobson radical of `A` is
`rad(A) := rad(A_*)^a` ([GR] §5.1, Definition 5.1.1, p. 103); for a `V`-algebra `R` and an ideal
`J ⊆ R`, `J^a ⊆ rad(R^a)` iff `mJ ⊆ rad(R)` ([GR] §5.1, Lemma 5.1.2, p. 103). A pair `(A, I)` with
`I ⊆ rad(A)` is henselian if `(A_*, m·I_*)` is a henselian pair (Mathlib
`HenselianRing A_* (m·I_*)`), and tight henselian if moreover `I` is tight ([GR] §5.1, Definition
5.1.8, p. 104); `(R^a, J^a)` is henselian iff `(R, mJ)` is ([GR] Lemma 5.1.10). If `A` is
`I`-adically complete (`A → lim A/Iⁿ` an isomorphism), `(A, I)` is henselian ([GR] §5.3, Remark
5.3.7, p. 112). Define `Almost.Algebra.IsTight` (`∃ m₀ ≤ S.ideal, m₀.FG ∧ ∃ n, Iⁿ ≤ m₀ • A`),
`Almost.Algebra.jacobson` (`rad(A) := (Ideal.jacobson ⊥ of A_*)^a`) and
`Almost.Algebra.IsHenselianPair` (`I ≤ rad A ∧ HenselianRing A_* (S.ideal • I_*)`); prove
`Almost.Algebra.isTight_span` (`ϖ • A` is tight for `ϖ ∈ S.ideal`), `Almost.Algebra.Ideal.IsTight.mono`
(subideals and powers of tight ideals are tight), `Almost.Algebra.Ideal.IsTight.map` (tightness is
preserved by base change), `Almost.Algebra.le_jacobson_toAlmost_iff`
(`J^a ≤ rad(R^a) ↔ S.ideal • J ≤ Ideal.jacobson ⊥`),
`Almost.Algebra.isHenselianPair_toAlmost_iff` (`IsHenselianPair (R^a) (J^a) ↔ HenselianRing R (S.ideal • J)`), `Almost.Algebra.isHenselianPair_of_isAdicComplete` and
`Almost.Algebra.Ideal.IsTight.elements_le_jacobson` (for `I` tight in `rad(A)`: `I_* ≤ rad(A_*)`, and
`(A_*, I_*)` is henselian if `(A, I)` is, [GR] Corollary 5.1.16). *Needs:* P0.8; P0.10; P0.11;
Mathlib `HenselianRing`, `Ideal.jacobson`, `IsAdicComplete`, `Submodule.FG`.

Two supporting results. The almost Nakayama lemma for tight ideals (*almost-nakayama*): for an
almost algebra `A` and `I ⊆ rad(A)` a tight ideal, (i) if `M` is an almost finitely generated
`A`-module with `IM = M`, then `M = 0` ([GR] §5.1, Lemma 5.1.6, p. 103); (ii) if `φ : N → M` is a
morphism of almost finite projective `A`-modules such that `φ ⊗_A A/I` is an isomorphism, then `φ`
is an isomorphism ([GR] §5.1, Corollary 5.1.7, p. 104); (iii) if `M` is almost finitely generated
and `M ⊗_A A/I = 0` then `M = 0` (Mathlib `Submodule.eq_bot_of_le_smul_of_le_jacobson_bot`). For
`A = K°a` and `I = ϖA`, an almost finitely generated `M` with `M = ϖM` is zero; the finiteness
hypothesis is needed, since `K = K°[1/ϖ]` satisfies `K = ϖK`. Idempotents lift uniquely along tight
henselian pairs (*tight-henselian-idempotent-lifting*): for `(A, I)` a tight henselian pair (e.g.
`A` `I`-adically complete with `I` tight), `A → A/I` induces a bijection from the idempotents of
`A_*` to the idempotents of `(A/I)_*` ([GR] §5.1, Proposition 5.1.17, p. 105; [GR] §2.5, Proposition
2.5.13(i), p. 31, for square-zero extensions; Mathlib `HenselianRing`, `IsIdempotentElem`). The
honest case (`S = classical V`, a henselian pair `(A, J)` or an `J`-adically complete `A`) is Tau
Ceti's `TauCeti.HenselianRing.exists_isIdempotentElem_sub_mem` and
`TauCeti.IsAdicComplete.exists_isIdempotentElem_eq`; the tight almost version is what is new. In the classical setup, for
`A = (V⟦X⟧)^a` and `I = XA` the idempotents of `A_*` and of `(A/I)_* = (V^a)_*` agree; with the rank
decomposition it lifts the decomposition `A/ϖ = ∏ Ā_i` to `A`
in *uniform-finite-projectivity-mod-varpi* (P0.28).

**Checks.**

- `Almost.Algebra.tight_test_principal` (computation) — for `ϖ ∈ S.ideal`, the ideal `ϖA` is tight
  (`m₀ = ϖV`, `n = 1`).
- `Almost.Algebra.IsTight_test_unit` (non-example) — for `S = ofValuationRing K°` and `A = K°a`, the
  unit ideal `A` is not tight: `A ⊆ m₀A` would force `m ⊆ m₀` for a finitely generated (hence
  principal) `m₀ = εK° ⊊ m`.
- `Almost.Algebra.adic_test_zero_ideal` (degenerate) — the zero ideal of any almost algebra is tight.
- `Almost.Algebra.IsHenselianPair_classical` (compatibility) — for `S = classical V`: every
  ideal is tight and `IsHenselianPair (R^a) (J^a) ↔ HenselianRing R J` (Mathlib), since then
  `m·J = J` and `(R^a)_* = R`.

The representative ideal `idealElements S A I` consists of those `x ∈ A_*` for which
`εx ∈ I` for every `ε ∈ m`. Thus `jacobson S A` is the saturation of the honest
Jacobson radical; the defining equation for `IsHenselianPair` uses `m·idealElements S A I`.
These are the representatives in [GR] Definition 5.1.1 and Lemma 5.1.10.

**Checks.**

- `idealElements_classical` — saturation fixes every ideal in the classical setup.
- `idealElements_zero` — the zero ideal of an almost-element ring is saturated.
- `idealElements_scalarIdeal` — saturating `mA_*` gives the unit ideal, including the zero setup.
- `jacobson_classical` — the classical radical is the ordinary Jacobson radical.
- `jacobson_field` — the radical of a classical field is zero.
- `jacobson_dual_numbers` — the nonzero nilpotent `ε` lies in the radical of classical rational dual numbers.
- `IsHenselianPair_zero` — every zero pair is henselian.
- `IsHenselianPair_unit` — if the unit ideal is tight, the unit pair is henselian exactly
  when its almost algebra is zero.
- `IsHenselianPair_valuation_unit` — for a henselian valuation domain with non-finitely-generated
  maximal ideal, the unit almost pair is henselian and nonzero; its unit ideal is not tight.
  Tightness is essential in the preceding Check ([GR] Corollary 5.1.16).
- `IsHenselianPair_classical` — henselianity agrees with Mathlib on ordinary pairs.

### P0.27 almost finite projective modules over complete algebras

**Almost finite projective modules over I-adically complete almost algebras.**
(*adic-almost-projective-modules*) Let `S` be a basic setup, `A` an almost algebra and `I ⊆ A` a
tight ideal with `A → lim_n A/I^{n+1}` an isomorphism, `A_n := A/I^{n+1}` (Gabber–Ramero's
conditions (a), (b) of [GR] Lemma 5.3.5 for the system `(A_n)`; [GR] §5.3, Lemma 5.3.5(ii), p. 112).
Then `M ↦ (M ⊗_A A_n)_n` is an equivalence from almost finite projective `A`-modules to compatible
systems `(M_n)` of almost finite projective `A_n`-modules (`M_{n+1} ⊗ A_n ≅ M_n`), with
quasi-inverse `(M_n) ↦ lim_n M_n` ([GR] §5.3, Theorem 5.3.24, p. 115; [GR] §5.3, (5.3.21), p. 115);
in particular `P ≅ lim_n P ⊗_A A_n` for every almost finite projective `P` ([GR] §5.3, Claim
5.3.25, p. 115). Moreover, if `M = lim_n M_n` for a compatible system of `A_n`-modules with
`M_{n+1} ⊗_{A_{n+1}} A_n ≅ M_n` (an `I`-c-adic module), and `M_0` is almost finitely generated,
then `M` is almost finitely generated and `M ⊗_A A_n ≅ M_n` for every `n`. For `A = K°a` and `I = ϖA`, almost finite
projective `K°a`-modules are the limits of compatible almost finite projective `(K°/ϖ^n)^a`-modules;
P0.28 uses this for the underlying modules of finite étale algebras. *Needs:* P0.26; P0.10; P0.11;
P0.6.

`Module.AdicModule` consists of the modules over `A/I^(n+1)` and adjacent
scalar-extension isomorphisms. Morphisms are compatible families of module maps.
`adicReduction` is quotient scalar extension on objects and morphisms;
`adicReduction_evaluation` fixes each evaluation functor. The equivalence on
almost finite projectives is `adic_almost_projective_equivalence`.
`adic_almost_finitely_generated` gives effectivity from almost finite generation
at level zero, under the same tightness and completeness hypotheses.

**Checks.** For `AdicModule`, `adicReduction`, and `evaluation`, `_zero_ideal`
gives the category of modules over `A`, `_zero` gives the zero module at every
stage, and `_two` gives the free rank-two module over each quotient. The
additional `AdicModule_unit_ideal` Check gives zero modules at every stage.

Supporting result: products, limits, `lim¹`, colimits, Tor and Ext of almost zero modules are
almost zero (*almost-zero-limits-and-colimits*): for a basic setup `S` and any family `(M_i)_{i∈I}`
of almost zero `V`-modules, `∏_i M_i` and `⊕_i M_i` are almost zero; the colimit of any diagram of
almost zero modules is almost zero; for an inverse system `(N_n)_{n∈ℕ}` of almost zero modules,
`lim_n N_n` and `lim¹_n N_n = coker(∏ N_n → ∏ N_n, (x_n) ↦ (x_n − φ_n(x_{n+1})))` are almost zero;
and for `M` almost zero, `Tor_i^V(M, N)`, `Ext^i_V(M, N)` and `Ext^i_V(N, M)` are almost zero for
all `i` and `N`; no finiteness or Mittag-Leffler hypothesis enters, the product being killed
elementwise. Consequently a map of inverse systems whose kernels and cokernels are termwise almost
zero induces almost isomorphisms on `lim` and on `lim¹` ([GR] §2.4, Lemma 2.4.2(i)-(ii), p. 22, the
case of the constant sequence `ε_n = m`). For `S = (𝒪_{ℂ_p}, 𝔪)` and `N_n = k[t]` (`k = V/m`) with
transition maps multiplication by `t`, `lim N_n = 0` and `lim¹ N_n = k⟦t⟧/k[t]`, which is nonzero
and almost zero (a `k`-vector space); with zero transition maps `lim¹` would vanish, the map
`(x_n) ↦ (x_n − φ_n(x_{n+1}))` being the identity, and with identity transition maps too (it is
`(x_n) ↦ (x_n − x_{n+1})`, surjective via `x_n = −Σ_{j<n} y_j`).
*Needs:* P0.5; DerivedModuleAlgebra §1.T.1 / *lim and lim¹ of a tower* and
*The six-term sequence of lim and lim¹*; Mathlib module products, colimits, Tor and Ext.

The ordinary tower is the supplied functor `N : ℕᵒᵖ ⥤ ModuleCat V`, with natural
transformations as morphisms. Use `ModuleTower.Product N`, `ModuleTower.shift N`,
`ModuleTower.lim N` and `ModuleTower.limOne N` directly; the displayed difference is
`ModuleTower.shift`, with the supplier's sign. The almost-isomorphism assertion applies to
the supplied `ModuleTower.limMap` and `ModuleTower.limOneMap`, using its six-term boundary
on the kernel, image and cokernel towers. These are ordinary modules before almostification:
the polynomial example has nonzero ordinary `lim¹` although its almostification is zero.

**Checks.**

- `lim1_test_zero_maps`, `lim1_test_identity_maps` — for zero transitions the difference map is
  the identity; for identity transitions it is surjective, solved by `x_n = −Σ_{j<n} y_j`.
- `lim1_test_sign` — for transitions `·2` on `ℤ` and `(x₀,x₁) = (1,2)`, the first difference is
  `1 − 2·2 = −3`; this fixes the sign in the displayed `lim¹` presentation.
- `lim1_test_polynomial` — transitions `·X` on `k[X]` give a nonsurjective difference map,
  so the `lim¹` term cannot be dropped.

The following Checks use the same supplied tower carrier and maps.

- `tower_test_zero_maps` — the difference map is the identity and both limits are zero.
- `tower_test_identity_maps` — the limit is the constant module and `lim¹` vanishes.
- `tower_test_polynomial` — transitions given by multiplication by `X` on `k[X]` have zero limit and nonzero `lim¹`.
- `tower_test_sign` — with transition multiplication by `2` on `ℤ` and `x_n = n + 1`, the first difference is `−3`.

### P0.28 finite étale lifting modulo a tight ideal

**Finite étale almost algebras lift uniquely modulo a tight ideal of a complete almost algebra.**
(*finite-etale-lifting-along-complete-flat-almost-algebras*) Let `S = (V, m)` be a basic setup
(`m̃` flat, needed for the cotangent complex behind the nilpotent lifting), `A` an almost
`V`-algebra and `I ⊆ A` a tight ideal (e.g. `I = ϖA` with `ϖ ∈ m`) such that `A` is `I`-adically
complete (`A → lim_n A/I^{n+1}` an isomorphism); flatness of `A` over `V^a` is not needed for the
equivalence itself. Then `B ↦ B ⊗_A A/I` is an equivalence of categories `A_fet ≃ (A/I)_fet`, with
quasi-inverse `B₀ ↦ lim_n B_n`, where `B_n` is the unique finite étale `A/I^{n+1}`-algebra lifting
`B₀` ([GR] §5.3, Theorem 5.3.27 with proof, p. 116). In particular, for `ϖ ∈ m` and `A`
`ϖ`-adically complete, `A_fet ≃ (A/ϖ)_fet`; this is [Sch12] Theorem 4.17 over an arbitrary basic
setup ([Sch12] §4, Theorem 4.17 with proof, p. 21, states it for `S = (K°, K°°)` and `A` flat over
`K°a`, citing [GR] Theorem 5.3.27 under the same number). Complements: in the case `I = ϖA`, if `A` is flat over `V^a`
then every `B ∈ A_fet` is flat over `V^a` and `ϖ`-adically complete,
by *finite-etale-lift-flat-and-complete*; for root setups `B` is uniformly finite projective over
`A` iff `B/ϖB` is over `A/ϖ` by *uniform-finite-projectivity-mod-varpi*. For `A = K°a` and
`ϖ ∈ K°°`, `K°a_fet ≃ (K°a/ϖ)_fet` is the middle link in
`K_fet ≃ K°a_fet ≃ (K°a/ϖ)_fet = (K♭°a/ϖ♭)_fet ≃ K♭°a_fet ≃ K♭_fet` ([Sch12] §4); for a finite étale
`(A/ϖ)`-algebra `B₀`, pairs `(B, ι : B/ϖB ≅ B₀)` with `B` finite étale over `A` exist and
are unique up to the unique isomorphism compatible with `ι`. The classical
case `m = V` with `A` `I`-adically complete (so `(A, I)` henselian) is included in
`Almost.Algebra.finiteEtaleBaseChange_isEquivalence`: take `BasicSetup.classical V`
and use `Almost.Algebra.classicalEquiv` to identify ordinary algebras. The theorem
over a general basic setup and tight ideal is owned by P0.28 ([GR] Theorem 5.3.27,
p. 116). *Needs:* P0.14; P0.26; P0.24; P0.27; Mathlib `CategoryTheory.Equivalence`.

Five supporting results. Unramifiedness lifts from the reduction modulo a tight ideal
(*unramified-lifts-from-complete-reduction*): for `A` an almost algebra and `I ⊆ A` tight with `A`
`I`-adically complete, (i) for arbitrary ideals `I₁, I₂` and `φ : A → B`, if `φ ⊗ A/I₁` and `φ ⊗ A/I₂` are
unramified then so is `φ ⊗ A/(I₁I₂)` (no completeness is used in this clause) ([GR] §5.2, Theorem 5.2.12(ii), p. 109); (ii) if `B` is an almost
finite projective `A`-algebra such that `B/IB` is unramified over `A/I`, then `B` is unramified over
`A` ([GR] §5.3, Claim 5.3.29, p. 116); (iii) more generally, for `I ⊆ rad(A)` tight and `B` almost
finite over `A` with `B/IB` unramified over `A/I`, `B` is unramified over `A` ([GR] §5.2, Corollary
5.2.15, p. 110); for `A = K°a` and `B = L°a` (`L/K` finite étale), unramifiedness of `L°a/K°a`
follows from that of `(L°/ϖ)^a` over `(K°/ϖ)^a`. Finite étale algebras over complete flat almost
algebras are flat and complete (*finite-etale-lift-flat-and-complete*): for `ϖ ∈ m`, `A` an almost
algebra which is `ϖ`-adically complete and `B ∈ A_fet`, `B` is `ϖ`-adically complete
(`B ≅ lim_n B/ϖⁿB`), and if `A` is flat over `V^a` then so is `B` ([Sch12] §4, Theorem 4.17, p. 21,
the flatness and completeness assertion); for `A = K°a` and `B = L°a` (`L/K` finite,
`K` perfectoid), `L°a` is flat and `ϖ`-adically complete, as `L°` is. Uniform finite projectivity
of a finite étale algebra can be checked modulo `ϖ` (*uniform-finite-projectivity-mod-varpi*): for
`S` the root setup of `(ϖ_n)` (P0.2; the root setup supplies, for each `k`, elements `ϖ_{k+1}` and
`t = ϖ_{k+1}^{p^{k+1}−2}` with `ϖ = ϖ_{k+1}²·t`), `ϖ = ϖ_0`, `A` an almost algebra which is
`ϖ`-adically complete and `B ∈ A_fet`, `B` is uniformly almost finitely generated over `A` with
bound `r` iff `B/ϖB` is uniformly almost finitely generated over `A/ϖ` with bound `r`; hence `B` is
uniformly finite projective over `A` iff `B ⊗_A A/ϖ` is uniformly finite projective over `A/ϖ`
([Sch12] §4, Theorem 4.17, last sentence, p. 21, over `(K°, K°°)`; the Nakayama step is [GR] §5.1,
Lemma 5.1.6, p. 103); layer P3 uses it for "`S°a` is uniformly finite projective over `R°a`".
`ϖ`-adic completeness of an almost module and of its almost elements
(*almost-completeness-via-almost-elements*): for `ϖ ∈ m` and `M` an almost `V`-module whose almost
elements `M_*` have no `ϖ`-torsion (for `S = (K°, K°°)` this is flatness of `M`, [Sch12] Lemma
5.3(i)), (i) `(ϖⁿM)_* = ϖⁿM_*` for all `n`; (ii) `M_*/ϖⁿM_* → (M/ϖⁿM)_*` is injective with almost
zero cokernel ([Bhatt] §4.4, Lemma 4.4.1(3) with proof, pp. 30–31); (iii) `M` is `ϖ`-adically
complete (`M → lim_n M/ϖⁿM` an isomorphism in `Almost.Module S`) iff `M_*` is `ϖ`-adically complete
(Mathlib `IsAdicComplete (Ideal.span {ϖ}) M_*`); the comparison
`lim M_*/ϖⁿ → lim (M/ϖⁿ)_* = (lim M/ϖⁿ)_*` is injective with almost zero cokernel, the `lim¹` of
the almost zero cokernels being almost zero by *almost-zero-limits-and-colimits* ([Sch12] §5, Lemma
5.3(iv), p. 22; [Bhatt] §4.4, Lemma 4.4.2, p. 31; [Bhatt] §4.4, Proposition 4.4.3, p. 32, the general
setup version via derived completeness, without torsion-freeness; Mathlib `IsAdicComplete`,
`AdicCompletion`). The torsion-freeness hypothesis is not decorative for (i): for
`K = ℚ_p(p^{1/p^∞})^`, `ϖ = p`, and
`M = (K°/ϖ²)^a` (`ϖ`-adically complete, with `M_* = Hom(K°°, K°/ϖ²)` having `ϖ`-torsion),
`(ϖM)_* = Hom(K°°, ϖK°/ϖ²K°) ≅ Hom(K°°, K°/ϖ)` strictly contains `ϖ·M_*`, which is the image of
`Hom(K°°, K°/ϖ²) → Hom(K°°, K°/ϖ)`: the non-honest almost element `(c_n)`, `c_n = Σ_{j<n} ϖ^{1−1/p^j}`, of `elements_test_not_identity` (P0.6) is not in that image, because any
`(d_n) ∈ Hom(K°°, K°/ϖ²) = lim K°/ϖ^{2−1/p^n}` has `d_n ≡ d_0 mod ϖ` for all `n`. For
`M = (K°/ϖ)^a` the two sides of (i) are both `0` and the failure is invisible. Complete torsion-free
complexes that are almost exact modulo `ϖ` are almost exact, with the `lim¹` term controlled
(*almost-exactness-passes-to-completion*): for `ϖ ∈ m` and `C^•` a cochain complex of `V`-modules
(`C^i = 0` for `i < 0`) such that every `C^i` is `ϖ`-torsion-free (for the exactness of
`0 → C/ϖ^{n−1} → C/ϖⁿ → C/ϖ → 0`) and `ϖ`-adically complete (`C^i = lim C^i/ϖⁿ`), if the cohomology
of `C^•/ϖC^•` is almost zero in every degree, then (i) `H^i(C^•/ϖⁿC^•)` is almost zero for all
`n ≥ 1` and all `i`; (ii) for every `i` there is an exact sequence
`0 → lim¹_n H^{i−1}(C^•/ϖⁿ) → H^i(C^•) → lim_n H^i(C^•/ϖⁿ) → 0` (the Milnor sequence of the tower
`C^• = lim_n C^•/ϖⁿ`, whose transition maps are termwise surjective), and both outer terms are
almost zero, so `H^i(C^•)` is almost zero; (iii) `C^•[1/ϖ]` is exact ([ECD] §8, proof of Proposition
8.8, p. 43; [Berkeley] Lecture 8, proof of Proposition 8.2.8, p. 60; [GR] §2.4, Lemma 2.4.2(i)-(ii),
p. 22; Mathlib `CategoryTheory.ShortComplex.ShortExact.homology_exact₂`, `IsAdicComplete`,
`HomologicalComplex.homology`). Neither the torsion-freeness nor the completeness can be dropped,
and the `lim¹` term is not discarded: it is almost zero because every `H^{i−1}(C^•/ϖⁿ)` is. For
`C^0 = V[1/ϖ]` (torsion-free, not `ϖ`-adically separated) and `C^i = 0` otherwise, `C/ϖ = 0` is
exact but `H^0(C) = V[1/ϖ]` is not almost zero; [ECD] Proposition 8.8 applies the lemma to
`0 → R⁺ → S⁺ → O⁺(Y ×_X Y) → ⋯` for a v-cover of a totally disconnected `X`.
Use the native `V`-module Milnor sequence of DerivedModuleAlgebra §1.T.1 on the tower
`(C^•/ϖ^{n+1})_n`, with its supplied injection from `ModuleTower.limOne` of the
degree-`i−1` cohomology tower and projection to `ModuleTower.lim` of the degree-`i`
cohomology tower. The identification `C^• ≅ lim_n C^•/ϖ^{n+1}` respects the canonical
projections, so these are the same Milnor maps, natural in maps of complete complexes.
P0.27 supplies their outer terms' almost annihilation.
*Needs:* P0.27 / *almost-zero-limits-and-colimits*;
DerivedModuleAlgebra §1.T.1 / *The Milnor sequence for towers of complexes*.

**Checks.**

- `completion_test_torsion` — the two-term complex `ℤ/4 → ℤ/2` in degrees `0,1`, with quotient
  differential, is `2`-adically complete and becomes the identity `ℤ/2 → ℤ/2` modulo `2`.
  Its original kernel contains the nonzero class `2`; the torsion-free hypothesis is necessary
  (take the classical setup over `ℤ`).
- `finiteEtale_test_swap` — for a field `k`, the finite étale algebra `k × k` has a nonidentity
  swap, taking `(1,0)` to `(0,1)`. Uniqueness of a lift means uniqueness respecting its specified
  reduction, even for the tight ideal `I = 0`.

The proposed Lean quotient interfaces take an ordinary ideal `I ⊆ A_*` representing the almost
ideal `I^a`; every almost ideal is represented this way by its almost elements.
`Algebra.IsTight A I` means that some finitely generated `m₀ ⊆ m` and some power `I^n`
satisfy `m I^n ⊆ m₀ A_*`, exactly the inclusion after almostification.
`IsNilpotentIdeal` similarly requires `m I^n = 0` for some positive `n`.
`adicSystem` has objects `(A_*/I^{n+1})^a`, with its transition maps pinned by
`adicSystem_map`; `adicMap` is the canonical cone map and `IsAdicComplete` says it is
invertible. `finiteEtaleBaseChange` is the pushout functor, and
`finiteEtaleBaseChange_isEquivalence` states the tight complete reduction theorem.
`nilpotentEtaleEquiv` states the general étale reduction equivalence of P0.24.

- `adic_test_zero_ideal` — zero is tight and nilpotent, every almost algebra is complete for it, and its quotient map is invertible.
- `adic_test_unit_ideal` — the quotient by the unit ideal is zero; nilpotence and completeness at that ideal each force the almost algebra to be zero.
- `tight_test_principal` — the ideal represented by `(ε)`, for `ε ∈ m`, is tight.
- `tight_test_classical` — every ideal is tight in the classical setup.
- `adic_test_dual_numbers` — the ideal generated by `ε` in the classical dual-number algebra is nilpotent, and that algebra is complete for it.
- `adic_test_nilpotent` — an almost nilpotent ideal is tight, the algebra is complete for it, and finite étale base change is an equivalence.

The module-valued reduction functor `modPowerComplex` is extension to `V/(ϖⁿ)` followed
by restriction of scalars. `almostExact_of_complete_mod` applies to nonnegative complexes
whose terms are complete and have injective multiplication by `ϖ`; it gives almost zero
homology at every positive power and before reduction, and exactness after inversion.

**Checks.** `modPowerComplex_zero_exponent` gives the zero complex at exponent zero;
`modPowerComplex_zero_parameter` gives the original complex for the zero parameter and a
positive exponent; `modPowerComplex_unit` gives zero for a unit parameter at every exponent.

### P0.29 almost sheaves and almost cohomology

**Almost sheaves, almost sheaf cohomology and almost Čech complexes on a site.**
(*almost-cech-and-sheaf-cohomology*) Let `(C, J)` be a site whose category of sheaves of
`V`-modules is abelian with enough injectives (Mathlib `sheafIsAbelian`, `HasSheafify`) and
`S = (V, m)` a basic setup. For a sheaf `F` of `V`-modules on `(C, J)` (Mathlib
`Sheaf J (ModuleCat V)`): `F` is almost zero if `m·F(U) = 0` for all `U`; almost zero sheaves form
a Serre class of the abelian category of sheaves, and the Serre localization `Almost.Sheaf J S` is
the category of sheaves of almost `V`-modules (a basic setup relative to the topos of `(C, J)`,
[GR] §3.3, (3.3.1)-(3.3.2) and the introduction, p. 47), with exact localization `F ↦ F^a`. For
`X ∈ C` and `n ≥ 0` the cohomology `H^n(X, F)` of the underlying abelian sheaf (Mathlib
`Sheaf.H'`) is a `V`-module through the action of `V` on `F`, and its almostification
`H^n(X, F)^a ∈ Almost.Module S` depends only on `F^a` (almost zero sheaves have almost zero
cohomology); `F` is almost acyclic on `X` if `H^n(X, F)^a = 0` for `n > 0`. For a family
`𝔘 = (U_i → X)` the Čech complex `Č(𝔘, F)` (Mathlib `cechComplexFunctor` on the underlying presheaf)
is a complex of `V`-modules with almostification `Č(𝔘, F)^a`, and the augmented complex
`F(X) → Č(𝔘, F)` is almost exact if its cohomology is almost zero ([Berkeley] Lecture 7, proof of
Theorem 7.4.4, p. 53). The Čech-to-derived comparison for almost acyclic bases is
*almost-cech-acyclicity-criterion*. Define `Almost.Sheaf.IsAlmostZero` (`∀ U, S.IsAlmostZero (F.val.obj U)`), `Almost.Sheaf` (the Serre localization of `Sheaf J (ModuleCat V)` at almost
isomorphisms of sheaves), `Almost.Sheaf.toAlmost` (the exact localization `F ↦ F^a`),
`Almost.Sheaf.cohomologyModule` (the `V`-module structure on `H^n(X, F)`), `Almost.Sheaf.almostH`
(`H^n(X, F)^a := (toAlmost S).obj (H^n(X, F))`), `Almost.Sheaf.IsAlmostAcyclic` (`∀ n > 0, IsZero (almostH F n X)`) and `Almost.Sheaf.almostCech` (`(toAlmost S).mapHomologicalComplex` of the Čech
complex); prove that almost zero sheaves form a Serre class, `Almost.Sheaf.almostH_eq_zero_of_isAlmostZero`
(almost zero sheaves have almost zero cohomology in all degrees; `almostH` factors through
`toAlmost`) and `Almost.Sheaf.almostH_localization` (for `ϖ ∈ S.ideal`: `almostH F n X ≅ 0 ⇒ H^n(X, F) ⊗ V[1/ϖ] = 0`, P0.16). *Needs:* P0.5; P0.16; Mathlib `CategoryTheory.Sheaf.H'`,
`CategoryTheory.Sheaf.H.map`, `CategoryTheory.cechComplexFunctor`, `CategoryTheory.sheafIsAbelian`,
`SerreClassLocalization.abelian`.

Supporting result: almost Čech acyclicity on a basis implies almost acyclicity
(*almost-cech-acyclicity-criterion*). Let `F` be a sheaf of `V`-modules and `𝔅` a full subcategory
of `C` stable under fibre products such that every covering of an object of `𝔅` is refined by a
covering by objects of `𝔅` (e.g. rational subsets of an affinoid adic space with finite rational
covers, or affinoid perfectoid objects of a pro-étale or v-site). If for every `U ∈ 𝔅` and every
covering `𝔘` of `U` by objects of `𝔅` the augmented Čech complex `F(U) → Č(𝔘, F)` is almost exact,
then `H^n(U, F)` is almost zero for all `U ∈ 𝔅` and `n > 0` (and `H^0(U, F) = F(U)`) ([ECD] §8,
proof of Proposition 8.8, printed p. 43 in
[v4](https://arxiv.org/pdf/1709.07343v4), for the affinoid perfectoid v-site; [Berkeley]
Lecture 7, proof of Theorem 7.4.4, p. 53). [Berkeley] Theorem 7.4.4 and [Sch12] Proposition 6.14
obtain `H^i(X, O⁺)^a = 0` for `i > 0` on a perfectoid affinoid this way.
`Almost.Sheaf.almostCech_acyclicity_criterion` states the general basis criterion.
Induct on the cohomological degree: a class vanishes on a covering, the basis hypothesis
refines that covering, and the Čech-to-cohomology spectral sequence reduces to lower
positive degrees and positive Čech cohomology. Products and finite extensions preserve
almost-zero modules because `m² = m`. For the unit ideal this gives Cartan's criterion;
DiamondsAndVStacks D0 uses this specialization.

**Checks.**

- `Almost.Sheaf.test_point` (degenerate) — for the site on the terminal category (one object, only its identity arrow) with the trivial
  covering, `Almost.Sheaf J S ≌ Almost.Module S` and `almostH F n X = 0` for `n > 0`.
- `Almost.Sheaf.test_constant_residue` (computation) — for `S = ofValuationRing K°` and the constant
  sheaf `k = K°/K°°` on any site, `F` is almost zero, so `almostH F n X ≅ 0` for all `n`, even when
  `H^n(X, k) ≠ 0`.
- `Almost.Sheaf.test_not_acyclic` (non-example) — on the site of open subsets of the circle `S¹`,
  the constant sheaf `k` has `H¹(S¹, k) = k ≠ 0` but is almost acyclic: `IsAlmostAcyclic` must not
  be defined as vanishing of `H^n`.
- `Almost.Sheaf.test_classical` (compatibility) — for `S = classical V`, `almostH F n X` is
  Mathlib's `Sheaf.H'` of the underlying abelian sheaf with its `V`-module structure, via
  `Almost.Module.classicalEquiv`.

- Sectionwise annihilation is the defining equation, the classical setup detects the zero
  sheaf, and the zero setup kills every sheaf. These also test `almostZeroSheaf`.
- `toAlmost_zero_iff`, `toAlmost_classical`, `toAlmost_residue` — the quotient functor kills
  exactly the almost zero sheaves, is an equivalence classically, and kills sheaves whose
  sections are annihilated by the setup ideal. These characterize the Serre quotient `Sheaf`.
- `cohomologyModule_injective`, `cohomologyModule_zero_sheaf`, `cohomologyModule_sections` —
  positive derived sections vanish for injectives, all degrees vanish for the zero sheaf,
  and degree zero is nonzero when the sections are nonzero. The functor `sections` is
  evaluation of the underlying presheaf; `cohomologyModule` is its right derived functor.
- `almostH_zero_setup`, `almostH_classical`, `almostH_residue` — almost cohomology vanishes
  in the zero setup, detects ordinary cohomology classically, and vanishes in every degree
  for sectionwise almost zero coefficients.
- `IsAlmostAcyclic_injective`, `IsAlmostAcyclic_zero_setup`, `IsAlmostAcyclic_classical` —
  injectives are almost acyclic, every sheaf is almost acyclic in the zero setup, and the
  classical setup requires ordinary vanishing in positive degrees.
- `almostCech_empty`, `almostCech_zero_setup`, `almostCech_classical` — an empty family
  gives the zero complex, the zero setup kills every term, and classical almostification
  detects the cohomology of Mathlib's Čech complex. On a family over `X`, use the same
  construction on the slice category over `X`, so products are fibre products over `X`.


### Examples

The worked examples of this layer are the checks above, which run through three setups: the
classical setup `(V, V)`, where almostification is an equivalence and every almost notion is the
Mathlib notion; the zero setup `(V, ⊥)`, where every module is almost zero; and the valuation
setups `(𝒪_{ℂ_p}, 𝔪)` and `(K°, K°°)` for `K` the completion of `ℚ_p(p^{1/p^∞})`, together with the
root setup `𝔽_p[t^{1/p^∞}]`. On the valuation setups the residue field is almost zero while `V/pV`
is not; `((K°°)^a)_* = K°` while `((K°/ϖ)^a)_* = Hom(K°°, K°/ϖ)` strictly contains `K°/ϖ`; `V^a` is
almost projective but not a projective object; and the quadratic extension `L = K(√p)`, `p ≠ 2`,
gives `L°a` uniformly almost finitely generated with bound `2`, almost finitely presented, finite
étale over `K°a` with `Tr(1) = 2` and perfect trace form, with a diagonal idempotent that is an
almost element but not an element of `L° ⊗_{K°} L°`, and with rank decomposition `r = 3`, `A = A_2`.
The non-examples are the discrete valuation ring `ℤ_p` (`m·m ≠ m`, `ℤ/p²` breaks closure under
extensions), the ideal `(t) ⊆ 𝔽_p[t]` without roots (which also satisfies (A) but not (B)), the
field, whose zero maximal ideal is idempotent and finitely generated, the root setup with
`ϖ_0 = 0` but `ϖ_1 ≠ 0`, the dual numbers `A[ε]/(ε²)` whose trace form `[[2, 0], [0, 0]]` is not
perfect, the module `V` which is not firm when `m ≠ V`, the proper ideal
`m` which is not a tensor unit before almostification, the infinite sum `⊕_ℕ K°` which is not almost
finitely generated, `V ×_k V → V` which shows `(−)_!!` does not preserve flatness, the non-faithful
functor `invert ϖ`, the naive cotangent complex of `A_* → B_*`, the unit ideal of `K°a` which is not tight,
the complex `V[1/ϖ]` which is almost exact modulo `ϖ` but not almost exact, and the constant sheaf
`k` on `S¹` which is almost acyclic with `H¹ ≠ 0`.

The two proved Lean examples for this layer show that the zero setup kills every module and
that its unit comparison `m̃ → V` fails to be surjective when `V` is nontrivial (P0.7). Named
comparison targets identify classical almost isomorphisms with bijections, classical almost
finite presentation with finite presentation, and classical almost finite projectivity with
finite projectivity. The other Checks above are specified in prose.

### Dependencies

Mathlib as listed in the supplier contracts: `IsIdempotentElem`, `Module.Flat`,
`Module.FaithfullyFlat`, `Module.Projective`, `Module.Finite`, `Module.FinitePresentation`,
`ValuationRing`, `IsBezout`, `IsSMulRegular`, the Serre-class localisation
(`CategoryTheory.ObjectProperty.isoModSerre`, `SerreClassLocalization.*`,
`Abelian.isLocalization_isoModSerre_kernel_of_leftAdjoint`, `MorphismProperty.HasLocalization`,
`MorphismProperty.Localization'`), `LocalizedMonoidal`, `ModuleCat.monoidalCategory`,
`ModuleCat.monoidalClosedHomEquiv`, `CategoryTheory.CommMon`, `CategoryTheory.Mod`,
`CategoryTheory.Abelian.Ext`, `LinearMap.trace`, `Algebra.trace`, `Algebra.traceForm`,
`Module.Invertible`, `ExteriorAlgebra.exteriorPower`, `Algebra.Etale`,
`Algebra.FormallyUnramified`, `Ideal.isIdempotentElem_iff_of_fg`, `LocalizedModule`,
`Localization.Away`, `KaehlerDifferential`, `Derivation`, `Algebra.Extension.Cotangent`,
`Algebra.Extension.cotangentComplex`, `Algebra.H1Cotangent`, `CategoryTheory.SimplicialObject`,
`DerivedCategory`, `comonadicExtendScalars`, `ModuleCat.extendScalars`,
`CategoryTheory.Pseudofunctor.DescentData`, `CommRingCat.isLimitForkPushoutSelfOfFaithfullyFlat`,
`HenselianRing`, `Ideal.jacobson`, `IsAdicComplete`, `AdicCompletion`,
`Submodule.eq_bot_of_le_smul_of_le_jacobson_bot`, `CategoryTheory.Sheaf.H'`,
`CategoryTheory.cechComplexFunctor` and `CategoryTheory.sheafIsAbelian`. The layer
is consumed by layers P1–P3 (the setups of a perfectoid Tate ring, the
almost cotangent complex of perfectoid algebras modulo `ϖ`, almost purity). The classical complete-pair case of P0.28 uses `BasicSetup.classical` and
`Almost.Algebra.classicalEquiv`. Other roadmaps, by subsection: P0.19 uses
DerivedModuleAlgebra §§0.1–0.2, 0.4–0.11, 0.13 and 0.25 for polynomial resolutions,
derived tensor products, the full ordinary cotangent complex and its square-zero deformation theory;
P0.20–P0.24 use this supplier through P0.19, with P0.22 using §0.25's normalized laws.
P0.27–P0.28 use DerivedModuleAlgebra §1.T.1's ordinary module towers, six-term sequence
and native module Milnor maps for almost annihilation and passage to completion.
P0.12 consumes PivotalSpherical §Layer 2's trace construction, cyclicity
and tensor law, its §Layer 7 standard symmetric pivotal/spherical structure, and AdicSpacesPartIII
§R3.8's *projective-module-trace* for the ordinary finite-free comparison. Tau Ceti:
`TauCeti.HenselianRing.exists_isIdempotentElem_sub_mem` and
`TauCeti.IsAdicComplete.exists_isIdempotentElem_eq` (the honest case of the idempotent lifting in
P0.26).

## Layer P1: perfectoid Tate rings, tilts and untilts

Here `ℚ_p^cycl` is the completion of `⋃_n ℚ_p(ζ_{p^n})`, and `ℤ_p^cycl` is its valuation ring
(P1.31). A trailing `^` on a valued field denotes its completion.

The layer defines perfectoid Tate rings and perfectoid fields with their characterisations, the
tilt with its integral subrings and the sharp map, the almost setup `(R°, R°°)` of a perfectoid
Tate ring, Fontaine's `θ` with its primitive kernel, untilts and their classification by primitive
ideals of degree one, integral perfectoid rings and the integral/Tate dictionary, the tilting
equivalence over an arbitrary perfectoid base through primitive ideals, and the mod-`ϖ` equivalence over a perfectoid field, completed filtered
colimits, perfected Tate algebras and the standard examples. No perfectoid base field is fixed
for the ring-level theory; the field results explicitly assume a complete nontrivially normed
ultrametric field. The zero Tate ring is allowed. Here “of characteristic `p`” means `pR = 0`
when zero rings are allowed; Lean’s `CharP R p` asserts characteristic exactly `p` and therefore
requires `R ≠ 0`. In particular `tilt.charP` has a `Nontrivial R` hypothesis, while the tilt of
the zero ring is zero (characteristic one). Statements using Mathlib’s `PreTilt` or
`fontaineTheta` assume the integral ring is nonzero; their zero-ring cases use the unique map.

<a id="p1-1"></a>

### P1.1 perfectoid Tate rings, perfectoid algebras and perfectoid Tate pairs

The layer opens with the perfectoid Tate ring itself, its characteristic-`p` form and the
pseudo-uniformizers with compatible `p`-power roots that every perfectoid Tate ring has.

**Perfectoid Tate rings, perfectoid algebras and perfectoid Tate pairs.**
(*perfectoid-tate-rings-and-algebras*) Fix a prime `p`. Let `R` be a commutative topological ring
which is a Tate ring (Tau Ceti `TauCeti.Huber.IsTateRing`), complete and Hausdorff (`CompleteSpace R`
and `T2Space R` for the uniform structure of the additive group), with power-bounded subring `R°`
(`TauCeti.Huber.powerBoundedSubring`). For a commutative ring `A` and ideals `I, J ⊆ A` with `p ∈ J`
and `x^p ∈ J` for every `x ∈ I`, write `Φ : A/I → A/J` for the ring homomorphism
`x mod I ↦ x^p mod J`. `R` is a perfectoid Tate ring if (a) `R` is uniform, i.e. `R°` is bounded in
`R` (`TauCeti.Huber.IsBounded`; the uniformity of AdicSpaces layer 4.2, [HK] Definition 2.3), and
(b) there is a pseudo-uniformizer `ϖ` of `R` (a topologically nilpotent unit,
`TauCeti.Huber.IsPseudoUniformizer`; necessarily `ϖ ∈ R°`) such that `ϖ^p` divides `p` in `R°`, not
merely in `R`, and `Φ : R°/ϖR° → R°/ϖ^pR°` is bijective. No perfectoid base field is fixed;
characteristic `0` and characteristic `p` are both allowed (for `pR = 0` the condition `ϖ^p | p` is
vacuous), and `p` may be a nonzero nonunit (as in a product `K × K♭`). Uniformity is not implied by
(b), and (b) with the divisibility cannot be replaced by surjectivity of Frobenius on `R°/p` alone
(`ℚ_p`). A perfectoid `R`-algebra is a perfectoid Tate ring `S` with a continuous ring homomorphism
`R → S`; morphisms are continuous `R`-algebra homomorphisms. A perfectoid Tate pair is a Huber pair
`(R, R⁺)` (`TauCeti.Huber.Pair`) with `R` perfectoid; the condition depends only on `R`
(*frobenius-surjective-on-plus-ring*, P1.17) ([ECD] §3, Definition 3.1, p. 14; [Berkeley] Lecture 6,
Definition 6.1.1, p. 41; [KL16] §3.3, Definition 3.3.1, p. 58; [AWS] §2.1, Definition 2.1.1, p. 49;
[BMS] §3.2, before Lemma 3.20, p. 26). In `Perfectoid`, define `IsPerfectoidTateRing` (the Prop-valued class on a complete
Hausdorff nonarchimedean Tate ring with fields `isBounded_powerBoundedSubring` and
`exists_pseudoUniformizer`); the second field states (b) elementwise: some `ϖ ∈ R°` is a
pseudo-uniformizer with `ϖ^p ∣ p` in `R°`, every `x ∈ R°` is a `p`-th power modulo `ϖ^p` (`Φ` is
surjective), and `ϖ^p ∣ y^p` forces `ϖ ∣ y` (`Φ` is injective). The ring homomorphisms packaging
`Φ` are `frobeniusQuot` (the ring homomorphism
`A ⧸ I →+* A ⧸ J`, `[x] ↦ [x^p]`, for a commutative ring `A`, prime `p`,
`p ∈ J`, and `x^p ∈ J` for every `x ∈ I`) and `frobeniusModPow`
(the case `I = (ϖ)`, `J = (ϖ^p)`, requiring `ϖ^p ∣ p` in `A`), with the simp rule `frobeniusQuot_mk`
(`frobeniusQuot I J _ _ (mk I x) = mk J (x ^ p)`); for a Huber pair use `IsPerfectoidTateRing p R`
directly. Prove `IsPerfectoidTateRing.of_surjective` (surjectivity of `Φ : R°/ϖ → R°/ϖ^p` suffices,
injectivity being automatic), that `Φ` is bijective for every
pseudo-uniformizer with `ϖ^p | p` in `R°`, `IsPerfectoidTateRing.iff_frobenius_surjective_modP`
(for `R` uniform: perfectoid iff some `ϖ` has `ϖ^p | p` in `R°` and Frobenius on `R°/p` is
surjective; uniformity cannot be dropped, since `ℂ_p[ε]/(ε²)` satisfies the right-hand side with
`ϖ = p^{1/p}` and is not perfectoid),
`IsPerfectoidTateRing.isReduced` (the perfectoid case of `TauCeti.Huber.IsUniform.isReduced`, the
supplied theorem for uniform Hausdorff Tate rings), `IsPerfectoidTateRing.iff_perfectRing` (for `CharP R p`:
perfectoid iff `PerfectRing R p`, P1.2), transport of the predicate along a
ring isomorphism that is a homeomorphism, via `TauCeti.Huber.map_powerBoundedSubring`,
`IsPerfectoidTateRing.prod` (`R × S` is perfectoid when `R`, `S` are: `(R × S)° = R° × S°` and
`(ϖ_R, ϖ_S)` is a pseudo-uniformizer; it rests on the instance `IsTateRing.instProd` making
`R × S` a Tate ring), `map_mem_powerBoundedSubring` (a continuous ring homomorphism
`f : A → S` from a Tate ring to a uniform Tate ring has `f(A°) ⊆ S°`, since `{x^N} ⊆ ϖ^{-k}A_0` for
some `k` and a ring of definition `A_0`, so `{f(x)^N}` lies in the bounded set `f(ϖ)^{-k}S°`),
and surjectivity of Frobenius on `R⁺/p` for every ring of integral elements. Check `IsPerfectoidTateRing.isUniform`
applies the field `IsPerfectoidTateRing.isBounded_powerBoundedSubring` to construct
`TauCeti.Huber.IsUniform`: condition (a) is the supplied uniformity predicate
([ECD] Definition 3.1, p. 14). Uniform Hausdorff Tate rings are reduced
(*perfectoid-tate-ring-is-reduced*): a nonzero nilpotent `x` gives `Rx ⊆ R°` (every `rx` is
nilpotent), and `Rx ∋ ϖ^{-n}x` for all `n` is unbounded (`ϖ^n·ϖ^{-n}x = x ↛ 0`), contradicting
boundedness, so every perfectoid Tate ring is reduced, while `ℂ_p[ε]/(ε²)` is not reduced, hence
not uniform ([Sch12] §5, Proposition 5.8, proof, pp. 23–24; [Berkeley] Proposition 6.1.6, proof,
p. 43). *Needs:* Tau Ceti `TauCeti.Huber.IsTateRing`, `TauCeti.Huber.IsHuberRing`,
`TauCeti.Huber.IsPseudoUniformizer`, `TauCeti.Huber.powerBoundedSubring`, `TauCeti.Huber.IsBounded`,
`TauCeti.Huber.Pair`, `TauCeti.Huber.IsRingOfIntegralElements.mem_of_isTopologicallyNilpotent`,
`TauCeti.Huber.map_powerBoundedSubring`; Mathlib `CompleteSpace`, `T2Space`, `Ideal.Quotient.mk`,
`IsReduced`.

**Checks.**

- `perfectoid_padic_negative` (non-example) — `¬ IsPerfectoidTateRing ℚ_[p]`: a
  pseudo-uniformizer of `ℚ_p` has `|ϖ| = p^{-k}` with `k ≥ 1`, so `|ϖ^p| < |p|` and `ϖ^p ∤ p` in
  `ℤ_p`, although Frobenius on `ℤ_p/p = 𝔽_p` is bijective ([ECD] Remark 3.3).
- Non-example: `R = ℂ_p[ε]/(ε²)` (product topology on
  `ℂ_p ⊕ ℂ_pε`) is a complete Tate ring with `ϖ = p^{1/p}`, `ϖ^p = p`, and Frobenius surjective on
  `R°/p = 𝒪_{ℂ_p}/p` (as `ℂ_pε ⊆ R°` and `ℂ_pε = p·ℂ_pε`), but `R° ⊇ ℂ_pε` is unbounded, so `R` is
  not perfectoid; a definition omitting uniformity would accept it.
- `perfectoid_zero` (degenerate) — the zero ring is a perfectoid Tate ring (`0` is a
  topologically nilpotent unit and all quotients vanish).
- Computation: for `ℂ_p` (Mathlib `PadicComplex p`) with
  `ϖ = p^{1/p}`: `ϖ^p = p` and `Φ : 𝒪/ϖ → 𝒪/p` is bijective because every element of `𝒪_{ℂ_p}` has
  a `p`-th root.
- Characterisation: for the `t`-adic completion `R`
  of the perfection of `𝔽_p((t))`: `IsPerfectoidTateRing R ↔ PerfectRing R p`, and both hold; for
  `𝔽_p((t))` itself (`TauCeti.Huber.LaurentSeries.isTateRing`) both fail.

For prime `p` and a complete Hausdorff Tate ring `R` with `UniformSpace R`,
`IsUniformAddGroup R` and `IsTopologicalRing R`, the predicate `IsPerfectoidTateRing p R` is
equivalent to the Frobenius-modulo-`p` formulation that P3.17a's
sousperfectoid frames use; both use P1.1's predicate itself, so the equivalence is the criterion
`IsPerfectoidTateRing.iff_frobenius_surjective_modP` together with the boundedness field.
Both sides require bounded `R°`, a pseudouniformizer with `ϖ^p ∣ p` in `R°`,
and the Frobenius condition of [ECD] Remark 3.2, p. 14.
`IsUniformAddGroup.rightUniformSpace_eq` identifies the completeness structures.
No nonzero-ring or base-field hypothesis is imposed.

**Checks.** Forward: a P1.1 instance gives bounded `R°` and the modulo-`p` criterion; reverse:
bounded `R°` with the modulo-`p` criterion recovers the P1.1 predicate; nonuniform: an unbounded
power-bounded subring is rejected, even if Frobenius modulo `p` is surjective. The equivalence
includes the zero ring tested by `perfectoid_zero` above.

The ring of [ECD] Example 3.4(iv), p. 15 is explicit. Starting with
`CyclotomicToricPolynomial`, the monoid algebra over the power-bounded ring of
`padicCyclotomic` on the two systems of nonnegative `p`-power exponents, form
`CyclotomicToricQuotient` by `TU = p`. Complete at `T` to obtain
`CyclotomicToricIntegral`, then invert `T` to obtain `CyclotomicToricRing`.
The monomials `cyclotomicToricMonomial`, their parameter `cyclotomicToricParameter`,
its completion `cyclotomicToricCompletedParameter`, and `cyclotomicToricRoot`
are given by the quotient, completion, and localization maps.

**Checks.** For each of these eight definitions, `_roots` checks both coherent
root systems, `_relation` checks `TU = p` and invertibility of `T`, and
`_nonfield` checks that `p` is not invertible while `T^(1/p)` is a
pseudo-uniformizer. The affinoid space is `IsAffinoidPerfectoid.spa_ecd34iv`.

Lean: `Perfectoid.IsPerfectoidTateRing`, `IsPerfectoidTateRing.of_surjective`,
`IsPerfectoidTateRing.iff_frobenius_surjective_modP`, `IsPerfectoidTateRing.isUniform`,
`IsPerfectoidTateRing.isReduced`, `IsPerfectoidTateRing.prod`, `IsTateRing.instProd`, and
`TauCeti.Huber.IsTateRing.of_nontriviallyNormedField` (the Huber/Normed compatibility signature:
a nontrivially normed ultrametric field is a Tate ring,
with ring of definition the closed unit ball);
`Perfectoid.not_isPerfectoidTateRing_padic` and `Perfectoid.isPerfectoidTateRing_of_subsingleton`
state the `ℚ_p` and zero-ring boundaries.

### P1.2 characteristic p: perfectoid equals perfect complete Tate

**In characteristic p, perfectoid Tate rings are exactly the perfect complete Tate rings.**
(*characteristic-p-perfectoid-iff-perfect*) Let `R` be a topological ring with `pR = 0`. Then `R`
is a perfectoid Tate ring iff `R` is a perfect complete Hausdorff Tate ring; for such `R`, `R°` and
every ring of integral elements `R⁺` are perfect, `Φ : R° → R°` is an automorphism, and every
pseudo-uniformizer satisfies the condition of P1.1, `ϖ^p | p = 0` being vacuous ([ECD] §3,
Proposition 3.5 and its proof, p. 15; [KL16] Remark 3.3.4, p. 59; [AWS] Example 2.1.2, p. 49;
[Sch12] Proposition 5.9, p. 24; [Bhatt] Example 3.1.2(3), p. 10). Hence the completed perfection of
`𝔽_p((t))` is a perfectoid field, `𝔽_p((t))` (a complete Tate ring,
`TauCeti.Huber.LaurentSeries.isTateRing`) is not perfectoid, and a nonarchimedean field of
characteristic `p` is a perfectoid field iff it is perfect. The Lean form
`IsPerfectoidTateRing.iff_perfectRing` is stated on a complete Hausdorff Tate ring `R` with
`CharP R p`, where it reads `IsPerfectoidTateRing p R ↔ PerfectRing R p`; the general topological
ring `R` of the prose is the special case once `R` is known to be complete Hausdorff Tate. *Needs:* P1.1;
*integral-elements-adically-complete*; *perfectoid-condition-independent-of-pseudouniformizer*;
*perfect-complete-tate-ring-is-uniform*; Mathlib `PerfectRing`, `frobeniusEquiv`, `CharP`; Tau Ceti
`TauCeti.Huber.LaurentSeries.isTateRing`.

Three supporting results. Rings of integral elements are adically complete
(*integral-elements-adically-complete*): let `R` be a complete Hausdorff uniform Tate ring,
`R⁺ ⊆ R°` a ring of integral elements (`TauCeti.Huber.IsRingOfIntegralElements`; `R⁺ = R°` allowed)
and `ϖ ∈ R⁺` a pseudo-uniformizer. Clause (a) states that the subspace topology of `R⁺` is the
`ϖ`-adic topology and `R⁺` is `ϖ`-adically complete and separated
(`IsAdicComplete (span {ϖ}) R⁺`). Uniformity makes `R°` bounded; since `R⁺ ⊆ R°` is open,
it is an open bounded subring, hence a ring of definition by
`TauCeti.Huber.exists_pairOfDefinition_ringOfDefinition_eq_iff`.
Apply AdicEtaleGeometry A1.0.2(i)'s `PairOfDefinition.isAdicComplete_of_completeSpace`
(*power-bounded-henselian-along-pseudouniformizer*) to this ring of definition and `ϖ`.
The general topology and completeness theorem is supplied there; this corollary uses the
boundedness/ring-of-definition bridge here.
The complements proved here: (b) `ϖ` is a nonzerodivisor of `R⁺` and `R = R⁺[1/ϖ]`; (c) if `ϖ | p`
in `R⁺`, then `R⁺` is `p`-adically complete and separated; (d) if `R ≠ 0` and `p` is topologically
nilpotent in `R`, then `p` is not a unit of `R⁺`. So for `R ≠ 0` perfectoid, `R⁺` satisfies the
hypotheses `Fact ¬IsUnit (p : R⁺)` and `IsAdicComplete (span {p}) R⁺` of Mathlib's `PreTilt.untilt`
and `WittVector.fontaineTheta` ([Stacks] Tag 090T (Lemma 10.96.8); [BMS] §3.1, before Definition
3.1, p. 19; [Wedhorn] Proposition 6.14, p. 49).
*Needs:* AdicEtaleGeometry A1.0.2(i); Tau Ceti
`IsUniform.isBounded_powerBoundedSubring`,
`exists_pairOfDefinition_ringOfDefinition_eq_iff`; Mathlib `IsAdicComplete`.

**Checks.**

- For `R = ℚ_p`, `R⁺ = ℤ_p` and `ϖ = p`, clause (a) gives `p`-adic completeness,
  (b) gives regularity and `ℚ_p = ℤ_p[1/p]`, and (d) excludes `p` as a unit of `ℤ_p`.
- For the zero ring all completeness and localization assertions hold, while `p` is a unit;
  clause (d) requires `R ≠ 0`.
- For the complete nonuniform Tate ring `ℂ_p⟨T⟩/(T²)`, its ring `R°` of integral elements
  contains every scalar multiple of `T`. Thus `T ∈ ⋂_n ϖⁿR°` for a scalar
  pseudo-uniformizer `ϖ`, and `R°` is not `ϖ`-adically separated; the boundedness bridge
  requires uniformity (AdicEtaleGeometry A1.0.2).

Injectivity of Frobenius is automatic and the perfectoid
condition is independent of the pseudo-uniformizer
(*perfectoid-condition-independent-of-pseudouniformizer*): for a complete Hausdorff Tate ring `R`,
(a) for every pseudo-uniformizer `ϖ` with `ϖ^p | p` in `R°`, `Φ : R°/ϖ → R°/ϖ^p` is injective;
(b) if `R` is uniform and `ϖ, ϖ'` are pseudo-uniformizers with `ϖ^p | p` and `ϖ'^p | p` in `R°`,
`Φ` is bijective for `ϖ` iff it is for `ϖ'`; equivalently `R` is perfectoid iff some `ϖ` has
`ϖ^p | p` in `R°` and Frobenius on `R°/p` is surjective (the surjectivity forms are
*frobenius-surjectivity-criterion*, P1.5), and then `Φ` is bijective for every such
`ϖ`; (c) the same for a ring of integral elements `R⁺`: if `ϖ ∈ R⁺`, `ϖ^p | p` in `R⁺` and
`x ∈ R⁺` with `x^p ∈ ϖ^pR⁺`, then `x ∈ ϖR⁺`. Parts (a) and (c) use only that `R°` (resp. `R⁺`) is
integrally closed in `R` and that `ϖ` is a unit of `R` (`TauCeti.Huber.isPowerBounded_of_isIntegral`,
`TauCeti.Huber.isIntegrallyClosedIn_powerBoundedSubring`); (b) uses uniformity and completeness
through *frobenius-surjectivity-criterion* ([ECD] §3, Remark 3.2, p. 14; [Berkeley] Remark 6.1.3,
p. 42; [BMS] Lemma 3.20, proof, p. 26; [Sch12] Proposition 5.5, proof, p. 23). In `ℚ_p^cycl`,
`ϖ = ζ_{p²} − 1` and `ϖ' = ζ_{p³} − 1` both satisfy `ϖ^p | p` and `Φ` is bijective for both; in
`ℂ_p` the conditions for `p^{1/p}` and `p^{1/p²}` coincide. A perfect complete Tate ring of
characteristic `p` is uniform (*perfect-complete-tate-ring-is-uniform*): for `R` complete,
Hausdorff and Tate with `pR = 0` and `PerfectRing R p`, `R°` is bounded, by the open mapping theorem
(`TauCeti.Huber.IsTateRing.isOpenMap`, `TauCeti.Huber.IsTateRing.exists_hasBasis_nhds_zero`) ([ECD]
Proposition 3.5, proof, p. 15; [Berkeley] Proposition 6.1.6, proof, p. 43; [KL16] Remark 3.0.3,
p. 54; [AWS] Example 2.1.2, p. 49). The `t`-adic completion of the perfection of `𝔽_p((t))` is
uniform with `R°` the completion of `𝔽_p[[t]]^{perf}`, while the non-uniform `𝔽_p((t))[ε]/(ε²)`
is not perfect (`ε` is nilpotent), consistently with the lemma.

Lean: `IsPerfectoidTateRing.iff_perfectRing`.

### P1.3 pseudo-uniformizers with compatible p-power roots and the spectral gauge

**Pseudo-uniformizers with compatible p-power roots and the spectral gauge.**
(*pseudouniformizer-with-compatible-roots*) Let `R` be a Tate ring. A pseudo-uniformizer
with compatible `p`-power roots is an element `ϖ♭ = (ϖ^{1/p^n})_{n≥0}` of the multiplicative monoid
`lim_{x↦x^p} R` (Mathlib `Perfection R p`) such that `ϖ := ϖ^{1/p^0}` is a pseudo-uniformizer of
`R` lying in `R°` with `ϖ^p | p` in `R°`. Then every `ϖ^{1/p^n}` is a pseudo-uniformizer in `R°`,
`ϖ^q := (ϖ^{1/p^n})^m` is well defined for `q = m/p^n ∈ ℤ[1/p]_{≥0}` (and for `q ∈ ℤ[1/p]` as a
unit of `R`), and, when `R` is perfectoid, `ϖ♭` is a pseudo-uniformizer of the tilt `R♭` with `(ϖ♭)♯ = ϖ`
(*tilt-integral-subrings*, P1.5). The gauge formula is defined for any such Tate ring; its norm properties require
`R` to be complete, Hausdorff and uniform. Attached to `ϖ♭` is the spectral gauge
`|x|_ϖ := inf{2^{-q} : q ∈ ℤ[1/p], ϖ^{-q}x ∈ R°}`, a power-multiplicative, submultiplicative,
ultrametric norm defining the topology of a nonzero `R`, with `|ϖ^qx|_ϖ = 2^{-q}|x|_ϖ`,
`R° = {x : |x|_ϖ ≤ 1}` and `R°° = {x : |x|_ϖ < 1}`; the normalisation `|ϖ|_ϖ = 1/2` is a
convention for nonzero `R`. On the zero ring the same infimum is zero; power-multiplicativity holds
there for positive exponents, and its exponent-zero form requires `R ≠ 0`. Changing `ϖ` gives an equivalent norm; it need not give a single real power of the old norm on a ring with several factors. The gauge is the spectral seminorm of
AdicSpacesPartII R0.4 (`Huber.spectralSeminorm`, Mathlib's `smoothingSeminorm`) applied to
its Tate norm datum `(R°, ϖ, 1/2)` (`Huber.TateNormDatum`). The compatible roots give the
`ℤ[1/p]`-exponent formula above; this formula and its `R°`-level identities are the added assertions. Existence is
*exists-pseudouniformizer-with-compatible-roots* (P1.4) ([ECD] §3, Lemma 3.10, p. 16; [Sch12]
Remark 3.5, p. 16; [Berkeley] Lemma 6.2.2, proof, p. 44). Define `PseudoUniformizerRoots` (structure
on `R`: `roots : Perfection R p` with `coeff 0 roots ∈ R°`, `IsPseudoUniformizer (coeff 0 roots)`
and `(coeff 0 roots)^p ∣ p` in `R°`), `PseudoUniformizerRoots.root`
(`root n := Perfection.coeffMonoidHom R p n ϖ♭ = ϖ^{1/p^n}`), `PseudoUniformizerRoots.rootPB` (`root n` in `R°`) and `PseudoUniformizerRoots.zpow`
(`zpow m n = (root n)^m` as a unit of `R`, with equal values whenever `m/p^n` agrees;
additivity in the exponent, `zpow_mul_p`, and membership in `R°` for `m ≥ 0`),
`PseudoUniformizerRoots.toTilt` (the underlying compatible sequence as an element of
`tilt R`; when `R` is perfectoid it is a pseudo-uniformizer with `sharp ϖ♭ = ϖ`) and `PseudoUniformizerRoots.gauge` (`|x|_ϖ ∈ ℝ≥0`); prove
the root recurrence using Mathlib’s `Perfection.coeffMonoidHom_pow_p'`,
that each `root n` is a pseudo-uniformizer in `R°`, and
`PseudoUniformizerRoots.nonempty` (every perfectoid Tate ring admits one). For a prescribed
pseudo-uniformizer `ϖ_1 ∈ R°` with `ϖ_1^p | p`, there is a compatible-root
parameter congruent to `ϖ_1` modulo `ϖ_1^p` ([ECD] Lemma 3.10, proof, p. 16). Prove that the gauge
is a norm (`|xy| ≤ |x||y|`, `|x + y| ≤ max(|x|, |y|)`, `|x| = 0 ↔ x = 0`), that it is
power-multiplicative (`|x^n| = |x|^n` for `n > 0`, or for all `n` when `R ≠ 0`), the simp rule
`|ϖ^qx| = 2^{-q}|x|`, the characterisations `x ∈ R° ↔ |x| ≤ 1` and
`IsTopologicallyNilpotent x ↔ |x| < 1`, and that the
`ϖ^qR°`, `q ∈ ℤ[1/p]`, form a neighbourhood basis of `0`. *Needs:* P1.1; Mathlib `Perfection`,
`Perfection.coeffMonoidHom`; Tau Ceti `TauCeti.Huber.IsPseudoUniformizer`,
`TauCeti.Huber.isPowerBounded_of_isIntegral`, `TauCeti.Huber.powerBoundedSubring`.

**Checks.**

- `gauge_padic_formula` (computation) — for `ℂ_p` and a compatible
  choice `ϖ♭ = (p^{1/p^{n+1}})_n`, `gauge x = 2^{-p v_p(x)}` for `x ≠ 0`, with `v_p(p) = 1`.
  Thus `gauge ϖ = 1/2`, `gauge (ϖ²) = 1/4`, `gauge p = 2^{-p}`, and `gauge 0 = 0`.
- `gauge_wrong_root` (non-example) — the condition `ϖ^p ∣ p` in `O_C`
  excludes `ϖ = p`: the purported quotient has norm `p^{p−1} > 1`, including at `p = 2`.
- `gauge_product` (non-example) — on `ℂ_p × ℂ_p`, take compatible
  zeroth roots `(a,a)` and `(a,a²)` with `a^{2p} = p`. Their gauges both give `1/2` at `(a,0)`;
  at `(0,a²)` they give `1/4` and `1/2`, respectively. No single real power relates the gauges.
- `roots_padic_empty` (non-example) — `IsEmpty (PseudoUniformizerRoots ℚ_[p])`:
  a pseudo-uniformizer of `ℚ_p` has valuation `k ≥ 1`, not divisible by every `p^n`, so it has no
  compatible system of `p`-power roots; and none satisfies `ϖ^p | p` in `ℤ_p`.
- `roots_charP_unique` (degenerate) — if `R` is a perfectoid Tate ring of characteristic `p`
  (equivalently perfect, P1.2), the compatible roots of a fixed `ϖ` are unique, so
  `PseudoUniformizerRoots` over `ϖ` is a subsingleton.
- Characterisation: `⋃_{q>0} ϖ^qR° = R°°`, the
  topologically nilpotent elements (*topologically-nilpotent-elements-as-root-ideal*, P1.9).

Lean: `PseudoUniformizerRoots`, `PseudoUniformizerRoots.root`, `PseudoUniformizerRoots.nonempty`.
The root-system and gauge Checks are specified in this README only.

### P1.4 the tilt of a perfectoid Tate ring

The tilt comes next: the ring `R♭ = lim_{x↦x^p} R`, its integral subrings, the sharp map, the
functoriality of tilting, the tilt of a uniform Tate ring that is not assumed perfectoid, the
comparison with Mathlib's `PreTilt`, and the almost setup `(R°, R°°)`.

**The tilt `R♭ = lim_{x↦x^p} R` of a perfectoid Tate ring.** (*tilt-of-perfectoid-tate-ring*) Let `R`
be a perfectoid Tate ring. Its tilt is `R♭ := lim_{x↦x^p} R`, the multiplicative monoid
`Perfection R p` of compatible sequences `(x^{(n)})_{n≥0}` with `(x^{(n+1)})^p = x^{(n)}`, with the
inverse-limit topology (induced from `∏_ℕ R`) and the uniformity of its additive topological group, pointwise multiplication and addition
`(x + y)^{(i)} = lim_{n→∞}(x^{(i+n)} + y^{(i+n)})^{p^n}`. The limit exists and `R♭` is a topological
ring of characteristic `p` which is perfect ([ECD] §3, Definition 3.9 and Lemma 3.10, p. 16; [AWS]
Lemma 2.7.1(b), p. 66; [Berkeley] Lemma 6.2.2, p. 44); the limit addition, the topology and the
ring axioms are those of the tilt `uniformTilt R` of the complete uniform Tate ring `R` with `p`
topologically nilpotent (*tilt-of-uniform-tate-ring*, P1.7, which needs no perfectoid hypothesis),
and what perfectoidness adds is the rest of this target. Concretely, for `ϖ♭` a pseudo-uniformizer
with compatible roots and `ϖ_0` any pseudo-uniformizer dividing `p` in `R°`
(for example `ϖ_0 = ϖ♭^{(0)}`), the submonoid `lim_{x↦x^p} R°` is identified with the ring
`lim_Φ R°/ϖ_0` by *inverse-limit-of-integral-elements-along-frobenius*, `R♭ = (lim R°)[1/ϖ♭]` as
monoids, and the ring structure of `R♭` is that of this localisation; it is given by the limit
formula and does not depend on `ϖ_0` or `ϖ♭` ([ECD] Lemma 3.10, proof, p. 16). The construction
uses `ϖ_0 | p` and the completeness of `R°`; additivity of the zeroth projection is not asserted
(P1.5). For a Huber pair `(R, R⁺)` with `R` perfectoid, the tilt pair is `(R♭, R♭⁺)` with
`R♭⁺ := lim_{x↦x^p} R⁺` (*rings-of-integral-elements-under-tilting*, P1.18). If `pR = 0` then
`x ↦ x^{(0)}` is an isomorphism `R♭ ≅ R`. Define `tilt` (`tilt R := Perfection R p`, a type synonym
carrying the ring structure above; notation `R♭`), `tilt.instCommRing` (`CommRing (tilt R)` with the
monoid structure of `Perfection R p`, together with `instTopologicalSpace` induced from `ℕ → R` and
`instIsTopologicalRing`), `tilt.charP` (`CharP (tilt R) p` for `R ≠ 0`; `PerfectRing (tilt R) p` also for `R = 0`),
`tilt.coeff` (`coeff n : tilt R →* R`, the `n`-th coordinate `Perfection.coeffMonoidHom R p n`, with
`coeff_pow : coeff (n + 1) x ^ p = coeff n x`), `tilt.sharp` (P1.5), `tilt.map` (P1.6) and
`Pair.tilt` (the Huber pair `(tilt R, lim_{x↦x^p} R⁺)`); prove extensionality using Mathlib’s `Perfection.extMonoid`, `tilt.coeff_add` (`Tendsto (fun m ↦ (coeff (i + m) x + coeff (i + m) y) ^ p ^ m) atTop (𝓝 (coeff i (x + y)))`), `tilt.coeff_zero`
(`coeff n 0 = 0`), `tilt.coeff_frobeniusEquiv_symm` (simp:
`coeff n ((frobeniusEquiv (tilt R) p).symm x) = coeff (n + 1) x`), `tilt.isPerfectoidTateRing`
(`tilt R` is a perfectoid Tate ring of characteristic `p`, *tilt-is-perfectoid-tate-ring*, P1.14),
`tilt.mem_powerBoundedSubring_iff` (`x ∈ (tilt R)° ↔ ∀ n, coeff n x ∈ R° ↔ coeff 0 x ∈ R°`),
`tilt.equivOfCharP` (for `CharP R p`: `tilt R ≃+* R`, `x ↦ coeff 0 x`, a homeomorphism) and
`tilt.integralEquivPreTilt` (`(tilt R)° ≃+* PreTilt R° p`, P1.8). *Needs:* P1.1; P1.3; P1.2;
P1.7's construction of `uniformTilt` and its ring and topology laws for the carrier;
*inverse-limit-of-integral-elements-along-frobenius*; *exists-pseudouniformizer-with-compatible-roots*;
Mathlib `Perfection`, `Perfection.coeffMonoidHom`, `Perfection.teichmuller_spec`,
`Localization.Away`, `PerfectRing`.

Check `tilt.coeff_mul` is `map_mul` applied to the monoid homomorphism `tilt.coeff`;
`map_one` likewise gives `coeff n 1 = 1`. These are the coordinatewise multiplicative
laws in [ECD] Definition 3.9, p. 16.

**Checks.**

- `tilt_cyclotomic_twice` (compatibility) — for `R = tilt (ℚ_p^cycl)`, `tilt R ≃+* R` via `coeff 0`, a homeomorphism
  (`tilt_cyclotomic_twice`). The field identification is P1.31.
- `tilt_cyclotomic_epsilon` (computation) — in `tilt ℚ_p^cycl`, `ε := (ζ_{p^n})_n` with
  `ζ_{p^{n+1}}^p = ζ_{p^n}`, `ζ_1 = 1`, satisfies `coeff n ε = ζ_{p^n}` and `ε ≠ 1`.
- `tilt_coordinate_nonadditive` (non-example) — in `tilt ℚ_p^cycl`,
  `coeff 0 (ε − 1) ≠ coeff 0 ε − coeff 0 1 = 0`: coordinatewise subtraction is not the ring
  structure.
- `tilt.nontrivial_iff` (degenerate) — `Nontrivial (tilt R) ↔ Nontrivial R`; the tilt of the zero
  ring is zero.
- `tilt.isPerfectoidTateRing_charP` (computation) — for `R = ℚ_p^cycl`, `tilt R` is the
  perfectoid Tate ring `𝔽_p((t^{1/p^∞}))^` of characteristic `p`, with pseudo-uniformizer
  `t = ε − 1` and `t^{1/p^n} = ε^{1/p^n} − 1`.

The carrier in the cyclotomic Check is `completedPerfectLaurentFp`: start with
`PerfectLaurentPolynomial := 𝔽_p[t^{1/p^∞}]`, whose `perfectLaurentMonomial n` is
`t^{1/p^n}`, form its t-adic completion `PerfectLaurentIntegral`, and invert the image
`perfectLaurentParameter` of t. `perfectLaurentRoot n` is the image of that monomial in
the localization. `perfectLaurentTopology_basis` specifies the topology by powers of t
multiplying the integral completion ([ECD] Example 3.4(ii), p. 15, and Lemma 3.10, p. 16).
**Checks.** Each of these six constructions has the same three checks, with names
`<definition>_roots`, `<definition>_parameter`, and `<definition>_characteristic`:
successive roots have p-th power the preceding root; t is a nonunit in the integral
completion and a pseudo-uniformizer unit after localization; p is zero while t is
neither zero nor one. These Checks record the equations in the relevant carriers.

Two supporting results. The inverse limit of integral elements along Frobenius
(*inverse-limit-of-integral-elements-along-frobenius*): let `R` be a nonzero complete Hausdorff
uniform Tate ring, `R⁺` a ring of integral elements and `ϖ_0 ∈ R⁺` a pseudo-uniformizer dividing
`p` in `R⁺`. Projection `(x^{(n)})_n ↦ (x^{(n)} mod ϖ_0)_n` is a multiplicative bijection
`lim_{x↦x^p} R⁺ → lim_Φ R⁺/ϖ_0R⁺` and a homeomorphism for the inverse-limit topologies (`R⁺/ϖ_0`
discrete); the inverse sends `(x̄_0, x̄_1, …)` to `(x^{(i)})_i` with
`x^{(i)} = lim_{n→∞} x_{n+i}^{p^n}` for arbitrary lifts `x_j ∈ R⁺`. The multiplicative bijection
also holds algebraically for `p` in place of `ϖ_0`, and the ring homomorphism
`lim_Φ R⁺/p → lim_Φ R⁺/ϖ_0` induced by the quotient map is an isomorphism;
`R ≠ 0` makes `R⁺/ϖ_0` of characteristic `p`. Both bijections are Mathlib's
`Perfection.quotientMulEquiv p I : Perfection R⁺ p ≃* Perfection (R⁺ ⧸ I) p` for `I = span {ϖ_0}`
and `I = span {p}` (`R⁺` is `I`-adically complete for both by *integral-elements-adically-complete*
(a), (c), and `R⁺/I` has characteristic `p`), and `Perfection (ModP R⁺ p) p` is `PreTilt R⁺ p`;
Mathlib gives multiplicative isomorphisms and the ring structure only on the `Perfection (R⁺/I) p`
side. Give `lim_Φ R⁺/p = PreTilt R⁺ p` the topology transported from `lim_Φ R⁺/ϖ_0`
with discrete `R⁺/ϖ_0` factors. This topology is independent of the choice of `ϖ_0`, makes the
modulo-`p` comparison a homeomorphism, and preserves the displayed reduction formula on every
coordinate. It is not defined by making `R⁺/p` discrete: `p` need not be a pseudo-uniformizer
or a unit of `R`, and may be zero. The transported ring structure and these topological
comparisons with the subspace topology on `lim_{x↦x^p} R⁺` are proved here ([ECD] Lemma 3.10, proof, p. 16; [BMS] Lemma
3.2(i), p. 19; [KL15] Remark 3.4.10, p. 82; [Sch12] Lemma 3.4(i), proof, p. 16). For a perfectoid
field `K` and `R⁺ = K°` this is [Sch12] Lemma 3.4(i). If `R` is also perfect of characteristic `p`,
the zeroth projection identifies `lim_{x↦x^p} R⁺` with `R⁺` as topological rings. Here `R⁺` is
perfect because a `p`-th root in `R` of an element of `R⁺` is integral over the integrally closed
subring `R⁺`. The Lean statement `bijective_coeffZero_integralElements_of_perfectRing` carries
`CharP R p` and `PerfectRing R p` and states the algebraic bijectivity of this projection.

**Checks for the inverse-limit comparison.**

- For `R = ℚ_p`, `R⁺ = ℤ_p` and `ϖ_0 = p`, both inverse limits are the Teichmüller
  representatives, with their ring structure identified with `𝔽_p`; perfectoidness is unnecessary.
- For `K` the completed perfection of `𝔽_p((t))`, the zeroth projection identifies
  `lim_Frob K°` with `K°` in its valuation topology, as in `uniformTilt_charP`.
- For `R = 𝔽_p((t))`, `R⁺ = 𝔽_p[[t]]` and `ϖ_0 = t`, the hypotheses hold and
  `t ∣ p = 0`, but `lim_Frob R⁺ ≅ 𝔽_p`: the support of a series with `p^n`-th roots for every
  `n` is divisible by every `p^n`, hence contains only exponent zero. The zeroth projection
  includes the constants and misses `t`, which has valuation one and no `p`-th root.

Every perfectoid Tate ring has a
pseudo-uniformizer with compatible `p`-power roots (*exists-pseudouniformizer-with-compatible-roots*):
for `R` perfectoid (the zero ring allowed) and `ϖ_1` a pseudo-uniformizer with `ϖ_1^p | p` in `R°`,
there is `ϖ♭ ∈ lim_{x↦x^p} R°` with `ϖ := ϖ♭^{(0)} ≡ ϖ_1 mod ϖ_1^pR°`; then `ϖ = ϖ_1·u` with
`u ∈ (R°)^×`, `ϖR° = ϖ_1R°`, `ϖ^p | p` in `R°`, and `ϖ♭` is a pseudo-uniformizer with compatible
`p`-power roots; if `pR = 0` every pseudo-uniformizer has unique compatible roots ([ECD] Lemma 3.10
and its proof, p. 16; [Berkeley] Lemma 6.2.2, proof, p. 44; [BMS] Lemma 3.9, p. 22; Mathlib
`Perfection.coeff_surjective`). For `R = ℂ_p` and `ϖ_1 = p^{1/p}` the construction returns a unit
multiple of `p^{1/p}` with compatible roots; in characteristic `p`, `ϖ = ϖ_1` and
`ϖ^{1/p^n} = Φ^{-n}(ϖ_1)`.

### P1.5 the sharp map

**The sharp map `♯: R♭ → R` (continuous, multiplicative, not additive).** (*sharp-map*) For a
perfectoid Tate ring `R`, the sharp map `♯ : R♭ → R`, `x ↦ x♯ := x^{(0)}` (Mathlib
`Perfection.coeffMonoidHom R p 0` on the underlying monoid `Perfection R p`) is continuous and
multiplicative with `0♯ = 0`, and is not additive in general. It satisfies `(x^{1/p^n})♯ = x^{(n)}`;
`x ∈ R♭°` iff `x♯ ∈ R°`; `x♯ = 0` iff `x = 0`; for `x, y ∈ R♭°`, `(x + y)♯ ≡ x♯ + y♯ mod pR°`;
every element of `R°` is congruent modulo `pR°` to some `y♯` with `y ∈ R♭°`; reduction of `♯`
modulo `ϖ` is the ring isomorphism `R♭°/ϖ♭ ≅ R°/ϖ` of *tilt-integral-subrings*; and `(ϖ♭)♯ = ϖ`.
If `pR = 0`, `♯` is the ring isomorphism `R♭ ≅ R` ([ECD] §3, after Lemma 3.10, p. 16; [Berkeley]
§6.2, p. 45; [AWS] Definition 2.3.14, p. 56 (`♯ = θ ∘ [·]`); [Sch12] Lemma 3.4(i), proof, p. 16).
Define `tilt.sharp` (`sharp : tilt R →* R := Perfection.coeffMonoidHom R p 0`; notation `x♯`) and
`tilt.sharp₀` (the same map as a `MonoidWithZeroHom`); prove `tilt.continuous_sharp`,
`tilt.sharp_iterate_frobeniusEquiv_symm_pow` (simp:
`sharp (((frobeniusEquiv (tilt R) p).symm)^[n] x) ^ p ^ n = sharp x`, mirroring Mathlib
`PreTilt.untilt_iterate_frobeniusEquiv_symm_pow`), `tilt.mem_powerBoundedSubring_iff_sharp`
(`x ∈ (tilt R)° ↔ sharp x ∈ R°`), `tilt.sharp_eq_zero_iff` (`sharp x = 0 ↔ x = 0`),
`tilt.sharp_add_sub_mem` (for `x, y ∈ (tilt R)°`: `sharp (x + y) − sharp x − sharp y ∈ p • R°`),
`tilt.exists_sharp_sub_mem` (`∀ x ∈ R°, ∃ y ∈ (tilt R)°, x − sharp y ∈ p • R°`),
`tilt.sharpQuotEquiv` (the ring isomorphism `(tilt R)° ⧸ (ϖ♭) ≃+* R° ⧸ (ϖ)` induced by `sharp`),
`tilt.sharp_toTilt` (simp: `sharp ϖ♭.toTilt = ϖ`), the gauge identity `|sharp x|_ϖ = |x|_{ϖ♭}`
(*spectral-norm-under-sharp*, P1.7) and `tilt.sharp_eq_untilt` (under `(tilt R)° ≃+* PreTilt R° p`,
`sharp` is Mathlib's `PreTilt.untilt`, P1.8). *Needs:* P1.4; *tilt-integral-subrings*;
*frobenius-surjectivity-criterion*; Mathlib `Perfection.coeffMonoidHom`,
`Perfection.coeff_surjective`.

**Checks.**

- `sharp_cyclotomic_two_terms` (non-example) — in `tilt ℚ_p^cycl`: `sharp (ε − 1) ≠ sharp ε − sharp 1 = 0`
  (a ring homomorphism `sharp` would fail this).
- `sharp_epsilon` (computation) — `sharp ε = 1` in `tilt ℚ_p^cycl`, and `sharp p♭ = p` in
  `tilt ℂ_p` for `p♭ = (p^{1/p^n})_n`.
- `sharp₀_zero`, `sharp₀_one`, `sharp₀_multiplication` — the zero-preserving form of sharp
  sends zero to zero and one to one and preserves products; `sharp₀_apply` identifies it
  with the zeroth coordinate. The cyclotomic subtraction Check also applies to this map.
- `tilt.sharp_ringEquiv_of_charP` (degenerate) — if `CharP R p`, `sharp` is additive and is the
  ring isomorphism `tilt.equivOfCharP`.
- `sharp_eq_zero_iff_test` (characterisation) — for `R = ℂ_p × ℂ_p♭`, a perfectoid Tate ring,
  `sharp x = 0` only for `x = 0`.

Two supporting results. The Frobenius-surjectivity criterion with the pseudo-uniformizer condition
kept (*frobenius-surjectivity-criterion*): let `R` be a complete Hausdorff uniform Tate ring and
`ϖ ∈ R°` a pseudo-uniformizer with `ϖ^p | p` in `R°` (a hypothesis, not a conclusion; `R°` is
`ϖ`-adically complete by *integral-elements-adically-complete*). The following are equivalent:
(i) `Φ : R°/ϖ → R°/ϖ^p` is surjective; (ii) Frobenius on `R°/p` is surjective; (iii) every element
of `R°/ϖpR°` is a `p`-th power. The same equivalences hold for any ring of integral elements `R⁺`
with `ϖ ∈ R⁺` and `ϖ^p | p` in `R⁺`. The divisibility is essential: surjectivity of Frobenius on
`R°/p` without any `ϖ` with `ϖ^p | p` does not give perfectoidness (`ℚ_p`). The criterion is a
statement about any ring `A` that is `π`-adically complete with `π^p | p`: surjectivity of the
relative Frobenius `A/π → A/π^p`, `[x] ↦ [x^p]`, of Frobenius on `A/p` and of Frobenius on `A/πp`
are equivalent, and for `A = R°` one takes `π = ϖ`. Negative control, separating the relative
Frobenius from the Frobenius endomorphism of `A/π`: for `A = ℤ_p[π]` with `π^p = p` (so `π = p^{1/p}`),
and `u` the class of `π` in `A/p`, the endomorphism `x ↦ x^p` of `A/π ≅ 𝔽_p` is bijective; the
relative Frobenius `A/π ≅ 𝔽_p → A/π^p = A/p ≅ 𝔽_p[u]/(u^p)` has image the constants `𝔽_p`; and
Frobenius on `A/p ≅ 𝔽_p[u]/(u^p)` also has image `𝔽_p`. So (i) and (ii) miss `u`, and Frobenius on
`A/πp` is not surjective either, `A/p` being a quotient of `A/πp`: all three conditions of the
criterion fail together, and bijectivity of the Frobenius endomorphism of `A/π` says nothing about
perfectoidness. The Check states that `u` is not a `p`-th power in `(ZMod p)[X]/(X^p)`
([ECD] Remark 3.2, p. 14; [BMS] Lemma 3.9 (i)–(iii), p. 22; [Berkeley] Remark 6.1.3, p. 42; [KL15]
Remark 3.4.9, p. 81; [AWS] Remark 2.1.5, p. 50; Mathlib `IsAdicComplete`). The integral subrings of
the tilt (*tilt-integral-subrings*): for `R` perfectoid with tilt `R♭`, `ϖ♭` with compatible roots
and `ϖ = (ϖ♭)♯` (so `ϖ^q | p` for `q ≤ p`), (a) `R♭° = lim_{x↦x^p} R° = {x ∈ R♭ : x^{(n)} ∈ R°` for
all `n} = {x : x^{(0)} ∈ R°}`, and for every pseudo-uniformizer `ϖ_0` of `R` dividing `p` in `R°`
the projection isomorphisms `R♭° ≅ lim_Φ R°/ϖ_0 ≅ lim_Φ R°/p` are
*inverse-limit-of-integral-elements-along-frobenius* (P1.4) for `R⁺ = R°`;
(b) `ϖ♭` is a pseudo-uniformizer of `R♭`, `R♭°` is an open and bounded subring whose subspace
topology is the `ϖ♭`-adic one, and `R♭ = R♭°[1/ϖ♭]`; (c) for `q ∈ ℤ[1/p]` with `0 < q ≤ p`,
`x ↦ x^{(0)} mod ϖ^qR°` induces a ring isomorphism `R♭°/ϖ♭^qR♭° ≅ R°/ϖ^qR°`, in particular
`R♭°/ϖ♭ ≅ R°/ϖ`; (d) the kernel of `R♭° → R°/ϖ^q` is `ϖ♭^qR♭°` ([ECD] Lemma 3.10 and the text
after it, p. 16; [Berkeley] Lemma 6.2.2, p. 44; [BMS] Lemma 3.10, proof, p. 23; [Sch12] Lemma
3.4(iii), p. 16; Tau Ceti `TauCeti.Huber.IsPowerBounded`, `TauCeti.Huber.isPowerBounded_of_isIntegral`;
Mathlib `Perfection.coeff_surjective`). For a perfectoid field `K`: `K♭° = lim K° ≅ lim_Φ K°/ϖ` and
`K♭°/ϖ♭ ≅ K°/ϖ`; for `R = K⟨T^{1/p^∞}⟩`: `R♭°/ϖ♭ = (K°/ϖ)[T^{1/p^∞}]` ([Sch12] Proposition 5.20,
proof).

### P1.6 functoriality of tilting

**Functoriality of tilting: `f ↦ f♭`.** (*tilt-functoriality*) A continuous ring homomorphism
`f : R → S` between perfectoid Tate rings induces `f♭ : R♭ → S♭`, `(x^{(n)})_n ↦ (f(x^{(n)}))_n`
(Mathlib `Perfection.mapMonoidHom p f` on the underlying monoids), a continuous ring homomorphism
with `(f♭x)♯ = f(x♯)`, `f♭(R♭°) ⊆ S♭°`, `id♭ = id` and `(g ∘ f)♭ = g♭ ∘ f♭`; on `R♭°/ϖ♭ ≅ R°/ϖ` it
induces the reduction of `f`. No condition on plus rings enters the ring-level statement. A
morphism of Huber pairs `(R, R⁺) → (S, S⁺)` (`TauCeti.Huber.Pair.Hom`) with `R, S` perfectoid
induces a morphism of the tilt pairs. Hence tilting is a functor from perfectoid Tate rings (resp.
pairs) to perfectoid Tate rings (resp. pairs) of characteristic `p`, and a perfectoid `R`-algebra
`S` gives a perfectoid `R♭`-algebra `S♭` ([Berkeley] Theorem 6.2.7, p. 45; [AWS] Definition 2.3.2,
p. 54; [ECD] Theorem 3.13, p. 17). Define `tilt.map` (`map f : tilt R →+* tilt S` for `f` continuous,
with underlying monoid hom `Perfection.mapMonoidHom p f`), `Pair.tiltHom` (`Pair.Hom P Q ↦ Pair.Hom P.tilt Q.tilt`, compatible with `id` and `comp`), `tilt.mapEquiv` (a ring isomorphism `R ≃+* S` that
is a homeomorphism induces `tilt R ≃+* tilt S`) and `tiltFunctor` (the functor from perfectoid Tate
rings with continuous ring homomorphisms to perfectoid Tate rings of characteristic `p`, and its
slice version from perfectoid `R`-algebras to perfectoid `R♭`-algebras); prove `tilt.coeff_map`
(simp: `coeff n (map f x) = f (coeff n x)`), `tilt.sharp_map` (simp: `sharp (map f x) = f (sharp x)`), `tilt.map_id`, `tilt.map_comp` (`map (g.comp f) = (map g).comp (map f)`),
`tilt.continuous_map`, `tilt.map_mem_powerBoundedSubring` (`x ∈ (tilt R)° → map f x ∈ (tilt S)°`)
and `tilt.mapQuot_comm` (the square `(tilt R)°/(ϖ♭) ≅ R°/(ϖ) → S°/(f ϖ) ≅ (tilt S)°/(map f ϖ♭)`
commutes). For `g ∈ Gal(ℚ_p^cycl/ℚ_p)` with cyclotomic character `χ(g)`, `g♭(ε) = ε^{χ(g)}`
([Berkeley] Example 6.2.4). *Needs:* P1.4; P1.5; P1.1; Mathlib `Perfection.mapMonoidHom`; Tau Ceti
`TauCeti.Huber.Pair.Hom`.

**Checks.**

- `map_id_test` (degenerate) — `tilt.map (RingHom.id ℂ_p) = RingHom.id`.
- `tilt.map_inclusion_epsilon` (computation) — for the continuous inclusion `ι : ℚ_p^cycl → ℂ_p`,
  `tilt.map ι ε = ε`.
- `tilt.map_equivOfCharP` (characterisation) — for `f : R → S` continuous between perfectoid rings
  of characteristic `p`, `tilt.equivOfCharP ∘ tilt.map f = f ∘ tilt.equivOfCharP`.
- `Pair.tiltHom_plus` (compatibility) — for a `Pair.Hom f : (R, R⁺) → (S, S⁺)`, `Pair.tiltHom f`
  carries `lim R⁺` into `lim S⁺`.

The equivalence `tilt.mapEquiv` is pinned by
`tilt.mapEquiv_coeff`: its n-th coordinate is the image under the original ring isomorphism.
**Checks.** `tilt.mapEquiv_identity` fixes every coordinate; `tilt.mapEquiv_swap` interchanges
the two factors of a product; `tilt.mapEquiv_charP` agrees with the original isomorphism under
the characteristic-p identification by the zeroth coordinate.

### P1.7 the tilt of a complete uniform Tate ring

**The tilt `B♭ = lim_{x↦x^p} B` of a complete uniform Tate ring in which `p` is topologically
nilpotent.** (*tilt-of-uniform-tate-ring*) Let `B` be a complete Hausdorff uniform Tate ring in
which `p` is topologically nilpotent (`B` is not assumed perfectoid) and `B⁺ ⊆ B°` a ring of
integral elements. Its tilt is `B♭ := lim_{x↦x^p} B` (Mathlib `Perfection B p`) with the
inverse-limit topology, pointwise multiplication, `x♯ := x^{(0)}`, and addition
`(x + y)^{(i)} := lim_{n→∞}(x^{(i+n)} + y^{(i+n)})^{p^n}`. Then: (a) the limit exists and makes `B♭`
a perfect topological ring of characteristic `p`; `♯` is continuous and multiplicative;
`B♭⁺ := lim_{x↦x^p} B⁺ = {x : x♯ ∈ B⁺}` is an open subring and `B♭° := lim_{x↦x^p} B° = {x : x♯ ∈ B°}`; `B♭` is complete and Hausdorff; (b) for every ideal of definition `I ⊆ B⁺` with `p ∈ I` (for
instance `I = (p, ϖ)` for a pseudo-uniformizer `ϖ ∈ B⁺`; Kedlaya asks `p ∈ I^p`), reduction
`x ↦ (x^{(n)} mod I)_n` is an isomorphism of topological rings `B♭⁺ ≅ lim_Φ B⁺/I` (discrete
factors), and, for `B ≠ 0` (so that `p` is not a unit of `B⁺`, as Mathlib's `PreTilt` requires),
`B♭⁺ ≅ PreTilt B⁺ p` algebraically, with topology on `PreTilt` transported from
`lim_Φ B⁺/I`, independently of `I`, rather than from discrete `B⁺/p` factors;
(c) if `B` has a pseudo-uniformizer `ϖ` with `ϖ^p | p` in `B°`
and compatible `p`-power roots, i.e. `ϖ = ϖ♭♯` for some `ϖ♭ ∈ B♭`, then `ϖ♭` is a pseudo-uniformizer
of `B♭`, `B♭°` is open and bounded with the `ϖ♭`-adic topology, `B♭ = B♭°[1/ϖ♭]`, the spectral
gauges satisfy `|x|_{ϖ♭} = |x♯|_ϖ`, and `B♭` is a complete uniform perfect Tate ring of
characteristic `p`, hence a perfectoid Tate ring; this applies whenever there is a continuous ring
homomorphism `f : A → B` from a perfectoid Tate ring `A`, with `ϖ♭ := f♭(ϖ_A♭)`; the closure in `B♭`
of a perfect subring containing `ϖ♭` and `ϖ♭⁻¹` is then a perfectoid Tate ring of characteristic
`p`; without compatible roots `B♭` need not be Tate; (d) a continuous ring homomorphism `g : B → C`
of such rings induces `g♭ : B♭ → C♭`, `(x^{(n)}) ↦ (g(x^{(n)}))`, a continuous ring homomorphism with
`(g♭x)♯ = g(x♯)`, functorially; (e) there is a unique ring homomorphism `θ_B : W(B♭⁺) → B⁺` with
`θ_B([x]) = x♯` (for `B ≠ 0` it is Mathlib `WittVector.fontaineTheta` for the `p`-adically
complete ring `B⁺`, in which `p` is not a unit; for `B = 0` it is the zero map),
natural in `B`: `θ_C ∘ W(g♭) = g ∘ θ_B`; it is not surjective in general; (f) if `B` is perfectoid,
`B♭` with `♯`, `g♭` and `θ_B` is the tilt of P1.4 with the sharp map, functoriality and `θ` of
P1.5, P1.6 and P1.18 ([AWS] Lemma 2.7.1 with (b), (c) and its proof, pp. 66–67; [AWS] Hypothesis
2.7.2, p. 67; [AWS] Lemma 2.8.4, proof, p. 70). Define `uniformTilt` (`uniformTilt B : Type := Perfection B p` with `CommRing` (limiting addition), `TopologicalSpace`, `IsTopologicalRing`,
`PerfectRing`, `CompleteSpace` and `T2Space` instances, and `CharP p` when `B ≠ 0`, for `B` complete Hausdorff
uniform Tate with `IsTopologicallyNilpotent (p : B)`), `uniformTilt.sharp` (`sharp : uniformTilt B →* B := Perfection.coeffMonoidHom B p 0`, continuous, with `sharp (x^{1/p^n})^{p^n} = sharp x`),
`uniformTilt.map` (`map g : uniformTilt B →+* uniformTilt C` for `g` continuous; continuous;
`sharp (map g x) = g (sharp x)`; `map_id`, `map_comp`) and `uniformTilt.theta` (`theta B : 𝕎 (lim B⁺) →+* B⁺` with `theta [x] = sharp x`, equal to `WittVector.fontaineTheta B⁺ p` under `plusEquiv`;
`theta C ∘ WittVector.map (map g) = g ∘ theta B`); prove `uniformTilt.coeff_add` (`coeff i (x + y) = lim_{n→∞} (coeff (i + n) x + coeff (i + n) y)^{p^n}`, in `Filter.Tendsto` form),
`uniformTilt.plusEquiv` (for an ideal of definition `I` of `B⁺` with `p ∈ I`: `lim B⁺ ≃+* Perfection (B⁺ ⧸ I) p`, a homeomorphism for discrete factors, from `Perfection.quotientMulEquiv`; and `lim B⁺ ≃+* PreTilt B⁺ p`), `uniformTilt.mem_powerBoundedSubring_iff` (`x ∈ (uniformTilt B)° ↔ sharp x ∈ B°`; likewise `x ∈ lim B⁺ ↔ sharp x ∈ B⁺`), `uniformTilt.isTateRing_of_sharp` (if `ϖ♭ : uniformTilt B` has `sharp ϖ♭` a pseudo-uniformizer of `B` with `(sharp ϖ♭)^p ∣ p` in `B°`, then `IsTateRing (uniformTilt B)`, `IsBounded (uniformTilt B)°`, `IsPerfectoidTateRing (uniformTilt B)`, and `gauge ϖ♭ x = gauge (sharp ϖ♭) (sharp x)`), `uniformTilt.isPerfectoidTateRing_closure` (under
`isTateRing_of_sharp`, the closure of the range of a ring homomorphism from a perfect ring whose
range contains `ϖ♭` and `ϖ♭⁻¹` is a closed subring that is a perfectoid Tate ring of characteristic
`p`) and `uniformTilt.equivTilt` (for `B` perfectoid, `uniformTilt B ≃+* tilt B`, the identity of
`Perfection B p`, a homeomorphism commuting with `sharp` and `map`). The limiting-addition construction and
its functoriality use complete uniform Tate rings alone; P1.4–P1.6 are its perfectoid
specialisations. *Needs:* the Huber-ring foundations; Mathlib `Perfection`, `Perfection.coeffMonoidHom`,
`Perfection.mapMonoidHom`, `Perfection.quotientMulEquiv`, `Perfection.teichmuller_spec`, `PreTilt`,
`WittVector.fontaineTheta`, `PerfectRing`; Tau Ceti `TauCeti.Huber.IsTateRing`,
`TauCeti.Huber.IsBounded`, `TauCeti.Huber.powerBoundedSubring`,
`TauCeti.Huber.isPowerBounded_of_isIntegral`, `TauCeti.Huber.IsRingOfIntegralElements`.

**Checks.**

- `uniformTilt_cyclotomic` (computation) — for `B = ℚ_p^cycl`, `uniformTilt B` and `tilt B` are
  the same topological ring `𝔽_p((t^{1/p^∞}))^`, `t = ε − 1`, and `sharp` computed in either sends
  `t` to `lim_n (ζ_{p^n} − 1)^{p^n}`.
- `uniformTilt_padic` (non-example) — `uniformTilt ℚ_p ≃+* 𝔽_p` (Teichmüller representatives
  `{0} ∪ μ_{p−1}`, the elements of `ℚ_p` with compatible `p`-power roots; for `p = 2` the set is
  `{0, 1}`, as `−1` has no square root in `ℚ_2` and `1 + 4ℤ_2 ≅ ℤ_2` has no nonzero `2`-divisible
  element) with the discrete topology: it is not a Tate ring, and `ℚ_p` has no
  pseudo-uniformizer `ϖ` with `ϖ^p | p`, so (c) does not apply; a construction asserting that the
  tilt of every uniform Tate ring is Tate fails here.
- `uniformTilt_charP` (degenerate) — if `B` is a perfect uniform Tate ring of characteristic `p`,
  `sharp : uniformTilt B ≃+* B` is a topological ring isomorphism; `uniformTilt` of the zero ring
  is zero.
- `uniformTilt.isPerfectoidTateRing_of_hom` (characterisation) — for `f : A → B` continuous with
  `A` perfectoid and `B` uniform: `IsPerfectoidTateRing (uniformTilt B)`, and, for `B ≠ 0`,
  `CharP (uniformTilt B) p`; `map f ϖ♭` is a pseudo-uniformizer with `sharp (map f ϖ♭) = f ϖ`.

Supporting result: the spectral gauge is preserved by the sharp map (*spectral-norm-under-sharp*).
Let `R` be a perfectoid Tate ring, `ϖ♭` a pseudo-uniformizer with compatible roots, `ϖ = (ϖ♭)♯`,
and `|·|_ϖ`, `|·|_{ϖ♭}` the spectral gauges of `R` and `R♭` (P1.3; the second for `R♭` with its
pseudo-uniformizer `ϖ♭` and roots `ϖ♭^{1/p^n}`), normalised by `|ϖ| = |ϖ♭| = 1/2` when `R ≠ 0` (both gauges are zero when `R = 0`). Then
`|x♯|_ϖ = |x|_{ϖ♭}` for all `x ∈ R♭`; it is the norm Kedlaya–Liu put on the tilt by pullback along
`♯` ([Sch12] Lemma 5.21, proof, p. 27; [KL16] Theorem 3.3.8, proof, p. 60). For `K = ℚ_p^cycl` and
`ε = (ζ_{p^n})_n`: `|(ε − 1)♯| = lim |ζ_{p^n} − 1|^{p^n} = |p|^{p/(p−1)}`, which is `|ε − 1|` in
`K♭`.

The proposed Lean construction is `uniformTilt p B := Perfection B p`, with the inherited
inverse-limit topology and the uniformity of the new additive group. Boundedness of `B°` and
topological nilpotence of `p` are explicit `Fact` arguments. `uniformTilt.coeff_add` pins the
addition; `plusEquiv_coeff` pins reduction on every coordinate. The algebraic `plusEquiv`
requires `IsAdicComplete I B⁺` and `CharP (B⁺/I) p`; the latter restricts this signature to a
nonzero quotient. `theta_unique`, `theta_eq_fontaineTheta`, `theta_natural`, `map_id`, `map_comp`
and `equivTilt_coeff` specify the maps, rather than just their source and target types.

`uniformTilt.gauge_sharp` compares the two gauges for root systems related in every
coordinate by `coeff m (ψ.root n) = ϖ.root (n+m)`. `equivTilt_plus`, `equivTilt_theta` and
`equivTilt_map` identify the perfectoid specialization on integral subrings, Fontaine maps and
continuous homomorphisms ([AWS] Lemma 2.7.1 and §2.7).

### P1.8 the integral tilt is Mathlib's PreTilt

**The integral tilt `R♭⁺` is Mathlib's `PreTilt R⁺ p`, and `♯` is `PreTilt.untilt`.**
(*tilt-comparison-with-mathlib-pretilt*) Let `R` be a nonzero perfectoid Tate ring and `R⁺` a ring
of integral elements (e.g. `R°`); `R ≠ 0` is needed, as otherwise `p` is a unit of `R⁺ = 0`. Then
`R⁺` satisfies Mathlib's standing hypotheses `Fact ¬IsUnit (p : R⁺)` and `IsAdicComplete (span {p}) R⁺` (*integral-elements-adically-complete*, P1.2), and: (a) the subring `R♭⁺ = lim_{x↦x^p} R⁺` of
`R♭` is isomorphic as a topological ring to Mathlib's `PreTilt R⁺ p = Perfection (ModP R⁺ p) p` via
`x ↦ (x^{(n)} mod p)_n`: its algebraic multiplicative equivalence is
`Perfection.quotientMulEquiv` for `I = (p)`, with inverse built from
`Perfection.teichmuller`. Choose an actual pseudo-uniformizer `ϖ_0 ∈ R⁺` dividing `p` and give
`PreTilt R⁺ p` the topology transported along the algebraic comparison
`PreTilt R⁺ p ≃+* lim_Φ R⁺/ϖ_0`, with discrete `R⁺/ϖ_0` factors, from
*inverse-limit-of-integral-elements-along-frobenius* (P1.4). This topology is independent of
`ϖ_0` and makes the displayed map a homeomorphism; no discreteness of `R⁺/p` is assumed.
(b) Under this isomorphism `x ↦ x♯` on `R♭⁺` is
Mathlib's `PreTilt.untilt : PreTilt R⁺ p →* R⁺` (which is `Perfection.teichmuller` for `I = (p)`),
`PreTilt.coeff 0` is `♯ mod p` (`PreTilt.mk_untilt_eq_coeff_zero`), and the root compatibility
`(x^{1/p^n})♯^{p^n} = x♯` is `PreTilt.untilt_iterate_frobeniusEquiv_symm_pow`; (c) `R♭ ≅ (PreTilt R⁺ p)[1/ϖ♭]` (`Localization.Away`), the topology on `lim_{x↦x^p} R⁺` and on this localisation being
the part Mathlib does not supply.
Mathlib's `PreTilt` is thus the integral tilt of this layer; Mathlib has no ring structure on
`lim_{x↦x^p} R` and no tilt of a non-field Tate ring, which this layer supplies (the field case,
Mathlib's `Tilt` and `PreTilt.val`, is compared in P1.15) ([BMS] Lemma 3.2(i), p. 19; [KL15] Remark
3.4.10, p. 82; [Bhatt] Theorem 6.2.7(2), p. 47). For `R = ℂ_p`, `PreTilt 𝒪_{ℂ_p} p ≅ (ℂ_p)♭°` and
`PreTilt.untilt p♭ = p`; `WittVector p (PreTilt R⁺ p)` is Mathlib's `A_inf(R⁺)`, the source of
`WittVector.fontaineTheta` (P1.17). *Needs:* *inverse-limit-of-integral-elements-along-frobenius*
(P1.4); P1.5; Mathlib `PreTilt`, `PreTilt.untilt`, `PreTilt.mk_untilt_eq_coeff_zero`,
`PreTilt.coeff`, `Perfection.quotientMulEquiv`, `Perfection.coeff_zero_symm_quotientMulEquiv`,
`ModP`, `Localization.Away`.

**Checks.**

- For `R = ℂ_p`, reduction of every coordinate modulo `p` gives the algebraic comparison
  `PreTilt 𝒪_{ℂ_p} p ≅ (ℂ_p)♭°`; `PreTilt.untilt p♭ = p` and `PreTilt.coeff 0 p♭ = 0`.
- For `K` the completed perfection of `𝔽_p((t))`, the transported topology identifies
  `PreTilt K° p` with the nondiscrete valuation ring `K°`: the nonzero sequence `t^n` tends to
  zero. If instead each factor `K°/p = K°` is discrete, the Frobenius inverse limit is discrete,
  since its bijective zeroth projection has open singleton fibres. These topologies differ.
- For a pseudo-uniformizer `ϖ ∈ R⁺` with `ϖ^p ∣ p`, using `ϖ` or `ϖ^p` gives the same
  transported topology: both inverse-limit comparisons are homeomorphisms with
  `lim_{x↦x^p} R⁺` in its subspace topology, and the transition induced by
  `R⁺/ϖ^p → R⁺/ϖ` preserves reduction of every coordinate.

### P1.9 the almost setup of a perfectoid Tate ring

**The almost setup `(R°, R°°)` of a perfectoid Tate ring.** (*perfectoid-almost-setup*) Let `R` be a
perfectoid Tate ring. The pair `(R°, R°°)` is a basic setup for almost mathematics (P0.1: `R°°`
idempotent and `R°° ⊗_{R°} R°° ≅ R°°` flat), and so is `(R⁺, R°°)` for every ring of integral
elements `R⁺`. For an `R⁺`-module `M` the following are equivalent: (i) `M` is almost zero,
`R°°M = 0`; (ii) `ϖ'M = 0` for every pseudo-uniformizer `ϖ'` of `R`; (iii) `ϖ^{1/p^n}M = 0` for all
`n`, for one (equivalently any) pseudo-uniformizer `ϖ` with compatible `p`-power roots ([ECD] §3,
Definition 3.21, p. 19; [AWS] Definition 2.9.4, p. 73). The almost-zero modules form the Serre class
of P0.4, the almost categories `R°a-Mod` and `R⁺a-Mod` are the Serre quotients of P0.5 ([ECD]
Definition 3.23, p. 19), and restriction of scalars along `R⁺ → R°`, which is injective with almost
zero cokernel ([ECD] Example 3.22(ii), p. 19), induces an equivalence `R°a-Mod ≃ R⁺a-Mod` ([ECD]
after Definition 3.23, p. 19). The construction depends only on `R`. For a perfectoid field `K` it
is the classical setup `(K°, m_K)`, with `K°/m_K` almost zero and every almost zero `K°`-module a
`k`-vector space ([ECD] Example 3.22(i), p. 19). Define `almostSetup` (`almostSetup R : Almost.BasicSetup` with ring `R°` and ideal `topologicallyNilpotentIdeal R`) and `almostSetupPlus`
(`almostSetupPlus R⁺` with ring `R⁺` and ideal `R°° ∩ R⁺ = R°°`); the ideal of `almostSetup R` is
`topologicallyNilpotentIdeal R` by definition, its two fields being
`isIdempotentElem_topologicallyNilpotentIdeal` and `flat_tilde_topologicallyNilpotentIdeal`. Prove
`almostSetup_isAlmostZero_iff`
(`M` almost zero `↔ ∀ ϖ', IsPseudoUniformizer ϖ' → ϖ' • M = 0`), the root form
(`↔ ∀ n, ϖ^{1/p^n} • M = 0`, for any `ϖ♭` with compatible roots), that
`R° ⧸ R⁺` is almost zero as an `R⁺`-module,
`almostModuleEquivPlus` (restriction of scalars is an equivalence of the almost categories of `R°`
and `R⁺`), that a topological isomorphism of perfectoid Tate rings induces an
isomorphism of basic setups (the setup does not depend on `ϖ`), the field case (for a
perfectoid field `K`, `almostSetup K` is `(K°, m_K)`, the field case of P0.5) and the comparison
modulo `ϖ` (`R°/ϖ ≅ R♭°/ϖ♭` carries `R°°/ϖ` onto `R♭°°/ϖ♭`, identifying the almost
categories of `R°/ϖ` and `R♭°/ϖ♭`, used in P1.28). Theorem-level consumers: `H^i(Spa(R, R⁺), O⁺)`
is almost zero for `i > 0` ([ECD] Theorem 3.24; layer P2). *Needs:*
*exists-pseudouniformizer-with-compatible-roots* (P1.4); P1.3;
*topologically-nilpotent-elements-as-root-ideal*; P0.1; P0.2; P0.5; P0.6; Tau Ceti
`TauCeti.Huber.topologicallyNilpotentIdeal`, `TauCeti.Huber.IsRingOfIntegralElements`; Mathlib
`CategoryTheory.ObjectProperty.IsSerreClass`.

**Checks.**

- `almost_residue` (computation) — for `K` a perfectoid field, `K° ⧸ m_K` is almost
  zero and `ϖ^{1/p^n}` acts by `0` on it for all `n`; in particular this holds for `K = ℂ_p`
  ([ECD] Definition 3.21 and Example 3.22(i), p. 19).
- `not_idempotent_padic` (non-example) — for `ℚ_p`, `(pℤ_p)² ≠ pℤ_p`, so `(ℤ_p, ℤ_p°°)` is not a
  basic setup; `ℤ_p/p` is killed by `p` but not by an idempotent ideal containing `p`.
- `almost_zero_base` (degenerate) — for `R = 0` every module is almost zero.
- `almost_principal_quotient` (characterisation) — for `K = ℂ_p`, the `𝒪`-module `𝒪/m` is almost
  zero while `𝒪/p𝒪` is not (`p^{1/p}` does not kill the class of `1`).
- `almostModuleEquivPlus_test` (compatibility) — for `K` the completed perfection of
  `𝔽_p(x)((t))` and `K⁺ ⊊ K°` the preimage of the valuation ring `𝔽_p[x]^{perf}_{(x)}` of the
  residue field `𝔽_p(x)^{perf}` (an open valuation subring of rank `2`, with `K°° ⊊ K⁺`),
  `K°a-Mod ≃ K⁺a-Mod`, and `K°/K⁺` is almost zero.
- `pair_eq_powerBounded_padicComplex` (non-example) — `ℂ_p` has no second ring of integral
  elements: an open integrally closed `K⁺ ⊆ 𝒪_{ℂ_p}` contains `m_{ℂ_p}` and corresponds to an
  integrally closed subring of the residue field `𝔽̄_p`, which is `𝔽̄_p` itself (every element is
  integral over `𝔽_p`), so `K⁺ = 𝒪_{ℂ_p}`; in particular `ℂ_p` has no open valuation subring of
  rank `2`, and a test of `almostModuleEquivPlus` on `ℂ_p` compares `K°` with itself.

Membership in the ideal of `almostSetupPlus` is topological nilpotence in `R`.
`almostModuleEquivPlus_restrict` identifies its equivalence functor, after almostification,
with ordinary restriction of scalars from `R°` to `R⁺` followed by almostification.

**Checks.**

- `almostSetupPlus_zero`, `almostModuleEquivPlus_zero`: over the zero ring the ideal and every almost module vanish.
- `almostSetupPlus_full`, `almostModuleEquivPlus_full`: for `R⁺ = R°`, the ideal is `R°°` and the equivalence functor is the identity.
- `almostSetupPlus_proper`, `almostModuleEquivPlus_proper`: a proper plus subring omits some power-bounded element, but every element of `R°°` carries it into the plus ring, and restriction is still an equivalence.

Supporting result: `R°° = ⋃ ϖ^{1/p^n}R°` is an idempotent flat ideal
(*topologically-nilpotent-elements-as-root-ideal*). Let `R` be a Tate
ring and `ϖ♭ = (ϖ^{1/p^n})_n` a pseudo-uniformizer of `R` with compatible `p`-power roots (for `R`
perfectoid, P1.3). Then the ideal `R°°` of topologically nilpotent elements of `R°`
(`TauCeti.Huber.topologicallyNilpotentIdeal`) equals `⋃_n ϖ^{1/p^n}R° = ⋃_{q∈ℤ[1/p]_{>0}} ϖ^qR°`;
it satisfies `(R°°)² = R°°` and is flat as an `R°`-module (a filtered union of the free rank-one
modules `ϖ^{1/p^n}R°`). For every ring of integral elements `R⁺`, `R°° ⊆ R⁺` and
`R°° = ⋃_n ϖ^{1/p^n}R⁺`. Hence `(R°, R°°)` and `(R⁺, R°°)` satisfy the hypotheses of
*root-ideal-basic-setup* (P0.2); the argument uses only that `R°` is open and integrally closed in
`R`, and no uniformity ([ECD] §3, Definition 3.21, p. 19, and §3 preamble, p. 14;
[Bhatt] Lemma 3.1.3(2), p. 10; Mathlib `Module.Flat`; Tau Ceti
`TauCeti.Huber.IsPowerBounded.isTopologicallyNilpotent_mul`,
`TauCeti.Huber.IsRingOfIntegralElements.mem_of_isTopologicallyNilpotent`). For a perfectoid field
`K`, `K°° = m_K` with `m_K² = m_K`; for `ℚ_p`, `ℤ_p°° = pℤ_p` and `(pℤ_p)² ≠ pℤ_p`: without
`p`-power roots the ideal of topologically nilpotent elements is not idempotent.

Lean: `isIdempotentElem_topologicallyNilpotentIdeal`, `flat_tilde_topologicallyNilpotentIdeal`
(*topologically-nilpotent-elements-as-root-ideal*), `almostSetup`, `almostSetup_isAlmostZero_iff`.

### P1.10 perfectoid fields

Perfectoid fields follow: the definition, its characterisation, the Gelfand-spectrum and uniform
Banach-field inputs behind Kedlaya's theorem that a perfectoid Tate ring which is a field is a
perfectoid field, the tilt of a perfectoid field against Mathlib's `Tilt`, primitive elements of
`W(R⁺)`, and the identification of Fontaine's `θ` with Mathlib's.

**Perfectoid fields.** (*perfectoid-field-definition*) A perfectoid field is a perfectoid Tate ring
`K` (P1.1) whose underlying topological ring is a nonarchimedean field: `K` is a field, complete
with respect to a nontrivial nonarchimedean absolute value `|·|` defining its topology (in Lean,
`NontriviallyNormedField K` with `IsUltrametricDist K` and `CompleteSpace K`; the Tate structure has
ring of definition `K°`, the closed unit ball, and any `0 < |ϖ| < 1` as pseudo-uniformizer; `K` is
nonzero, so `p` is not a unit of `K°`; residue characteristic `p` is forced by `|p| < 1`) ([ECD] §3,
Definition 3.6, p. 15). Equivalently (P1.11; [Sch12] Definition 3.1, p. 15; [Bhatt] Definition 3.1.1,
p. 9): `K` is a nonarchimedean field which is not discretely valued, `|p| < 1`, and Frobenius is
surjective on `K°/p`. Its value group is `p`-divisible (*perfectoid-field-value-group-p-divisible*,
P1.15). In characteristic `p` a perfectoid field is the same as a complete perfect nonarchimedean
field ([Sch12] §3, after Lemma 3.2, p. 15). A perfectoid Tate ring whose underlying ring is a field
is automatically a perfectoid field (P1.14); `ℚ_p` and all discretely valued fields are not
perfectoid. Define `IsPerfectoidField` (`class IsPerfectoidField (K) [NontriviallyNormedField K] [IsUltrametricDist K] [CompleteSpace K] : Prop`, extending `IsPerfectoidTateRing p K`) and
use Tau Ceti's Huber/Normed instance `TauCeti.Huber.IsTateRing.of_nontriviallyNormedField`
(every nontrivially normed ultrametric field, without a completeness hypothesis, is a Tate ring
with ring of definition the closed unit ball). The power-bounded subring is the closed unit ball
and the topologically nilpotent ideal is the open unit ball, using
`TauCeti.Huber.IsUniform.of_normedDivisionRing`, `isPowerBounded_iff_norm_le_one` and
`isBounded_closedBall_zero`. The compatibility signature at pin `a91d3aaf` and its canonical
supplier are specified in the internal-dependency note above;
prove
`IsPerfectoidField.toIsPerfectoidTateRing`, `IsPerfectoidField.iff` (`IsPerfectoidField K ↔`
(valuation not discrete, in Lean: norms accumulate at `1`) `∧ ‖p‖ < 1 ∧ Surjective (frobenius (K°/p) p)`, P1.11),
the bound `‖p‖ < 1`, the existence of `ϖ` with `0 < ‖ϖ‖ < 1 ∧ ‖p‖ ≤ ‖ϖ‖ ^ p`, `IsPerfectoidField.valueGroup_pDivisible` (every element of `|K^×|` has a `p`-th root
in it), `IsPerfectoidField.iff_perfectRing` (for `CharP K p`: `IsPerfectoidField K ↔ PerfectRing K p`, P1.2), the statement that a perfectoid Tate ring which is a field is a perfectoid field (P1.14,
from `exists_absoluteValue_of_isField`), `tiltField.instIsPerfectoidField`
(the tilt of a perfectoid field is a perfectoid field of characteristic `p`, P1.15) and
`isPerfectoidField_padicComplex` (`ℂ_p`, Mathlib `PadicComplex p`, is a perfectoid field). *Needs:*
P1.1; Mathlib `NontriviallyNormedField`, `IsUltrametricDist`, `CompleteSpace`; Tau Ceti
`TauCeti.Huber.IsTateRing`, `TauCeti.Huber.Padic.isTateRing`.

**Checks.**

- `field_padic_negative` (non-example) — `¬ IsPerfectoidField ℚ_[p]`: `ℚ_p` is discretely
  valued ([ECD] Remark 3.3), although Frobenius on `ℤ_p/p = 𝔽_p` is surjective.
- `field_padicComplex` (computation) — `IsPerfectoidField (PadicComplex p)`: the
  value group is `|p|^ℚ` and every element of `𝒪_{ℂ_p}` has a `p`-th root.
- Characterisation: for the completed perfection `K` of
  `𝔽_p((t))`, `IsPerfectoidField K` holds; `𝔽_p((t))` is not perfect and not a perfectoid field.
- `not_isPerfectoidField_trivial` (degenerate) — the perfect field `𝔽̄_p` with the trivial absolute
  value is not a nonarchimedean field in this sense (the absolute value must be nontrivial), so it
  is not a perfectoid field.
- Compatibility: for `K` a field with
  `IsPerfectoidTateRing K`, `IsPerfectoidField K` (Kedlaya, P1.14).

Mathlib’s proposed [perfectoid-field interface](https://github.com/leanprover-community/mathlib4/pull/26389)
uses a rank-one valuation. The normed-field predicate here fixes a real-valued norm;
a comparison theorem, specified in this README only, matches it with the valuation formulation after choosing
a compatible absolute value. The comparison preserves topology, not a prescribed normalisation.

Lean: `IsPerfectoidField`, `isPerfectoidField_padicComplex`, `IsPerfectoidField.valueGroup_pDivisible`,
and the Huber/Normed compatibility signature `TauCeti.Huber.IsTateRing.of_nontriviallyNormedField`
(the Tate structure, also listed under P1.1). The `ℚ_p` boundary is
`Perfectoid.not_isPerfectoidTateRing_padic`; the two proved field examples are `ℂ_p` and
`padicCyclotomic`.

### P1.11 characterisation of perfectoid fields

**Characterisation of perfectoid fields.** (*perfectoid-field-characterisation*) Let `K` be a
nonarchimedean field (complete with a nontrivial rank-one nonarchimedean absolute value). Then `K`
is a perfectoid field iff (i) `K` is not discretely valued, (ii) `|p| < 1`, and (iii) Frobenius
`Φ : K°/p → K°/p` is surjective ([ECD] §3, Proposition 3.8 with (i)–(iii), p. 15; [Berkeley]
Proposition 6.1.9, proof, p. 43). Condition (i) cannot be dropped: `ℚ_p` and its unramified
extensions satisfy (ii) and (iii) ([ECD] Remark 3.3, p. 14; [Sch12] §3, after Definition 3.1,
p. 15). Hence [Sch12] Definition 3.1 agrees with [ECD] Definition 3.6, and a nonarchimedean field is
a perfectoid ring in Kedlaya's sense iff it is a perfectoid field ([AWS] Corollary 2.3.11, p. 55).
`ℂ_p`, `ℚ_p^cycl` and the completion of `ℚ_p(p^{1/p^∞})` satisfy (i)–(iii) (P1.32a). *Needs:* P1.10;
Mathlib `NontriviallyNormedField`, `IsUltrametricDist`.

Lean: `IsPerfectoidField.iff`, with (i) stated as the accumulation of the norms at `1`.

### P1.12 Gelfand spectrum of a Banach ring

**The unit criterion on the Gelfand spectrum of a Banach ring.**
(*gelfand-spectrum-units-and-maximum-formula*) The Gelfand spectrum `M(A)` of a commutative
nonarchimedean Banach ring `A` (complete for a submultiplicative norm `|·|` with
`|a + b| ≤ max(|a|, |b|)`; not assumed to be over a field) — the bounded multiplicative seminorms
`α : A → ℝ_{≥0}` with the topology of pointwise convergence — together with its compactness, its
nonemptiness for `A ≠ 0` and Berkovich's maximum formula `|f|_sp = max_{α ∈ M(A)} α(f)` for the
spectral seminorm, is AdicSpacesPartIII R3.9 (*gelfand-spectrum-banach-ring*, API
`GelfandSpectrum`, `GelfandSpectrum.compactSpace_nonempty`, `GelfandSpectrum.smoothingFun_eq_sSup`);
it is used
here for arbitrary nonarchimedean Banach rings, with the hypotheses stated there. The one clause
absent there is this target's content: `f ∈ A` is a unit iff `α(f) ≠ 0` for every `α ∈ M(A)`
([KBF] Lemma 2.4, proof, p. 4). For a complete Tate ring the carrier `M(A)` is the Berkovich
spectrum that DiamondsAndVStacks D5 uses downstream. The unit criterion and the maximum formula are the form
used by Kedlaya's Banach-field theorem (P1.13) and by the pointwise sharp-approximation lemma
(*sharp-approximation-lemma*, P1.23). *Needs:* AdicSpacesPartIII R3.9
(*gelfand-spectrum-banach-ring*); Mathlib `MulRingSeminorm`, `RingSeminorm`,
`NonarchimedeanAddGroup`.

The proposed Lean form of the unit criterion, with no declaration in `Suggested.lean` yet, is
`gelfandSpectrum_isUnit_iff`, for a complete ultrametric
normed commutative ring. Its spectrum argument is written as the subtype of multiplicative
seminorms bounded by `C‖·‖` for some `C > 0`, the defining subtype of the spectrum owned by
AdicSpacesPartIII R3. No second spectrum definition is introduced.

### P1.13 uniform Banach fields over a nondiscretely valued field

**A uniform Banach field over a nondiscretely valued field is a nonarchimedean field.**
(*uniform-banach-field-over-nondiscrete-field*) Let `F` be a nonarchimedean field whose value group
`|F^×|` is dense in `ℝ_{>0}` (the discretely valued case is not covered by the proof, [KBF] Remark
3.9), and let `A` be a uniform Banach `F`-algebra (complete, with topology defined by a
power-multiplicative norm; equivalently the spectral seminorm `|·|_sp` is a norm defining the
topology) whose underlying ring is a field. Then `|·|_sp` is multiplicative, so `A` is a
nonarchimedean field ([KBF] Theorem 3.7 and its proof, pp. 10–11). The criterion used: a uniform
Banach field `A` is a nonarchimedean field iff `|t|_sp|t^{-1}|_sp = 1` for all `t ∈ A^×` ([KBF]
Remark 3.3, p. 8), because then `|x|_sp = |xy·y^{-1}|_sp ≤ |xy|_sp/|y|_sp`; the proof uses the
Berkovich spectrum (AdicSpacesPartIII R3.9 through P1.12: `M(A) ≠ ∅`, the unit criterion and
`|x|_sp = max_α α(x)`; [KBF] Lemma 2.4, proof, p. 4). Uniformity is essential: [KBF] Example 2.15, pp. 6–7, completes
`F(T_1,T_2,…)` for the norm
`β(x) = inf_{x=Σ_i x_i} max_i 2^{f(x_i)} α(x_i)`, where `α` is the multiplicative Gauss norm
and `f(x)` is the least number of initial variables needed to express `x`. This completion is
a nonuniform Banach field. The completion for `α` itself is a nonarchimedean field. The new
unit-criterion target of P1.12 is `GelfandSpectrum.isUnit_iff`; the norm target here is
`UniformBanachField.spectralNorm_mul` under the hypotheses of this paragraph. *Needs:* P1.12; Mathlib `NontriviallyNormedField`, `IsUltrametricDist`,
`CompleteSpace`.

### P1.14 a perfectoid Tate ring that is a field is a perfectoid field

**A perfectoid Tate ring whose underlying ring is a field is a perfectoid field.**
(*perfectoid-ring-field-is-perfectoid-field*) Let `A` be a perfectoid Tate ring (in the sense of
P1.1, with no perfectoid base field: such `A` need not be an algebra over any perfectoid field,
[KBF] Remark 4.3, p. 13) whose underlying ring is a field. Then the topology of `A` is defined by a
multiplicative absolute value, so `A` is a perfectoid field ([KBF] Theorem 4.2 and its proof (the
characteristic-`p` case via the base field `𝔽_p((z))^{perf}`, then the tilt), p. 13; answers [ECD]
§3, Remark 3.7, p. 15; [Berkeley] Remark 6.1.8, p. 43; [AWS] Theorem 2.9.1, p. 73). The theorem is
needed when the topology is given only by a ring of definition, e.g. for the field quotients
`ℚ_p^cycl` and `ℚ_p(p^{1/p^∞})^` of Kedlaya's perfectoid ring over `ℚ_p` that lies over no
perfectoid field. *Needs:* P1.13; *tilt-is-perfectoid-tate-ring*; *perfectoid-field-iff-tilt-field*;
P1.5.

Two supporting results. The tilt of a perfectoid Tate ring is a perfectoid Tate ring of
characteristic `p` (*tilt-is-perfectoid-tate-ring*): for `R` perfectoid, `R♭` is a complete
Hausdorff uniform perfect Tate ring of characteristic `p`, hence a perfectoid Tate ring; `ϖ♭` is a
pseudo-uniformizer, `(R♭°, ϖ♭R♭°)` is a pair of definition (`TauCeti.Huber.PairOfDefinition`), and
`R♭ ≠ 0` iff `R ≠ 0` ([ECD] Lemma 3.10, p. 16; [Berkeley] Lemma 6.2.2, p. 44; [AWS] Lemma 2.7.1(b),
p. 66). A perfectoid Tate ring is a perfectoid field iff its tilt is (*perfectoid-field-iff-tilt-field*): for
`R` perfectoid with `ϖ♭` with compatible roots, `R` is a perfectoid field (a nonarchimedean field)
iff `R♭` is, and then the absolute values satisfy `|x|_{R♭} = |x♯|_R` when normalised by
`|ϖ♭|_{R♭} = |ϖ|_R`, the spectral gauges being those of P1.3 ([Sch12] Lemma 5.21 and its proof,
pp. 27–28; [KL15] Lemma 3.5.2(a), p. 82, and Example 3.3.8, p. 78; [AWS] Corollary 2.3.11). For a
perfectoid field `K`, `K × K♭` is perfectoid but not a field, and its tilt `K♭ × K♭` is not a field.

`UniformBanachField.spectralNorm_mul` states the spectral multiplicativity assertion
on a complete ultrametric normed `F`-algebra whose ring is a field. It assumes boundedness of the power-bounded elements for the given
norm and a nondiscrete value group on the complete nontrivially normed field `F`, as in [KBF]
Theorem 3.7, pp. 10–11. `exists_absoluteValue_of_isField` starts with the complete
Hausdorff perfectoid Tate topology and an algebraic field structure; its conclusion supplies an
ultrametric absolute value, a value strictly between zero and one, and the original neighbourhood
basis. It does not assume a multiplicative norm on that field ([KBF] Theorem 4.2, p. 13).

Lean: `exists_absoluteValue_of_isField`.

### P1.15 the tilt of a perfectoid field

**Tilt of a perfectoid field; comparison with Mathlib's Tilt and PreTilt.val.**
(*tilt-of-perfectoid-field*) Let `K` be a perfectoid field. Its tilt `K♭` (P1.4) is a perfectoid
field of characteristic `p` with absolute value `|x|_♭ := |x♯|`; `K♭° = lim_{x↦x^p} K° ≅ lim_Φ K°/ϖ`
is its valuation ring for any `ϖ ∈ K` with `|p| ≤ |ϖ| < 1` (so `ϖ | p` in `K°`); `|K♭^×| = |K^×|`;
`K♭°/ϖ♭ ≅ K°/ϖ` and `K♭°/m♭ ≅ K°/m`; `K♭ = K` if `char K = p`. Replacing `ϖ` by `(ϖ♭)♯` one may
assume `ϖ` has compatible `p`-power roots ([Sch12] Lemma 3.4(iii)–(iv) and Remark 3.5, p. 16;
[Berkeley] Remark 6.2.3, p. 44; [KL15] Lemma 3.5.2(a), p. 82). For `v : Valuation K ℝ≥0` the
absolute value, `O = K°` and `hv : v.Integers O`, the tilt is isomorphic to Mathlib's
`Tilt K v O hv p = FractionRing (PreTilt O p)`, and Mathlib's valuation `PreTilt.val K v O hv p`
satisfies `PreTilt.val x = v(x♯)` for `x ∈ PreTilt O p ≅ K♭°`. Prove
`tiltField.instIsPerfectoidField` (`tilt K` is a perfectoid field with this absolute
value), `tiltField.norm_def` (`‖x‖_{K♭} = ‖x♯‖_K`), `tiltField.valueGroup_eq` (`|K♭^×| = |K^×|` as
subgroups of `ℝ_{>0}`), `tiltField.integerEquiv` (`K♭° ≃+* lim_Φ K°/ϖ` for `ϖ | p`),
`tiltField.quotEquiv` (`K♭° ⧸ (ϖ♭) ≃+* K° ⧸ (ϖ)`), `tiltField.residueFieldEquiv`,
`tiltField.equivMathlibTilt` (`tilt K ≃+* Tilt K v O hv p`, restricting to `(tilt K)° ≃+* PreTilt O p`), `tiltField.preTiltVal_eq` (`PreTilt.val K v O hv p x = v (PreTilt.untilt x)`),
`tiltField.equivSelfOfCharP` (for `CharP K p`, `tilt K ≃+* K` isometrically) and `tiltField.map`
(a continuous embedding induces a continuous embedding of tilts; it is isometric if the
original embedding preserves the chosen absolute values, P1.6). For
`K = ℚ_p(p^{1/p^∞})^`, `K♭ = 𝔽_p((t^{1/p^∞}))^` with `t = (p, p^{1/p}, …)` and `t♯ = p`. *Needs:*
P1.4; P1.5; P1.8; *tilt-is-perfectoid-tate-ring*; *perfectoid-field-iff-tilt-field*; Mathlib
`Tilt`, `PreTilt.val`, `PreTilt.isDomain`, `Perfection.teichmuller_spec`, `Valuation.Integers`.

**Checks.**

- `tiltField_padicComplex_pflat` (computation) — in `tilt ℂ_p`, `p♭ = (p^{1/p^n})_n` has
  `‖p♭‖ = ‖p‖` and `‖ε − 1‖ = ‖p‖^{p/(p−1)}`.
- `tiltField.equivMathlibTilt` (compatibility) — for `K = ℂ_p`: `tilt ℂ_p ≃+* Tilt ℂ_p v 𝒪_{ℂ_p} hv p` and `PreTilt.val` corresponds to the absolute value of the tilt.
- `tiltField.equivSelfOfCharP` (degenerate) — for `K` the completed perfection of `𝔽_p((t))`,
  `tilt K ≃+* K`.
- `uniformTilt_padic` (non-example) — for `ℚ_p` (not perfectoid), `PreTilt ℤ_p p ≃+* Perfection 𝔽_p p ≃+* 𝔽_p` and Mathlib's `Tilt ℚ_p v ℤ_p hv p ≃+* 𝔽_p` carries the trivial
  valuation. Here `v : Valuation ℚ_p ℝ≥0`, `hv : v.Integers ℤ_p`, and `v(p) ≠ 1` are Mathlib's
  inputs. The Check specifies the extension of `PreTilt.val` to the fraction field and its
  value `1` at every nonzero element. Thus the tilt lacks a nontrivial absolute value without
  the perfectoid hypothesis ([Sch12] Definition 3.1 and Lemma 3.4, pp. 15–16;
  Mathlib `PreTilt`, `PreTilt.val`, `Tilt` in `Mathlib/RingTheory/Perfection.lean`).

Two supporting results. The value group of a perfectoid field is `p`-divisible
(*perfectoid-field-value-group-p-divisible*): for `K` perfectoid with `Γ = |K^×| ⊆ ℝ_{>0}`, `Γ` is
`p`-divisible, and in characteristic `0` it is generated by the `|x|` with `|p| < |x| ≤ 1`; in
characteristic `p` this follows from perfectness ([Sch12] Lemma 3.2 and its proof, p. 15; [Bhatt]
Lemma 3.1.3(1), p. 10). `|ℂ_p^×| = |p|^ℚ` is `p`-divisible; `|ℚ_p^×| = |p|^ℤ` is not. Continuous
valuations of `K` and `K♭` correspond (*continuous-valuations-under-tilting*): for a perfectoid
field `K` with tilt `K♭`, `|·| ↦ |·|♭`, `|x|♭ := |x♯|`, is a bijection between continuous
valuations of `K` up to equivalence and continuous valuations of `K♭` up to equivalence, higher
rank included; continuity is Huber's (`TauCeti.ValuationSpectrum.cont`), and continuous valuations
correspond to open valuation subrings `K⁺ ⊆ K°`, which contain `m`, i.e. to valuation subrings of
the common residue field `K°/m = K♭°/m♭` ([Sch12] Proposition 3.6 and its proof, p. 17; Mathlib
`Valuation.IsEquiv`). For a prime `p` and a complete nontrivially normed ultrametric field
`K` satisfying `IsPerfectoidField p K`, the comparison of continuous valuations states that
there is a homeomorphism from `spa (fieldMinimalPlus K)` to
`spa (Pair.tilt p (fieldBasePair p K).pair).plus` whose forward map is `spaTilt`.
The minimal plus ring is contained in every open valuation subring, so these carriers
include all continuous valuation classes, with no rank restriction. The target
*continuous-valuations-under-tilting* follows `fieldBasePair` in P2 and uses
`spaTiltHomeomorph` for that pair ([Sch12] Corollary 6.7(iii), pp. 34–35;
[Wedhorn] Remark 7.15(2), p. 60).

The normed presentation `tiltField p K` uses the compatible-sequence carrier of
`tilt p K`. `tiltField.toTilt_apply` identifies its ring operations, `norm_def` specifies its
norm, and `continuous_toTilt` identifies the topology. The global signatures `valueGroup_eq`,
`integerEquiv`, `residueFieldEquiv`, `equivMathlibTilt_integral` and `preTiltVal_eq` retain the
specified valuation and integral ring; Mathlib comparisons explicitly assume the library's
nonunit and adic-completeness hypotheses. The residue quotients use the ideals `‖x‖ < 1` in
the two power-bounded subrings. `equivSelfOfCharP` is an isometry for the specified norm.

`isField_iff_tilt_isField` states the field criterion without choosing a norm, including the
zero-ring boundary on both sides. The bound `‖p‖ < 1` and
`IsPerfectoidField.valueGroup_pDivisible` state the strict p bound and divisibility as an existence of a unit
whose norm has the prescribed p-th power ([Sch12] Lemmas 3.4 and 5.21).

### P1.16 primitive elements and ideals of degree one

**Primitive (distinguished) elements and ideals of degree 1 in `W(R⁺)`.**
(*primitive-degree-one-ideals*) Let `(R, R⁺)` be a perfectoid Tate pair of characteristic `p`. An
element `ξ = Σ_{n≥0} p^n[ξ_n] ∈ W(R⁺)` is primitive of degree `1` (distinguished) if `ξ_0` is
topologically nilpotent and `ξ_1` is a unit of `R⁺`; equivalently `ξ = [ξ_0] + p·u` with
`u ∈ W(R⁺)^×` ([AWS] Definition 2.3.4, p. 54; [KL16] Definition 3.2.3, p. 57). An ideal `J ⊆ W(R⁺)`
is primitive of degree `1` if it is principal with a primitive generator. Equivalently ([ECD] §3,
Definition 3.15, p. 17), `J` is generated by an element `p + [ϖ♭]α` with `ϖ♭ ∈ R⁺` a
pseudo-uniformizer and `α ∈ W(R⁺)`: every such element is primitive, and every primitive element is
a unit multiple of one of this form, because a topologically nilpotent element of `R⁺` is divisible
by a pseudo-uniformizer ([KL16] Lemma 3.2.2, p. 56; [AWS] Exercise 2.3.6, p. 54). `ξ_0 = 0` is
allowed ([KL16] Remark 3.2.5, p. 57): `ξ = p` is primitive and corresponds to the untilt `R`
itself; `p²` and `[ϖ♭]` are not primitive when `R ≠ 0`; over the zero ring every Witt vector is primitive. In the notation of an untilt `S`, `R = S♭` and
`R⁺ = S♭⁺`. The coordinate predicates themselves are defined for any commutative topological ring `S`;
the equivalences and regularity results below concern nonzero characteristic-`p` perfectoid
pairs (`CharP R p` in Lean) with perfect plus rings as above. The zero coordinate predicates
are tested separately.
Define `IsPrimitive` (for `ξ : WittVector p R⁺`: `IsTopologicallyNilpotent (ξ.coeff 0 : R) ∧ IsUnit (ξ.coeff 1)`) and `IsPrimitiveIdeal` (`J : Ideal (WittVector p R⁺)` with `∃ ξ, IsPrimitive ξ ∧ J = Ideal.span {ξ}`), together with `IsDistinguished` (a Witt vector whose first
Witt coordinate is a unit, the notion of [BMS] Remark 3.11, p. 23, used for `ker θ` in P1.18);
prove the Teichmüller form (`IsPrimitive ξ ↔ ∃ u : (W R⁺)ˣ, ξ = teichmuller p (ξ.coeff 0) + p * u ∧ IsTopologicallyNilpotent (ξ.coeff 0)`), that
`IsPseudoUniformizer ϖ → IsPrimitive (p + teichmuller p ϖ * α)` for every `α`, that every primitive
`ξ` is associated to such an element (`IsPrimitive ξ → ∃` unit `u`, pseudo-uniformizer `ϖ`, `α`,
`ξ = u * (p + [ϖ♭] α)`), that associates of primitive elements are primitive, the unit criterion
(`IsPrimitive ξ → IsPrimitive ξ' → ξ' = y * ξ → IsUnit y`,
*distinguished-element-criterion*, P1.18), that primitive elements are
nonzerodivisors ([Berkeley] Lemma 6.2.10, p. 46; [ECD] Lemma 3.16, p. 17),
`IsPrimitive.map` (for a morphism of perfectoid pairs of characteristic `p`,
`WittVector.map f` carries primitive elements to primitive elements, hence `J` to generators of
`J·W(R'⁺)`), that `p : W R⁺` is primitive (coefficients `(0, 1, 0, …)`),
`IsPrimitive.frobenius` (`φ(ξ)` and `φ^{-1}(ξ)` are primitive) and that a generator of `ker θ` is
primitive iff it is distinguished in the sense of [BMS] Remark 3.11, p. 23 (`IsDistinguished`). *Needs:* P1.2;
*witt-vectors-of-perfect-plus-ring*; Tau Ceti `TauCeti.Huber.IsPseudoUniformizer`; Mathlib
`WittVector`, `WittVector.teichmuller`.

**Checks.**

- `primitive_mixed_generators`, `primitive_two_coordinates` (computation) — in `W(𝒪_{ℂ_p♭})`, `p − [p♭] = p + [p♭]·(−1)` is
  primitive: its Witt coordinates are `(−p♭, 1, …)` for `p` odd and, since `−[x] = (x, x², x⁴, …)`
  for `p = 2`, `(p♭, 1 + (p♭)², …)` for `p = 2`; in both cases `ξ_0` is topologically nilpotent
  and `ξ_1` a unit. So is `Σ_{i=0}^{p−1}[ε]^{i/p}` (its image in `W(𝔽̄_p)` is `p`); in the proposed
  Lean form the element `ε` of `PrimitiveTests` is normalised by `coeff n ε = ζ_{p^{n+1}}`, i.e. it is
  the `ε^{1/p}` of P1.4, so the sum reads `Σ_{i<p} [ε]^i` there.
- `primitive_p_squared_and_teichmuller` (non-example) — for `R ≠ 0`, `p²` is not primitive (its Witt coordinate `ξ_1` is `0`),
  and neither is `[ϖ♭]`; a definition requiring only `ξ ∈ (p, [ϖ♭])` would accept them.
- `primitive_p` (degenerate) — `p` is primitive, and the untilt along `(p)` is `(R, R⁺)`
  itself.
- On any topological commutative ring of characteristic `p`, `(p)` is a primitive ideal.
  Over a nonzero such ring, `(p²)` and the zero ideal are not: every element of either ideal
  has first Witt coordinate zero, so neither ideal has a primitive generator.
- Characterisation: in `W(𝒪_{ℂ_p♭})`, `Σ_{i<p}[ε]^{i/p}` is a
  unit multiple of an element `p + [ϖ♭]α`.

Supporting result: `A_inf = W(R⁺)` for a perfectoid pair of characteristic `p`
(*witt-vectors-of-perfect-plus-ring*). Let `(R, R⁺)` be a nonzero perfectoid Tate pair of characteristic
`p` (so `R` and `R⁺` are perfect, P1.2) and `ϖ♭ ∈ R⁺` a pseudo-uniformizer. Then `R⁺` is perfect,
`ϖ♭`-adically complete and `ϖ♭`-torsion free, and `A_inf(R, R⁺) := W(R⁺)` satisfies: (a) `W(R⁺)` is
`p`-torsion free and `p`-adically complete with `W(R⁺)/p = R⁺` (a strict `p`-ring; Mathlib
`WittVector.eq_zero_of_p_mul_eq_zero` and `WittVector.isAdicCompleteIdealSpanP`, using
the perfectness of `R⁺` from P1.2);
(b) every element is uniquely `Σ_{n≥0} p^n[x_n]` with
`x_n ∈ R⁺`; (c) `[ϖ♭]` is a nonzerodivisor, `W(R⁺)` is `[ϖ♭]`-adically complete and separated, and
`W(R⁺)/(p,[ϖ♭]) = R⁺/ϖ♭` — the `(p,[ϖ♭])`-adic completeness and separatedness of `W(R⁺)`, and the
Huber-ring structure of the `(p,[ϖ♭])`-adic topology, are Tau Ceti's
`WittVector.isAdicComplete_span_p_teichmuller`, `isHausdorff_span_p_teichmuller`,
`completeSpace_and_t2Space_adicTopology_span_p_teichmuller` and
`TauCeti.Huber.WittVector.isHuberRing_adicTopology_span_p_teichmuller` (for `R⁺` perfect and
`ϖ♭`-adically complete), and `isTateRing_adicTopology_span_p_teichmuller_iff` says that `W(R⁺)` with
this topology is not Tate; (d) `Σ p^n[x_n]` is a unit iff `x_0` is a unit
of `R⁺`; (e) the `(p,[ϖ♭])`-adic topology does not depend on `ϖ♭`; (f) `W(R⁺)[1/[ϖ♭]]` is the ring
`W^b(R)` of Witt vectors with bounded Teichmüller coordinates, independent of `ϖ♭` and of `R⁺` up to
the evident inclusions; (g) for every `p`-adically complete ring `S` and ring homomorphism
`R⁺ → S/p` there is a unique ring homomorphism `W(R⁺) → S` lifting it; Frobenius `W(Φ)` is an
automorphism ([Bhatt] Example 6.1.6, p. 43; [BMS] Lemma 3.2(ii), p. 19; [AWS] Definition 2.2.1,
Lemma 2.2.7 and Theorem 2.2.10, pp. 51–52; [Sch12] Remark 5.14, p. 25; Mathlib
`WittVector.teichmuller`, `Localization.Away`; Tau Ceti `WittVector.isAdicComplete_span_p_teichmuller`,
`TauCeti.Huber.WittVector.isHuberRing_adicTopology_span_p_teichmuller`). For `R = F` the perfect
field of AdicSpaces layer 6 and `R⁺ = 𝒪_F` the completeness clause is the library statement for
`A_inf = W(𝒪_F)` (P1.33).
One statement, specified in this README only, records `ϖ♭`-adic completeness and torsion freeness
of `R⁺`, the quotient `W(R⁺)/p ≃ R⁺`, and the three assertions in (c).

Lean: `IsDistinguished`, `IsPrimitive`, `IsPrimitiveIdeal`; the proved Lean example shows that
`p` is distinguished in characteristic `p`. The primitive-element and primitive-ideal Checks
are specified in prose.

### P1.17 ECD's `θ` is Mathlib's `fontaineTheta`

**ECD's `θ` is Mathlib's `WittVector.fontaineTheta`; surjectivity from `surjective_fontaineTheta`.**
(*fontaine-theta-comparison-with-mathlib*) Let `R` be a nonzero perfectoid Tate ring and `R⁺` a
ring of integral elements. Mathlib's `WittVector.fontaineTheta R⁺ p : 𝕎(PreTilt R⁺ p) →+* R⁺`
(defined under `[Fact ¬IsUnit (p : R⁺)]` and `[IsAdicComplete (span {p}) R⁺]`, both supplied by
*integral-elements-adically-complete*, P1.2) corresponds, under `R♭⁺ ≅ PreTilt R⁺ p` (P1.8), to
ECD's `θ : W(R♭⁺) → R⁺`, `Σ_n p^n[r_n] ↦ Σ_n p^n r_n♯`. In particular `θ([x]) = x♯`
(`WittVector.fontaineTheta_teichmuller`) and `θ mod p` is `x ↦ x♯ mod p`
(`WittVector.mk_fontaineTheta`). Mathlib's `surjective_fontaineTheta` applies because Frobenius on
`R⁺/p` is surjective by *frobenius-surjective-on-plus-ring*, so `θ` is surjective. `θ` is the
unique ring homomorphism `W(R♭⁺) → R⁺` lifting `x ↦ x♯ mod p` ([AWS] Lemma 2.2.7 and Definition
2.7.3, p. 67; [ECD] §3, Lemma 3.14(i) and its proof (the ring-map check through the ghost map modulo
`ϖ^m`), p. 17; [KL16] Definition 3.3.6, p. 59; [BMS] Definition 3.5, p. 21). Mathlib's
`fontaineThetaInvertP` and `BDeRhamPlus R⁺ p` are built on this `θ`. For `R = ℂ_p`,
`fontaineTheta` sends `[p♭]` to `p` and `[ε]` to `1`; for `R` of characteristic `p`, `θ` is
reduction modulo `p`, `W(R⁺) → R⁺` ([Bhatt] Proposition 6.1.8, proof). *Needs:* P1.8;
*frobenius-surjective-on-plus-ring*; Mathlib `WittVector.fontaineTheta`, `surjective_fontaineTheta`,
`WittVector.fontaineTheta_teichmuller`, `WittVector.mk_fontaineTheta`, `fontaineThetaInvertP`,
`BDeRhamPlus`, `PreTilt.untilt`, `WittVector.ghostComponent`.

Supporting result: every ring of integral elements of a perfectoid Tate ring satisfies the
Frobenius condition (*frobenius-surjective-on-plus-ring*). Let `R` be a perfectoid Tate ring and
`R⁺ ⊆ R°` any ring of integral elements (open and integrally closed, so `R°° ⊆ R⁺`,
`TauCeti.Huber.IsRingOfIntegralElements.mem_of_isTopologicallyNilpotent`). Then (a) Frobenius on
`R⁺/p` is surjective, and every element of `R⁺/ϖpR⁺` is a `p`-th power for every pseudo-uniformizer
`ϖ ∈ R⁺` with `ϖ^p | p` in `R⁺`; (b) there is a pseudo-uniformizer `ϖ ∈ R⁺` with `ϖ^p | p` in `R⁺`
and `p/ϖ^p ∈ R°°`, and for every pseudo-uniformizer `ϖ ∈ R⁺` with `ϖ^p | p` in `R⁺`,
`Φ : R⁺/ϖ → R⁺/ϖ^p` is bijective; (c) consequently `(R, R⁺)` satisfies [ECD] Definition 3.1 with
`R⁺` in place of `R°`, which is the formulation of [AWS] Definition 2.1.1 and [KL16] Definition
3.3.1, and these definitions agree with ECD's: the perfectoid condition on a Huber pair depends on
`R` alone ([BMS] Lemma 3.20, proof, p. 27; [AWS] Remark 2.3.12 and Corollary 2.3.10, p. 55; [KL16]
Proposition 3.3.3, p. 58). For `K` a perfectoid field and `K⁺ ⊆ K°` an open valuation subring of
rank `2` (one exists iff the residue field of `K` is not algebraic over `𝔽_p`, e.g. for the
completed perfection of `𝔽_p(x)((t))`; `𝒪_{ℂ_p}` is the only ring of integral elements of `ℂ_p`),
Frobenius on `K⁺/p` is surjective.

Lean: `Perfectoid.frobenius_surjective_plus` and
`Perfectoid.surjective_fontaineTheta_of_isPerfectoidTateRing` state the Frobenius and Fontaine
surjections on the pinned `ModP`, `PreTilt` and Witt-vector carriers.

### P1.18 Fontaine's `θ`: the primitive kernel

Fontaine's map `θ` is treated next: its primitive kernel for any perfectoid Tate
pair, the untilt along a primitive ideal, and the proof that the untilt is a perfectoid Tate pair
whose tilt is the pair one started from.

**Fontaine's `θ: W(S♭⁺) → S⁺` has kernel generated by a primitive nonzerodivisor.**
(*fontaine-theta-and-primitive-kernel*) Let `(S, S⁺)` be a perfectoid Tate pair (any
characteristic, no perfectoid base field) with tilt `(S♭, S♭⁺)`, and `θ : W(S♭⁺) → S⁺`,
`Σ[r_n]p^n ↦ Σ r_n♯p^n`, Fontaine's ring homomorphism. (a) `θ` is surjective, by Mathlib's
`surjective_fontaineTheta` through the identification of P1.17. (b) `ker θ` is generated by a
primitive element of degree `1`, which can be taken of the form `ξ = p + [ϖ♭]α` with `ϖ♭ ∈ S♭⁺` a
pseudo-uniformizer such that `(ϖ♭♯)^p | p` in `S⁺`, and `α ∈ W(S♭⁺)`; `ξ` is a nonzerodivisor.
(c) An element of `ker θ` generates `ker θ` iff it is primitive ([BMS] Remark 3.11, p. 23). (d) `θ`
extends to a surjection `W(S♭⁺)[1/[ϖ♭]] → S` with kernel `ξW(S♭⁺)[1/[ϖ♭]]`, and
`W(S♭⁺)/(ξ, [ϖ♭]) = S♭⁺/ϖ♭ ≅ S⁺/ϖ♭♯`. If `pS = 0` then `ker θ = (p)` ([ECD] §3, Lemma 3.14(ii) and
its proof (the auxiliary `f` with `f♯ ≡ p mod pϖR⁺`, reduction to an isomorphism modulo `[ϖ♭]`),
pp. 17–18; [Berkeley] Lemma 6.2.8, proof, p. 46; [AWS]
Lemma 2.7.5, p. 67; [KL15] Lemma 3.6.3, p. 88, for `ℚ_p`-algebras; [Bhatt] Proposition 6.1.8,
p. 43, the field case). For `S = ℂ_p`: `ker θ = (p − [p♭])`, and `Σ_{i<p}[ε]^{i/p}` is another
generator. *Needs:* P1.17; P1.16; *tilt-integral-subrings* (P1.5); P1.5;
*rings-of-integral-elements-under-tilting*; *distinguished-element-criterion*;
*quotient-by-primitive-element*.

Three supporting results. Rings of integral elements correspond under tilting
(*rings-of-integral-elements-under-tilting*): let `R` be perfectoid with tilt `R♭`, `ϖ♭` with
compatible roots satisfying `ϖ^p | p` in `R°`, where `ϖ = (ϖ♭)♯`; rings of integral elements are open, integrally closed and contained
in `R°`. (a) `R⁺ ↦ R♭⁺ := lim_{x↦x^p} R⁺ = {x ∈ R♭ : x♯ ∈ R⁺}` is an inclusion-preserving bijection
from rings of integral elements of `R` to those of `R♭`; the inverse sends `R♭⁺` to the preimage in
`R°` of the image of `R♭⁺` in `R♭°/ϖ♭ ≅ R°/ϖ`. (b) The isomorphism `R♭⁺/ϖ♭ ≅ R⁺/ϖ` of
*inverse-limit-of-integral-elements-along-frobenius* (P1.4) is compatible with `R♭°/ϖ♭ ≅ R°/ϖ`.
(c) On both sides, rings of integral elements correspond to integrally closed subrings of the
common quotient `R°/ϖ ≅ R♭°/ϖ♭` (equivalently of `R°/R°° ≅ R♭°/R♭°°`). Hence a perfectoid Tate pair
`(R, R⁺)` has tilt pair `(R♭, R♭⁺)`, and `R° ↦ R♭°` ([ECD] §3, Lemma 3.11, p. 17; [Berkeley] Lemma
6.2.5, p. 45; [Sch12] Lemma 6.2, proof, p. 30; [AWS] Lemma 2.7.10, p. 69; Tau Ceti
`TauCeti.Huber.IsRingOfIntegralElements`,
`TauCeti.Huber.mem_of_isTopologicallyNilpotent_of_isIntegrallyClosedIn`; Mathlib `integralClosure`,
`IsIntegrallyClosedIn`). For an open valuation subring `K⁺ ⊆ K°` of a perfectoid field, `K♭⁺` is
the open valuation subring of `K♭` with the same image in `K°/m = K♭°/m♭`; for `R = K⟨T^{1/p^∞}⟩`
and `R⁺ = R°`, `R♭⁺ = K♭°⟨T^{1/p^∞}⟩`. Primitive elements are nonzerodivisors and divide each
other only up to units (*distinguished-element-criterion*): for `(R, R⁺)` a perfectoid Tate pair
of characteristic `p` (`R⁺` perfect, `ϖ`-adically separated and `ϖ`-torsion free,
*witt-vectors-of-perfect-plus-ring*), (a) every primitive `ξ ∈ W(R⁺)`, in particular every
`p + [ϖ♭]α` with `ϖ♭` a pseudo-uniformizer of `R⁺` and `α ∈ W(R⁺)`, is a nonzerodivisor ([ECD] §3,
Lemma 3.16 and its proof (division by `ϖ♭` and induction), p. 17, where the printed `α ∈ W(R⁺)` is
a misprint for `α ∈ W(R♭⁺)` in ECD's notation; [Berkeley] Lemma 6.2.10, p. 46, in characteristic-`p`
notation; [BMS] Lemma 3.10(i), p. 22); (b) if `ξ, ξ'` are primitive and `ξ' = yξ` with `y ∈ W(R⁺)`,
then `y` is a unit ([AWS] Remark 2.3.5, p. 54), so two primitive elements generating comparable
ideals are associate. In `W(𝒪_{ℂ_p♭})`, `p − [p♭]` and `ξ_cyc = Σ_{i<p}[ε]^{i/p}` both generate
`ker θ`, hence are associates ([Sch13] Lemma 6.3); `μ = [ε] − 1 = ([ε]^{1/p} − 1)·ξ_cyc` lies in
`ker θ` but does not generate it, the factor `[ε]^{1/p} − 1` having zeroth Witt coefficient
`ε^{1/p} − 1` in the maximal ideal, so membership in `ker θ` alone does not imply primitivity.
`W(R⁺)/ξ` is `[ϖ]`-torsion free and `[ϖ]`-adically complete (*quotient-by-primitive-element*): for
`(R, R⁺)` a perfectoid Tate pair of characteristic `p`, choose a primitive generator
`ξ = p + [ϖ]α` with `ϖ ∈ R⁺` a pseudo-uniformizer and `α ∈ W(R⁺)` (P1.16), `A⁺ := W(R⁺)/ξ` satisfies: (a) the image of
`[ϖ]` is a nonzerodivisor; (b) `A⁺` is `[ϖ]`-adically complete and separated, and its `[ϖ]`-adic
and `(p,[ϖ])`-adic topologies coincide; (c) `A⁺/[ϖ] ≅ W(R⁺)/(p,[ϖ]) = R⁺/ϖ`; (d) `A⁺` is
`p`-adically complete; (e) `ξW(R⁺)[1/[ϖ]] ∩ W(R⁺) = ξW(R⁺)`. These are the two properties [ECD]
Lemma 3.14 uses without proof ([ECD] Lemma 3.14, proof, p. 18; [BMS] Lemma 3.2(ii), p. 19; [Stacks]
Tag 090T). For `ξ = p − [p♭]` in `W(𝒪_{ℂ_p♭})`, `W/ξ ≅ 𝒪_{ℂ_p}`; for `ξ = p`, `W(R⁺)/p = R⁺`.

`tilt.sharpQuotEquiv` uses a small integral tilted element `t`, its sharp in `R⁺`, and the
hypothesis `t♯ ∣ p` in `R⁺`; `sharpQuotEquiv_apply` identifies the quotient map on every element.
`Pair.tilt_bijective` and `Pair.tilt_plus_le_iff` give the bijection and its order compatibility.
`tilt.primitive_iff_generates_kernel` requires membership in the Fontaine kernel before testing
primitivity. The Teichmüller form, the associate form, the stability under unit multiples, the unit
criterion and `IsPrimitive.frobenius` of P1.16 are over a perfect characteristic-p Tate pair.
`primitive_quotient_complete` specifies the same Teichmüller parameter in regularity,
adic completeness, p-adic completeness and the small quotient ([Berkeley] Lemmas 6.2.5,
6.2.8 and 6.2.10; [AWS] Remark 2.3.5; [BMS] Lemma 3.2(ii)).

Lean: `ker_fontaineTheta` (the kernel of Mathlib's `WittVector.fontaineTheta` is generated by an
`IsDistinguished` element) and `Perfectoid.exists_regular_generator_ker_fontaineTheta`
(the generator can be chosen to be a nonzerodivisor).

### P1.19 the untilt along a primitive ideal

**The untilt of `(R, R⁺)` along a primitive ideal `J`: `(W(R⁺)[1/[ϖ]]/J, W(R⁺)/J)`.**
(*untilt-construction*) Let `(R, R⁺)` be a perfectoid Tate pair of characteristic `p`, `J ⊆ W(R⁺)`
a primitive ideal of degree `1` and `ϖ ∈ R⁺` a pseudo-uniformizer. The untilt of `(R, R⁺)` along
`J` is the pair `(A, A⁺)` with `A⁺ := W(R⁺)/J` and `A := A⁺[1/π]`, `π` the image of `[ϖ]`,
topologised so that `A⁺` is open with its `π`-adic (equivalently `(p,[ϖ])`-adic) topology;
equivalently `A = W^b(R)/J·W^b(R)` with `W^b(R) = W(R⁺)[1/[ϖ]]`. `A` is a complete Hausdorff Tate
ring with pseudo-uniformizer `π` and ring of definition `A⁺`, independent of the choice of `ϖ`; the
construction is functorial for morphisms of pairs `f` with `W(f)(J) ⊆ J'` (then
`J' = W(f)(J)·W(R'⁺)`), and comes with the multiplicative map `♯_J : R → A`, given on
`x ∈ R⁺` by `θ_J([x])` (`θ_J` the quotient map followed by localization), and extended by
inverting a pseudo-uniformizer, with `♯_J(ϖ) = π`. That `(A, A⁺)` is a perfectoid Tate pair with tilt `(R, R⁺)`
is P1.20 ([ECD] §3, Theorem 3.17, p. 18; [Berkeley] Theorem 6.2.11, p. 47; [AWS] Theorem 2.3.9,
p. 55, and Remark 2.3.3, p. 54; [Sch12] Remark 5.19, p. 27). Define `untilt` (`untilt (P : Pair R) J : Type`, the ring `W(R⁺)[1/[ϖ]] ⧸ J·W(R⁺)[1/[ϖ]]`, with the choice-free description
`W^b(R)/JW^b(R)`), `untiltPlus` (the subring `A⁺`, the image of `W(R⁺) ⧸ J` in `A`),
`untilt.instIsTateRing` (`IsTateRing`, `CompleteSpace`, `T2Space` and `NonarchimedeanRing`
instances, with pair of definition `(A⁺, πA⁺)`), the canonical `θ_J : W(R⁺)[1/[ϖ]] →+* untilt P J` from Mathlib’s
`IsLocalization.map` applied to `Ideal.Quotient.mk J`; its surjectivity and kernel are
`IsLocalization.map_surjective_of_surjective` and `IsLocalization.ker_map`, `untilt.sharp` (`sharp_J : R →* untilt P J`,
`x ↦ θ_J([x])`, continuous and multiplicative; `sharp_J ϖ` is a pseudo-uniformizer) and
`untilt.map` (for arbitrary ideals `J,J'`, a pair morphism with `W(f)(J) ⊆ J'`
induces a ring homomorphism; when both ideals are primitive this map is continuous;
`untilt.map_id`, `untilt.map_comp`, `untilt.continuous_map`); prove `untilt.quotEquiv` (`untiltPlus ⧸ (sharp_J ϖ) ≃+* R⁺ ⧸ (ϖ)`, for a
choice of `ϖ` for which `J` has a generator `p + [ϖ]α` as in *quotient-by-primitive-element*),
`untilt.congr_pseudoUniformizer` (canonical isomorphism between the constructions for two choices
of `ϖ`), `untilt.isPerfectoidTateRing` (the untilt is perfectoid with ring of integral elements
`untiltPlus`, P1.20), `untilt.tiltEquiv` (`tilt (untilt P J) ≃+* R`, identifying the plus rings and
`ker θ` with `J`, P1.20) and `untilt_p_recovery` (`untilt P (span {p}) ≃+* R`). *Needs:* P1.16;
*witt-vectors-of-perfect-plus-ring*; *quotient-by-primitive-element*; Mathlib `Localization.Away`,
`WittVector`; Tau Ceti `TauCeti.Huber.IsTateRing`, `TauCeti.Huber.PairOfDefinition`,
`TauCeti.Huber.IsPseudoUniformizer`.

For every characteristic-`p` perfectoid Tate pair `P` and primitive ideal `J`,
`untilt.sharp_apply` specifies `sharp_J x = algebraMap (mk J [x])` for every `x : P.plus`.
For every pseudo-uniformizer `t : P.plus`, `untilt.nhds_zero_hasBasis` specifies the
neighborhoods of zero on the raw carrier `untilt p P J`: they have basis `π^n · untiltPlus`,
`n ∈ ℕ`, where `π = algebraMap (mk J [t])`. Thus the term at `n = 0` is the plus ring and
the term at `n = 1` is its principal ideal generated by `π`. For a morphism `f : P → P'`
of characteristic-`p` perfectoid Tate pairs and arbitrary ideals with `W(f)(J) ⊆ J'`,
`untilt.map_algebraMap` requires the induced map to send `algebraMap (mk J w)` to
`algebraMap (mk J' (W(f) w))` for every Witt vector `w`; no primitivity is needed for this
equation. For primitive `J`, `untilt.sharp_tiltEquiv` gives
`sharp_J (untilt.tiltEquiv x) = tilt.sharp x` for every `x : tilt p (untilt p P J)`.
`untilt.tiltEquiv_plus_kernel` specifies the restriction of this same equivalence to the
plus rings and requires its induced Witt-vector isomorphism to carry the Fontaine kernel
to `J` ([ECD] §3, Theorem 3.17, p. 18, with Lemma 3.14, pp. 17–18).

**Checks.**

- `untilt_large_quotient_negative` (non-example) — for `C = ℂ_p`, the class of `p` in
  `O_C/(p²)` is nonzero since `|p| > |p²|`. This quotient does not have characteristic `p`,
  whereas every quotient of `O_{C♭}` does. Thus reduction modulo the sharp of an arbitrary
  pseudo-uniformizer, such as `(p♭)²`, cannot give the comparison; the choice in the theorem
  is essential, also at `p = 2`.


- `untilt_p_recovery` (degenerate) — `untilt (R, R⁺) (span {p}) ≅ R` as topological rings, with
  `untiltPlus = R⁺`.
- `untilt_pflat` (computation) — for `R` the completed perfection of `𝔽_p((t))` and
  `J = span {p − [t]}`: the untilt is the completion of `ℚ_p(p^{1/p^∞})` with `sharp_J t = p`
  ([AWS] Example 2.3.7).
- `untilt_cyclotomic_recovery` (computation) — for `J = span {Σ_{i<p}[1 + t]^{i/p}}` the untilt is
  `ℚ_p^cycl` ([AWS] Example 2.3.8).
- `untilt_not_primitive` (non-example) — for `R ≠ 0` and `J = span {p²}` (not primitive) the
  analogous quotient `W(R⁺)[1/[ϖ]]/(p²)` is not reduced: `p² = 0` there while `p ≠ 0`, because
  `W(R⁺)[1/[ϖ]]/(p) = R⁺[1/ϖ] = R ≠ 0`; hence it is not perfectoid (for `R = 0` every quotient is
  the zero ring, which is perfectoid).
- `untilt_quotEquiv_test` (characterisation) — for `J = span {p − [p♭]} ⊆ W(𝒪_{ℂ_p♭})` and
  `ϖ = p♭`: `sharp_J(p♭) = θ_J([p♭]) = p` (as `p − [p♭] ∈ J`) and `untiltPlus = 𝒪_{ℂ_p}`, so
  `untilt.quotEquiv` reads `𝒪_{ℂ_p}/(p) ≃+* 𝒪_{ℂ_p♭}/(p♭)`.

### P1.20 the untilt is a perfectoid Tate pair with tilt `(R, R⁺)`

**The untilt along a primitive ideal is a perfectoid Tate pair with tilt `(R, R⁺)`.**
(*untilt-is-perfectoid-pair*) For `(R, R⁺)` a perfectoid Tate pair of characteristic `p` and
`J ⊆ W(R⁺)` primitive of degree `1`, the untilt `(A, A⁺)` of P1.19 is a perfectoid Tate pair: `A`
is a perfectoid Tate ring, `A⁺ = W(R⁺)/J` is a ring of integral elements of `A` (open, integrally
closed, contained in `A°`), `A°/A⁺` is almost zero, and the identification `R⁺ ≅ lim_{x↦x^p} A⁺`
of *untilt-tilt-identification* extends to an isomorphism of perfectoid Tate pairs
`(R, R⁺) ≅ (A♭, A♭⁺)` under which `ker θ_A = J` ([AWS] Lemmas 2.6.14 and 2.7.9 and Theorem 2.3.9,
p. 55; [KL16] Theorem 3.3.8, p. 60; the converse functor of [ECD] Theorem 3.17 (stated before it as
straightforward to check, p. 18) and of [Berkeley] Theorem 6.2.11,
`(R, R⁺, J) ↦ (W(R⁺)[[ϖ]⁻¹]/J, W(R⁺)/J)`; [BMS] Lemma 3.21, p. 27). For `R` the completed perfection of `𝔽_p((t))` and
`J = (p − [t])` this gives `(ℚ_p(p^{1/p^∞})^, ℤ_p[p^{1/p^∞}]^)`; `J = (p)` returns `(R, R⁺)`.
*Needs:* P1.19; *untilt-tilt-identification*; *perfectoid-tate-ring-from-integral-perfectoid*;
*quotient-by-primitive-element*; *frobenius-surjectivity-criterion*; P1.1. The supporting
*perfectoid-tate-ring-from-integral-perfectoid* uses P1.24's `IsIntegralPerfectoid` and
integral Frobenius criterion, constructed before this supporting result.

Two supporting results. The tilt of `W(R⁺)/J` is `R⁺`, and `θ` for `W(R⁺)/J` is the quotient map
(*untilt-tilt-identification*): in the situation of P1.19, with `J = ξW(R⁺)` primitive and
`A⁺ = W(R⁺)/J` `p`-adically complete (*quotient-by-primitive-element* (d)), let `θ_J : W(R⁺) → A⁺`
be the quotient map. Then (a) `A⁺/p ≅ R⁺/ξ_0R⁺` (`ξ_0` the zeroth Witt coordinate), Frobenius on
`A⁺/p` is surjective, and `x ↦ (θ_J([x^{1/p^n}]))_n` is a multiplicative bijection
`R⁺ → lim_{x↦x^p} A⁺` whose composite with reduction is a ring isomorphism
`R⁺ ≅ lim_Φ A⁺/p = PreTilt A⁺ p`; (b) under this identification Mathlib's `WittVector.fontaineTheta A⁺ p` is `θ_J`, so `ker θ_{A⁺} = J`; (c) `R = R⁺[1/ϖ] ≅ lim_{x↦x^p} A` as monoids ([KL16] Lemma
3.3.7 and Remark 3.3.9, p. 60; [AWS] Definition 2.3.14 and Remark 2.3.16, p. 56; [KL15] Lemma
3.4.2, p. 79; Mathlib `Perfection.quotientMulEquiv`, `WittVector.fontaineTheta`, `PreTilt`). For
`J = (p)` the identification is the identity; for `J = (p − [p♭]) ⊆ W(𝒪_{ℂ_p♭})`, `A⁺ = 𝒪_{ℂ_p}`
and this is `PreTilt 𝒪_{ℂ_p} p ≅ 𝒪_{ℂ_p♭}`. An integral perfectoid ring with a nonzerodivisor `π`
gives a perfectoid Tate ring (*perfectoid-tate-ring-from-integral-perfectoid*): let `R_0` be
`π`-adically complete and `π`-torsion free for some `π` with `π^p | p` in `R_0`, with Frobenius
surjective on `R_0/p` and `Φ : R_0/π → R_0/π^p` bijective (equivalently `R_0` integral perfectoid in
the sense of P1.24 with `π` a nonzerodivisor, [BMS] Lemma 3.10). Then `R := R_0[1/π]`, with `R_0`
open and `π`-adic, is a complete Hausdorff perfectoid Tate ring; after replacing `π` by a unit
multiple with compatible `p`-power roots `π^{1/p^n} ∈ R_0` ([BMS] Lemma 3.9, p. 22),
`R° = {x ∈ R : π^{1/p^n}x ∈ R_0` for all `n}` (the almost elements of `R_0`), `R_0 ⊆ R°`,
`πR° ⊆ R_0`, and `R°/R_0` is killed by every `π^{1/p^n}` ([BMS] Lemma 3.21 and its proof, p. 27;
[Sch12] Lemmas 5.6–5.7, p. 23; Tau Ceti `TauCeti.Huber.PairOfDefinition`, `TauCeti.Huber.IsTateRing`,
`TauCeti.Huber.powerBoundedSubring`). For `R_0 = ℤ_p^cycl`, `π = ζ_{p²} − 1`: `R = ℚ_p^cycl` and
`R° = R_0`; for `R_0` the `p`-adic completion of `ℤ_p^cycl[T^{1/p^∞}]`: `R = ℚ_p^cycl⟨T^{1/p^∞}⟩`
with `R° = R_0` ([ECD] Example 3.4(iii)).

**Checks.** The following use the actual localized Witt quotient and the comparison
of [ECD] Theorem 3.17, p. 18; [Berkeley] Theorem 6.2.11, p. 47.

- `untilt.sharp_zero_base` — over a zero characteristic-`p` base, every primitive
  untilt is zero and sharp takes every element to zero.
- `untilt.sharp_radical` — for an integral pseudo-uniformizer `t` and primitive
  `J = (p−[t])`, sharp sends `t²` to `p²` in the localized quotient.
- `untilt.sharp_nonadditive` — over `ℂ₂♭`, choose integral `t` with original
  sharp image `2` and primitive `J = (2−[t])`. Then `untilt.sharp` sends `t+t`
  to `0`, whereas the sum of its two images is `4 ≠ 0`. This excludes additivity.
- `untilt.map_zero_target` — over a nonzero characteristic-`p` base, enlarging
  `(p)` to the unit ideal kills the nonzero sharp image of any unit.
- `untilt.map_projection_values` — for `K = ℂ_p♭`, untilt the first projection
  `K×K → K` along `(p)`. It sends the sharp images of `(1,0)` and `(0,1)` to
  `1` and `0`, respectively, distinguishing the two projections.
- `untilt.map_noninjective` — over a nonzero characteristic-`p` base, the map
  from the quotient along `(p²)` to that along `(p)` kills the nonzero class
  of `p`, so the algebraic untilt map need not be injective. Its domain uses
  the algebraic quotient; `(p²)` is not primitive.
- `untilt.tiltEquiv_zero_base` — the tilt of an untilt over a zero base is
  zero, and every element has zero image under the specified comparison.
- `untilt.tiltEquiv_two_roots` — for compatible pseudo-uniformizer roots
  `ϖ_n` in the characteristic-`p` base, form the two tilt elements with
  coordinates `sharp(ϖ_n)` and `sharp(ϖ_{n+1})`. The comparison sends their
  sum to `ϖ_0+ϖ_1`.
- `untilt.tiltEquiv_not_frobenius` — for the same first root sequence over a
  nonzero base, its image is not `ϖ_0^p`. A nonzero topologically nilpotent
  unit cannot equal its `p`th power, so this excludes a Frobenius twist.

### P1.21 untilts are classified by primitive ideals

The classification of untilts follows, with marked untilts, the behaviour of strict inclusions
under tilting, integral perfectoid rings and the integral/Tate dictionary, and the flat deformation
of perfectoid algebras modulo `ϖ`.

**Untilts are classified by primitive ideals of degree 1.**
(*untilts-classified-by-primitive-ideals*) The functors `(S, S⁺) ↦ (S♭, S♭⁺, ker θ_S)` and
`(R, R⁺, J) ↦ (W(R⁺)[1/[ϖ]]/J, W(R⁺)/J)` are mutually quasi-inverse equivalences between
(i) perfectoid Tate pairs `(S, S⁺)` with morphisms of Huber pairs and (ii) triples `(R, R⁺, J)`
with `(R, R⁺)` a perfectoid Tate pair of characteristic `p` and `J ⊆ W(R⁺)` primitive of degree
`1`, a morphism being a morphism `f` of pairs with `W(f)(J) ⊆ J'` (then `J' = W(f)(J)W(R'⁺)`). No
perfectoid base field is fixed and both characteristics are allowed. The unit is induced by
`θ_S : W(S♭⁺)[1/[ϖ♭]]/ker θ ≅ S`, the counit is the identification of P1.20, and both are natural.
Characteristic-`p` pairs correspond to `J = (p)` ([ECD] §3, Theorem 3.17 with (ii), p. 18; [KL16]
Theorem 3.3.8, p. 60; [AWS] Theorem 2.3.9, p. 55, and Remark 2.3.18, p. 57; [KL15] Theorem 3.6.5,
p. 89, the characteristic-`0` case; [Berkeley] Definition 6.2.9, Lemma 6.2.10 and Theorem 6.2.11,
pp. 46–47). For `(K♭, K♭°)` with `K` a perfectoid field and `J = ker θ`, the construction returns
`(K, K°)`; over `𝔽_p((t^{1/p^∞}))^`, `J = (p)` gives the field itself, `J = (p − [t])` the
completion of `ℚ_p(p^{1/p^∞})`, `J = (Σ_{i<p}[1+t]^{i/p})` the field `ℚ_p^cycl`, and `φ(J)` gives
Frobenius-twisted markings. *Needs:* P1.20; *untilt-tilt-identification*; P1.18; P1.19; P1.6;
*theta-kernel-base-change*; Mathlib `CategoryTheory.Equivalence`.

Supporting result: `ker θ` is compatible with base change (*theta-kernel-base-change*). Let
`f : (R, R⁺) → (S, S⁺)` be a morphism of perfectoid Tate pairs (`f` continuous with `f(R⁺) ⊆ S⁺`)
with tilt `f♭`. Then `θ_S ∘ W(f♭) = f ∘ θ_R` and `ker θ_S = W(f♭)(ker θ_R)·W(S♭⁺)`; every generator
`ξ` of `ker θ_R` maps to a generator of `ker θ_S`. For `R = K` a perfectoid field this says that
one `ξ ∈ A_inf(K, K⁺)` generates `ker θ` for every perfectoid `(K, K⁺)`-algebra ([AWS] Remark
2.3.17, p. 56; [BMS] Remark 3.11, p. 23). For `R = ℂ_p` and `S = ℂ_p⟨T^{1/p^∞}⟩`,
`ker θ_S = (p − [p♭])`; for `S = R/I` a perfectoid quotient, `ker θ_S` is generated by the image of
`ξ`.

The categorical carriers are `PairCat p` and `PrimitiveTriple p`. `PairCat` stores
an actual `TauCeti.Huber.Pair` with the complete Hausdorff perfectoid Tate structure; its morphisms
are exactly `Pair.Hom`. A triple stores `p = 0` on its base and a primitive Witt ideal; a morphism
carries that ideal into the target ideal. `classificationFunctor` uses the actual Fontaine
kernel, and `primitiveUntiltingFunctor` uses the localized Witt quotient. The equations
`PrimitiveTriple.untilt_plus`, `untilt_nhds`, `tiltIso_sharp`,
`primitiveUntiltingFunctor_map`, `classificationEquivalence_unit` and
`classificationEquivalence_counit` specify the plus ring, topology, maps and natural comparisons
of `classificationEquivalence` ([Berkeley] Theorem 6.2.11, p. 47).

### P1.22 marked untilts

**Marked untilts of a perfectoid pair of characteristic p.** (*marked-untilt*) Let `(R, R⁺)` be a
perfectoid Tate pair of characteristic `p`. A marked untilt of `(R, R⁺)` is a triple `(S, S⁺, ι)`
with `(S, S⁺)` a perfectoid Tate pair and `ι : (S♭, S♭⁺) ≅ (R, R⁺)` an isomorphism of Huber pairs
(the marking is part of the data); a morphism `(S, S⁺, ι) → (S', S'⁺, ι')` is a morphism `f` of
Huber pairs with `ι' ∘ f♭ = ι`. By P1.21, `(S, S⁺, ι) ↦ W(ι)(ker θ_S)` is an equivalence from
marked untilts to the discrete category of primitive ideals `J ⊆ W(R⁺)`: a marked untilt has no
nontrivial automorphisms and isomorphic marked untilts are uniquely isomorphic. Marked untilts pull
back along morphisms of pairs `(R, R⁺) → (R', R'⁺)` (`J ↦ J·W(R'⁺)`); marked untilts of `(R, R⁺)`
and of `(R, R°)` correspond (plus rings correspond, *rings-of-integral-elements-under-tilting*);
`pS = 0` iff `J = (p)`, the marked untilt `(R, R⁺, id)` ([Berkeley] Lemma 6.2.8 and the text after
it, pp. 45–46; [AWS] Definition 2.3.13, p. 55; [ECD] §3, after Theorem 3.13, p. 17). Define
`MarkedUntilt` (`MarkedUntilt (P : Pair R)`: a perfectoid Tate ring `S` with instances, a ring of
integral elements `S⁺`, and `ι : Pair.tilt (S, S⁺) ≅ P` as Huber pairs), `MarkedUntilt.Hom`
(`Pair.Hom f` with `ι' ∘ Pair.tiltHom f = ι`; `Category` instance), `MarkedUntilt.primitiveIdeal`
(`W(ι)(ker θ_S)`, a primitive ideal of `W(R⁺)`, P1.18), `MarkedUntilt.ofPrimitive`
(`J ↦ (untilt P J, untiltPlus, tiltEquiv)`, P1.20), `MarkedUntilt.pullback` (for a morphism of
pairs `P → P'` of characteristic `p`: `MarkedUntilt P → MarkedUntilt P'`, `J ↦ J W(R'⁺)`,
compatible with `id` and `comp` up to unique isomorphism) and `MarkedUntilt.self` (the
characteristic-`p` untilt `(R, R⁺, id)` with `primitiveIdeal = span {p}`); prove
`MarkedUntilt.primitiveIdeal_ofPrimitive` (simp: `primitiveIdeal (ofPrimitive J) = J`),
`MarkedUntilt.iso_iff_primitiveIdeal_eq` (two marked untilts are isomorphic exactly when their primitive ideals agree, yielding the bijection on isomorphism classes),
`MarkedUntilt.subsingleton_hom` (`Subsingleton (X ⟶ Y)`; in particular `Unique (Aut X)`),
`MarkedUntilt.plusEquiv` (an equivalence of the categories of marked untilts of
`(R,R⁺)` and `(R,R°)`, or of their sets of isomorphism classes) and
`MarkedUntilt.charP_iff` (`pS = 0 ↔ primitiveIdeal = span {p}`; exact `CharP S p` additionally requires `S ≠ 0`). *Needs:* P1.21; P1.6; P1.19;
*rings-of-integral-elements-under-tilting*; Tau Ceti `TauCeti.Huber.Pair`,
`TauCeti.Huber.Pair.Hom`.

**Checks.**

- `marked_self_kernel` (degenerate) — for `P = (ℚ_p^cycl)♭ = 𝔽_p((t^{1/p^∞}))^`,
  `primitiveIdeal (MarkedUntilt.self P) = span {p}` in `W(P⁺)`, with `W(P⁺)/(p) = P⁺` recovering
  `P` itself, whereas the untilt `ℚ_p^cycl` has primitive ideal generated by
  `([ε] − 1)/([ε^{1/p}] − 1)`.
- `marked_padicComplex_kernel` (computation) — for `P = (ℂ_p♭, 𝒪_{ℂ_p♭})` and the
  canonical marking of `ℂ_p`: `primitiveIdeal = span {p − [p♭]}`.
- `marked_automorphism` (characterisation) — every automorphism of a marked untilt is the
  identity.
- `marked_frobenius_twist` (non-example) — for the marked untilt `(ℂ_p, ι)`, the marking
  `φ_R ∘ ι` gives primitive ideal `φ(J) ≠ J`, so `(ℂ_p, ι)` and `(ℂ_p, φ ∘ ι)` are non-isomorphic
  marked untilts with isomorphic underlying rings: the marking cannot be dropped. An explicit
  instance: for `R = 𝔽_p((t^{1/p^∞}))^` and the mixed-characteristic `J = (p − [t])`, whose untilt
  is `ℚ_p(p^{1/p^∞})^` with `t♯ = p` (P1.19), `φ(J) = (p − [t]^p) ≠ J`, since the untilt map sends
  `p − [t]^p` to `p − p^p ≠ 0`; so the Frobenius-twisted marking is a different marked untilt of the
  same field. Control: the characteristic-`p` ideal `J = (p)` is Frobenius-stable, and its twisted
  marking is isomorphic to the original one through the Frobenius of `R`.

In the proposed Lean form, specified in this README only, the base of `MarkedUntilt p P` is a
bundled `PairCat p`; its marking is
an isomorphism of pairs. `MarkedUntilt.classification hp` maps to the discrete category of
primitive ideals, where `hp : (p : P) = 0`. Its `IsEquivalence` theorem includes full faithfulness
and essential surjectivity. `ofPrimitive`, `self`, `primitiveIdeal_ofPrimitive`, `self_marking`
and `pullback` specify the object constructions. The zero base is allowed: the characteristic
criterion uses `p = 0`, not exact prime characteristic.

### P1.23 strict inclusions under tilting

**Strict inclusions correspond under tilting.** (*strict-inclusions-under-tilting*) A continuous
ring homomorphism `f : A → B` of complete Hausdorff Tate rings is a strict inclusion if it is
injective and a homeomorphism onto its image with the subspace topology (equivalently, as `A` is
complete, a topological ring isomorphism onto a closed subring); strictness is a topological
condition and injectivity alone does not suffice. Let `A` be a perfectoid Tate ring, `ϖ♭ ∈ A♭` with
compatible roots, `ϖ = ϖ♭♯`, and `f : A → B` continuous. (a) If `B` is a perfectoid Tate ring, `f`
is a strict inclusion if and only if `f♭ : A♭ → B♭` (P1.6) is ([AWS] Theorem 2.4.3 and its proof
(completed by Lemma 2.8.1), p. 57). (b) More generally, let `B` be a complete Hausdorff uniform Tate ring (`p` is topologically
nilpotent in `B` because it is in `A`) and `B♭` its tilt (P1.7; `f(ϖ) = (f♭ϖ♭)♯` has compatible
roots and `f(ϖ)^p | p` in `B°`). For the spectral gauges `|·|_ϖ`, `|·|_{f(ϖ)}`, `|·|_{ϖ♭}`,
`|·|_{f♭ϖ♭}`: `f` is a strict inclusion iff `|f(a)| = |a|` for all `a ∈ A`; `f♭` is a strict
inclusion iff it is isometric; and `f` is a strict inclusion iff `f♭` is ([AWS] Lemma 2.8.1, p. 69,
whose second right-hand side should read `zW^b(A♭)`; [AWS] Lemma 2.8.4, proof, p. 70; [KL16] Lemma
3.3.23, p. 66). The analogous statement for injections that are not strict is [AWS] Corollary
2.9.13 and is not stated here. A finite extension `K ⊆ L` of perfectoid fields is a strict
inclusion, and so is `K♭ ⊆ L♭`; the continuous injection `K⟨T^{1/p^∞}⟩ → K⟨T^{1/p^∞}⟩`,
`T^{1/p^n} ↦ (ϖT)^{1/p^n}`, is not strict (`|T^N| = 1` while `|(ϖT)^N| = 2^{-N}`), and neither is
its tilt `T ↦ ϖ♭T`. *Needs:* P1.6; P1.7; P1.1; P1.3; *metric-perfectoid-criterion*;
*sharp-approximation-lemma*; Tau Ceti `TauCeti.Huber.IsBounded`,
`TauCeti.Huber.powerBoundedSubring`.

Two supporting results. The metric criterion (*metric-perfectoid-criterion*): let `B` be a complete
Hausdorff uniform Tate ring and `ϖ ∈ B°` a pseudo-uniformizer with `ϖ^p | p` in `B°` admitting a
compatible system of `p`-power roots `ϖ^{1/p^n} ∈ B°`, and let
`|x| := inf{2^{-q} : q ∈ ℤ[1/p], ϖ^{-q}x ∈ B°}` be the spectral gauge of `ϖ` (the formula of P1.3
on a ring not assumed perfectoid, i.e. the spectral seminorm of AdicSpacesPartII R0.4 for the
norm datum `(B°, ϖ, 1/2)`; `B` may have characteristic `0`, `p` or neither). Then (a) `|·|`
is an ultrametric, submultiplicative, power-multiplicative norm defining the topology of `B`, with
`|ϖ^qx| = 2^{-q}|x|` and `B° = {x : |x| ≤ 1}`; (b) `B` is a perfectoid Tate ring if and only if
there is `c ∈ (0, 1)` such that for every `x ∈ B` there is `y ∈ B` with `|x − y^p| ≤ c|x|`; (c) if
`B` is perfectoid, every `c ∈ (2^{-p}, 1) = (|ϖ|^p, 1)` has this property, and `y` can be taken in
`B°` when `x ∈ B°`. Kedlaya states the criterion for a uniform analytic Huber pair with an ideal of
definition `I`, `p ∈ I^p`, and the norm attached to `I`; this is its Tate-ring form with the gauge
of a pseudo-uniformizer with compatible roots, which every perfectoid Tate ring has and which every
Tate ring receiving a continuous map from a perfectoid Tate ring has ([AWS] §2.8, Lemma 2.8.3 with
its proof and the text before it, p. 70; [KL15] Proposition 3.6.2(c), p. 87; Tau Ceti
`TauCeti.Huber.IsBounded`, `TauCeti.Huber.isPowerBounded_of_isIntegral`,
`TauCeti.Huber.IsPseudoUniformizer.hasBasis_nhds_zero`). For `B = ℂ_p` with `ϖ = p^{1/p}` every
`c ∈ (2^{-p}, 1)` works; for a perfectoid field `K`, `B = K⟨T⟩` fails the criterion at `x = T`
(the reduction of `y^p` in `(K°/K°°)[T]` is a `p`-th power, so `|T − y^p| ≥ 1 = |T|`), and `B` is
not perfectoid; the criterion passes along isometric continuous maps with dense image (layer P2).
Euclidean division by a primitive element and approximation by sharp elements
(*sharp-approximation-lemma*): let `A` be a perfectoid Tate ring, `ϖ♭ ∈ A♭` with compatible roots,
`ϖ = ϖ♭♯`, and `|·|` the spectral gauges of `A` and `A♭` (P1.3), so that `|y♯| = |y|` by
*spectral-norm-under-sharp*. Let `z = [z̄_0] + p·z_1` (`z_1 ∈ W(A♭°)^×`) be a primitive generator
of `ker(θ : W(A♭°) → A°)` (P1.18; all such differ by units), extend `θ` to
`W^b(A♭) := W(A♭°)[1/[ϖ♭]] → A`, and give `W^b(A♭)` the Gauss norm `|Σ_n p^n[x_n]| := sup_n |x_n|`;
an element `x = Σ_n p^n[x_n]` is prepared if `|x_0| ≥ |x_n|` for all `n ≥ 1`. Then: (a) (Euclidean
division) for `x ∈ W^b(A♭)` put `x′ := p^{-1}(x − [x_0])` and
`r(x) := x − z_1^{-1}x′z = [x_0] − [z̄_0]z_1^{-1}x′`; if `x ∉ zW^b(A♭)`, the terms of the sequence
`x, r(x), r(r(x)), …` are congruent to `x` modulo `z` and are prepared from some term on; no nonzero
element of `zW^b(A♭)` is prepared; (b) (norms) `|θ(x)| ≤ |x|` for every `x`, with equality when `x`
is prepared; hence the quotient norm of the Gauss norm on `W^b(A♭)/(z) ≅ A` is the spectral gauge,
and `|z̄_0| = |p|` (`|p| < 1`, and `|p| = 0` iff `pA = 0`); (c) (sharp approximation) for every
`a ∈ A` there is `y ∈ A♭` with `|a − y♯| ≤ |p|·|a|`, so `|y| = |a|`; for `a ∈ A°` one can take
`y ∈ A♭°`; (d) (pointwise form) `α ↦ α ∘ ♯` maps the Gelfand spectrum `M(A)` of bounded
multiplicative seminorms on `(A, |·|)` into `M(A♭)`, and for every `a ∈ A` and `ε > 0` there is
`y ∈ A♭` with `α(a − y♯) ≤ α(p)·max{α(a), ε}` for all `α ∈ M(A)` simultaneously; with
`|·| = max_{α ∈ M(A)} α` (P1.12; [AWS] Lemma 1.5.21, p. 25) this refines (c) up to `ε`; (e)
(quasi-additivity of `♯`) `|(x + y)♯ − x♯ − y♯| ≤ |p|·max{|x|, |y|}` for `x, y ∈ A♭`. Part (d) uses
the Gelfand spectrum and the maximum formula of P1.12; parts (a)–(c) and (e) do not ([AWS]
Definition 2.6.5, Lemma 2.6.7 and Lemma 2.6.9, p. 64; [AWS] Lemma 2.8.3, proof, and Lemma 2.8.5,
proof, p. 70; [KL16] Lemma 3.2.7(a) with proof, p. 57, and Theorem 3.3.18, proof of case (ii),
p. 65; Mathlib `WittVector`, `WittVector.teichmuller`, `WittVector.fontaineTheta_teichmuller`). For
`A = ℂ_p`, `z = p − [p♭]`: every `a ∈ ℂ_p` is `y♯` up to `|p||a|`, and `a − y♯` need not vanish; in
characteristic `p`, `z = p`, `z̄_0 = 0`, `|p| = 0`, `r(x) = [x_0]` is prepared after one step and
`a = y♯` exactly. Parts (c)–(e) are used by layer P4.

The supporting statements are the gauge laws of P1.3 (submultiplicativity, the ultrametric
inequality, separation, power-multiplicativity, the scaling rule `|ϖ^qx| = 2^{-q}|x|`, the
descriptions `x ∈ R° ↔ |x| ≤ 1` and `IsTopologicallyNilpotent x ↔ |x| < 1`, and the neighbourhood
basis) and the gauge identity under sharp of P1.5; uniformity is explicit for the gauge laws.
Power-multiplicativity requires a positive exponent or a nonzero ring, retaining the zero-ring
failure at exponent zero.
`metric_perfectoid_criterion`, `sharp_approximation`, `sharp_quasi_additive` and
`tilt.strict_inclusion_iff` use the complete Hausdorff topology; strictness means a topological
embedding, not preservation of an arbitrarily normalized norm ([AWS] §§2.6–2.8).

### P1.24 integral perfectoid rings

**Integral perfectoid rings.** (*integral-perfectoid-rings*) A commutative ring `R` is integral
perfectoid if there is `π ∈ R` with `π^p | p` such that `R` is `π`-adically complete and separated,
the Frobenius `R/p → R/p` is surjective, and the kernel of Fontaine's map `θ : W(R♭) → R` is
principal, where `R♭ = lim_{x ↦ x^p} R/p` is the tilt (Mathlib's `PreTilt`) ([BMS] §3.2, Definition
3.5, p. 21). Equivalent forms: for a nonzerodivisor `π` with `π^p | p` and `R` `π`-adically
complete, `R` is integral perfectoid iff Frobenius `R/π → R/π^p` is an isomorphism; every integral
perfectoid `R` admits a unit multiple of `π` with a compatible system of `p`-power roots ([BMS]
Lemma 3.9 and Lemma 3.10, p. 22). Integral perfectoid rings may have `p`-torsion and need not be
domains. Define `IsIntegralPerfectoid` with an admissible adic element, surjective Frobenius,
and principal kernel of the supplied Fontaine map. In Lean the last condition quantifies over
proofs that `p` is a nonunit and that the ring is `p`-adically complete. The latter follows from
the adic presentation by [Stacks] Tag 090T; if `p` is a unit, that presentation forces the zero
ring, and the unique map has principal kernel. Thus the condition includes the zero ring without
an impossible `Fact ¬IsUnit p` instance. Prove that
for every `π` with `π^p | p` and `R` `π`-adically complete, Frobenius `R/π → R/π^p`
is bijective, with no nonzerodivisor hypothesis ([BMS] Lemma 3.10(i), p. 22), and that
for `π^p ∣ p` with `R` `π`-adically
complete, a unit multiple of `π` has compatible `p`-power roots. The zero ring satisfies the
definition using the unique Fontaine map; for nonzero rings use Mathlib’s `PreTilt` and `θ`. *Needs:* Mathlib `IsAdicComplete`, `PreTilt`, `WittVector.fontaineTheta`.

**Checks.**

- Non-example: `ℤ_p` is not integral perfectoid: Frobenius
  on `ℤ_p/p = 𝔽_p` is bijective and `ker θ = 0` is principal, but no admissible `π` exists: `π^p | p`
  forces `p·v_p(π) ≤ 1`, so `π` is a unit (`π = p` fails, `p^p ∤ p`), and `ℤ_p` is not `π`-adically
  complete for a unit `π` (`lim ℤ_p/π^n = 0 ≠ ℤ_p`).
- Computation: the `p`-adic completion of
  `ℤ_p[p^{1/p^∞}]` is integral perfectoid with `π = p^{1/p}`.
- `integral_perfect_charP` (degenerate) — a perfect `𝔽_p`-algebra that is `π`-adically
  complete for some `π` is integral perfectoid (`π^p | p = 0` trivially).

The proposed Mathlib [integral-perfectoid interface](https://github.com/leanprover-community/mathlib4/pull/26385)
uses a specified topological ring and pseudo-uniformizer. The definition here is the
choice-independent algebraic predicate of [BMS] Definition 3.5. A comparison target, specified
in this README only, compares these formulations when the given topology
is the complete separated `π`-adic topology and the chosen element satisfies `π^p ∣ p`; the
comparison must include the principal Fontaine-kernel condition.

`IntegralAdicPresentation p A π` records the divisibility, completeness, Frobenius and
Fontaine-kernel conditions with `π` specified. The comparison
relates it to the algebraic predicate under `π^p ∣ p` and completeness for that same `π`.
It uses the pinned Fontaine map, including its nonunit hypothesis. This specifies the algebraic
content of the proposed topological interface; a topology identified as `π`-adic supplies the
chosen presentation.

Lean: `IsIntegralPerfectoid`, with fields `exists_adic_element`, `frobenius_surjective` and
`theta_kernel_principal`; `Perfectoid.IsIntegralPerfectoid.of_perfectRing` states that every
perfect ring of characteristic `p` is integral perfectoid. The `𝒪_{ℂ_p}` and `ℤ_p` Checks are
specified in prose.

### P1.25 perfectoid Tate rings and integral perfectoid rings

**`R perfectoid Tate ⇔ R⁺ integral perfectoid and bounded`.** (*integral-perfectoid-comparison*) Let
`R` be a complete Hausdorff Tate ring and `R⁺` a ring of integral elements. If `R` is a perfectoid
Tate ring, then `R⁺` (and `R°`) is an integral perfectoid ring in the sense of P1.24: `π`-adically
complete for some `π ∈ R⁺` with `π^p | p` in `R⁺` (exact divisibility, as in P1.24; such a `π`
exists by *frobenius-surjective-on-plus-ring* (b), and may be taken a pseudo-uniformizer of `R`),
Frobenius surjective on `R⁺/p`, and the kernel of `θ : A_inf(R⁺) → R⁺` principal. Conversely, if
`R⁺` is integral perfectoid and bounded in `R`, then `R` is a perfectoid Tate ring; boundedness of
`R⁺` is necessary for the converse (a non-bounded `R⁺` cannot occur inside a uniform `R`; conversely
`ϖR° ⊆ R⁺` for a bounded open integrally closed `R⁺`, so `R` is uniform), and the point of the
converse is to produce a pseudo-uniformizer `ϖ'` of `R` with `ϖ'^p | p`: the given `π` is one when
`p` is a unit of `R`, but need not be a unit of `R` in general (`K × K♭` with `R⁺ = K° × K♭°` is
integral perfectoid for `π = (π_K, 0)`, and a perfect `𝔽_p`-algebra for `π = 0`) ([BMS] §3.2, Lemma 3.20 with its proof and the text before it, pp. 26–27; [BMS]
Definition 3.5, p. 21; [BMS] Lemma 3.10(ii), p. 22). The forward
direction is *frobenius-surjective-on-plus-ring* (P1.17) together with the primitive-kernel theorem
(P1.18); what is proved here is the converse. Together with
*perfectoid-tate-ring-from-integral-perfectoid* ([BMS] Lemma 3.21, P1.20) this compares the integral
notion with the Tate-ring notion of this layer.
`ℤ_p^cycl`, `𝒪_{ℂ_p}` and the `p`-adic completion of `ℤ_p^cycl[T^{1/p^∞}]` are integral perfectoid
([BMS] Example 3.6) and `ℚ_p^cycl`, `ℂ_p`, `ℚ_p^cycl⟨T^{1/p^∞}⟩` are perfectoid Tate rings; perfect
`𝔽_p`-algebras are integral perfectoid with `π = 0`, but with the discrete topology they are not
Tate, so the comparison needs `R` Tate. *Needs:* P1.1; P1.24; *perfectoid-tate-ring-from-integral-perfectoid*;
*frobenius-surjective-on-plus-ring*; *integral-elements-adically-complete*.

The reverse criterion of P1.24 holds
with a nonzerodivisor `π`, `π^p ∣ p`, and completeness for that same `π` ([BMS] Lemma 3.10(ii)):
bijectivity of Frobenius `R/π → R/π^p` then gives integral perfectoidness.
`isPerfectoidTateRing_iff_integral P` states the comparison for a complete Hausdorff Tate ring
and its actual ring of integral elements, with boundedness retained on the integral side.

Lean: `isPerfectoidTateRing_iff_integral`.

### P1.26 flat deformation of perfectoid algebras modulo `ϖ`

**Unique flat deformation of perfectoid algebras modulo `ϖ` over a perfectoid field.**
(*deformation-lifting-of-perfectoid-mod-varpi-algebras*) Let `K` be a complete nontrivially
normed ultrametric perfectoid field, and choose a pseudo-uniformizer `ϖ = (ϖ♭)♯` with compatible
roots and `ϖ^p ∣ p` in `K°`. The element `ϖ` is a nonzerodivisor on `K°`, and
`ϖ^{n−1}K°/ϖ^nK° ≅ K°/ϖ` is square-zero in `K°/ϖ^n` for `n ≥ 2`. Reduction modulo `ϖ`,
`A ↦ A/ϖ`, is an equivalence between (i) `ϖ`-adically complete `K°a`-algebras `A` that are
(almost) flat over `K°a` with `Φ : A/ϖ^{1/p} ≅ A/ϖ`, and (ii) (almost) flat
`(K°/ϖ)^a`-algebras `B` with `Φ : B/ϖ^{1/p} ≅ B`. Concretely `B` lifts uniquely, through the
square-zero extensions `(K°/ϖ^n)^a → (K°/ϖ^{n−1})^a`, to a compatible system of flat
`(K°/ϖ^n)^a`-algebras `B_n` with `B_n/ϖ^{n−1} ≅ B_{n−1}`, and `A = lim B_n`; two flat
deformations with the same reduction are uniquely isomorphic compatibly with the reduction.
The limit has no `ϖ`-torsion on almost elements, which implies flatness over the valuation
ring `K°` and almost flatness over `K°a` ([Sch12] Lemma 5.3(i), p. 22).
The almost deformation theory used (obstructions in `Ext²`, torsors under `Ext¹`, automorphisms
`Hom`, lifting of morphisms) is P0.22. This is [Sch12] Theorem 5.10, p. 24, with proof on p. 26,
the step `K°a-Perf ≃ (K°a/ϖ)-Perf` of Theorem 5.2, p. 22. Base flatness is retained in the
categories; the equivalence has field scope, as in P1.28. P1.27 supplies the arbitrary-base
tilting equivalence via primitive ideals.
This is Scholze's deformation-theoretic route: the tilting equivalence (P1.27) and the mod-`ϖ`
equivalence (P1.28) are derived from P1.21 and do not depend on it. `K⟨T^{1/p^∞}⟩` and
`K♭⟨T^{1/p^∞}⟩` have the same reduction `(K°/ϖ)[T^{1/p^∞}]` and are therefore tilts of each other
([Sch12] Proposition 5.20). *Needs:* P0.22; P0.20; P0.16; P0.19; *cotangent-complex-vanishing-mod-varpi*; Mathlib
`IsAdicComplete`, `Module.Flat`.

Supporting result: the Frobenius criterion for vanishing of the (almost) cotangent complex
(*cotangent-complex-vanishing-mod-varpi*). (a) Let `A_0 → B` be a map of `𝔽_p`-algebras such that
the relative Frobenius `A_0(Φ) ⊗^L_{A_0} B → B(Φ)` is an isomorphism in `D(A_0)` (a hypothesis on
the derived relative Frobenius, not on `Ω¹` alone). Then `L_{B/A_0} ≃ 0`; in particular
`L_{B/𝔽_p} ≃ 0` for `B` perfect, and `L_{B/A_0} ≃ 0` whenever `A_0 → B` is flat and relatively
perfect. (b) The almost version holds over any basic setup `(V, m)` with `V` an `𝔽_p`-algebra: if
the relative Frobenius of a map of almost algebras is an isomorphism in the almost derived
category, then the almost cotangent complex `L^a` (P0.20) vanishes. (c) For `R` a perfectoid Tate
ring, `ϖ = (ϖ♭)♯` with compatible roots, and `A` a `(R°/ϖ)^a`-algebra which is (almost) flat over
`(R°/ϖ)^a` with `Φ : A/ϖ^{1/p} ≅ A`, `L^a_{A/(R°/ϖ)^a} ≃ 0`; for `R = K` a perfectoid field this is
[Sch12] Corollary 5.16 ([GR] arXiv v3, Lemma 6.5.9(i) with proof, pp. 176–177 (LNM 1800 Lemma
6.5.13(i)); [Sch12] §5, Proposition 5.13(i)–(ii) with proof, p. 25, and Corollary 5.16, proof,
p. 26; [Bhatt] Proposition 6.1.4, p. 42). `Ω¹_{R/𝔽_p} = 0` for perfect `R`, since
`dx = d(y^p) = py^{p−1}dy = 0` ([Sch12] Remark 5.14); the degree `0` and `1` parts are Mathlib's
`KaehlerDifferential` and `Algebra.H1Cotangent`. The classical cotangent complex with its
simplicial description, transitivity and Tor-independent base change is P0.19.

The categories `IntegralAlmostPerfectoid p R ϖ` and `ModAlmostPerfectoid p R ϖ` are full
subcategories of the almost algebras of P0.8. Their predicates specify base flatness,
completeness on almost elements on the integral side, and the relative Frobenius condition.
`IsAlmostBijective` tests annihilation of the kernel and cokernel of the underlying semilinear
ring map by the almost ideal. The quotient-side setup is the base change to `R°/ϖ`, so its
flatness is over that quotient. `almostReductionFunctor_obj` fixes reduction on almost-elements
representatives, and `almostReductionEquivalence` states the flat deformation equivalence.
`almostReductionFunctor_map` supplies a single family of these object isomorphisms
such that every map `f : A → A′` becomes the almostification of the quotient map
`A_*/ϖ → A′_*/ϖ` induced by `elements.map f`. Its equation is
`F.map f ≫ e(A′).hom = e(A).hom ≫ toAlmost.map (quotientMap f_*)`;
the isomorphisms are chosen simultaneously for all objects and maps
([Sch12] Theorem 5.10, p. 24).
The reduction functor, `almostReductionFunctor_obj` and `almostReductionEquivalence` assume
`IsPerfectoidField p K`, with `K` complete, nontrivially normed and ultrametric. The predicates
and their full subcategories remain defined for arbitrary perfectoid Tate rings `R`.
`almost_mod_cotangent_vanishes` uses the full almost cotangent complex of P0.20 over such a
base, by the almost Frobenius criterion above; [Sch12] Corollary 5.16, p. 25, with proof on
p. 26, is its field specialization.

### P1.27 the tilting equivalence

The layer closes with the tilting equivalence over an arbitrary perfectoid base, the mod-`ϖ`
equivalence over a perfectoid field, completed filtered colimits, perfected Tate algebras, the
cyclotomic field, the standard examples and the compatibility with AdicSpaces' perfect field `F`.

**Tilting equivalence `Perf_R ≃ Perf_{R♭}` over an arbitrary perfectoid Tate ring, with the explicit
tilt.** (*tilting-equivalence-and-explicit-tilt*) Let `R` be a perfectoid Tate ring with tilt `R♭`
and `J_R := ker(θ_R : W(R♭°) → R°)`, a primitive ideal (P1.18). No perfectoid base field is fixed;
continuous homomorphisms between Tate rings preserve power-bounded elements into uniform targets
(`map_mem_powerBoundedSubring`, P1.1), so `R`-algebras are pairs over `(R, R°)`. The tilting
functor `S ↦ S♭` (`R♭ → S♭` the tilt of `R → S`, P1.6) is an equivalence from perfectoid
`R`-algebras to perfectoid `R♭`-algebras, with quasi-inverse
`T ↦ T♯ := W(T°)[1/[ϖ♭]]/J_R·W(T°)[1/[ϖ♭]]`, the untilt along `J_R·W(T°)` (P1.19); the unit
`S ≅ (S♭)♯` is induced by `θ_S` and the counit `(T♯)♭ ≅ T` by P1.20, both natural. Explicitly
`S♭ = lim_{x↦x^p} S`, `S♭° = lim_{x↦x^p} S° ≅ lim_Φ S°/ϖ` and `S♭°/ϖ♭ ≅ S°/ϖ`; the functor is
independent of choices, extends to pairs `(S, S⁺) ↦ (S♭, S♭⁺)`, and `S` is a perfectoid field iff
`S♭` is, by *perfectoid-field-iff-tilt-field*. For `R = K` a perfectoid field,
`T♯ = W(T°) ⊗_{W(K♭°)} K°[1/ϖ]` ([Sch12] Remark 5.19, p. 27; [Bhatt] Remark 6.2.8), and the functor
agrees with Scholze's chain `K-Perf ≃ K°a-Perf ≃ (K°a/ϖ)-Perf = (K♭°a/ϖ♭)-Perf ≃ K♭°a-Perf ≃ K♭-Perf` ([ECD] §3, Theorem 3.13 with its proof ("the proof works in general", here supplied by the
untilt classification) and the text after it, p. 17; [Berkeley] Theorem 6.2.7, p. 45; [AWS] Remark
2.3.17, p. 56; [Sch12] Theorem 5.2, p. 22, and Proposition 5.17, p. 26, over a perfectoid field).
The tilt of `R⟨T_1^{1/p^∞}, …, T_n^{1/p^∞}⟩` is `R♭⟨T_1^{1/p^∞}, …, T_n^{1/p^∞}⟩` with `T_i♯ = T_i`
(P1.30; [Sch12] Proposition 5.20; [AWS] Example 2.3.15). *Needs:* P1.21; *theta-kernel-base-change*;
P1.6; P1.19; P1.20; P1.1; Mathlib `CategoryTheory.Equivalence`.

`PairCat.sliceTiltFunctor p R` acts on `CategoryTheory.Under R`, sending the
structural pair map to its tilt. `sliceTriple` extends the base Fontaine kernel along the Witt
map of the structural morphism, and `sliceUntiltFunctor` uses its localized quotient.
`sliceTiltingEquivalence` has these fixed functor and inverse fields, with natural unit and
counit. Thus its signature applies to all perfectoid pairs, including zero pairs, without
requiring a pre-existing equivalence as an argument.

### P1.28 the equivalence through the mod-`ϖ` category

**`Perf_K ≃ K°a-Perf ≃ (K°a/ϖ)-Perf` over a perfectoid field.**
(*perfectoid-mod-varpi-equivalence*) Let `K` be a complete nontrivially normed ultrametric
perfectoid field, and choose compatible roots of a pseudo-uniformizer `ϖ` with `ϖ^p ∣ p` in
`K°`. Use the almost setup `(K°, K°°)`. A perfectoid `K°a`-algebra is a flat, `ϖ`-adically
complete almost algebra `A` with Frobenius `A/ϖ^{1/p} → A/ϖ` an isomorphism. A perfectoid
`(K°a/ϖ)`-algebra is a flat algebra over that almost base with Frobenius
`B/ϖ^{1/p} → B` an isomorphism. The functors `S ↦ S°a` and `A ↦ A/ϖ` are equivalences between
these categories and perfectoid `K`-algebras ([Sch12] Definition 5.1(ii)–(iii), Theorem 5.2,
pp. 21–22, and Theorem 5.10, p. 24, with proof on p. 26). Identifying the base reductions gives the chain of
equivalences with the corresponding categories over `K♭`.

The source scope here is a field. An unrestricted version over a perfectoid Tate ring `R`,
with almost flatness over `Λ = 𝔽_p[x^{1/p^∞}]/(x)` replacing base flatness, requires a reference
proving that exact category equivalence, including the inverse-limit lifting and almost-elements
claims. [ECD] Proposition 9.3, p. 44, supplies the totally disconnected application, with the
base-flatness input of Proposition 7.23; it does not supply the unrestricted statement. P1.27
retains the arbitrary-base tilting equivalence via primitive ideals.

Two supporting results have the same field scope. The *almost-integral-dictionary* sends
`S` to `S°a` and sends `A` to `A_*[1/ϖ]`, with `A_*` open and `ϖ`-adic. For a torsion-free
`K°`-module `N`, its almost elements are the elements `x ∈ N[1/ϖ]` such that
`ϖ^{1/pⁿ}x ∈ N` for every `n`. Reduction of almost elements gives an inclusion
`A_*/ϖ^qA_* ⊆ (A/ϖ^qA)_*`, not an equality in general; its cokernel is almost zero
([Sch12] Lemma 5.3, Remark 5.4, and Lemmas 5.6–5.7, pp. 22–23).

For *frobenius-inverse-limit-lifting-in-characteristic-p*, assume `K` has characteristic `p`.
A perfectoid reduction `B` lifts to `lim_Φ B`; its `K°a`-action is
`a·(b_n)_n = (a^{1/pⁿ}b_n)_n`. The lift is perfect, `ϖ`-adically complete and torsion-free,
and reduction recovers `B` ([Sch12] Remark 5.18, p. 26; [KL15] Lemma 3.4.2, p. 79;
[KL16] Lemma 3.3.7, p. 60). The first two scalar coordinates are `ab₀` and `a^{1/p}b₁`;
using `a` in both coordinates does not commute with Frobenius. The examples `B=K°/ϖ`
and `B=(K°/ϖ)[X^{1/p^∞}]` lift respectively to `K°` and `K°⟨X^{1/p^∞}⟩`.
*Needs:* P0.6, P0.8, P0.16, P0.22; P1.9; P1.26; Mathlib `Perfection` and
`Perfection.quotientMulEquiv`.

`FieldAlgebras p K` uses the maximal integral subring and continuous structural maps from
`K`. `fieldIntegralFunctor_obj` identifies its integral almost algebra with `S°a`, and
`fieldIntegralEquivalence_inverse` identifies the inverse ring with `A_*[1/ϖ]`.
For the same complete nontrivially normed ultrametric perfectoid field `K` and
chosen roots `ϖ`, `fieldIntegralFunctor_map` gives a simultaneous family of
object isomorphisms under which every continuous `K`-algebra map is sent to the
almostification of its restriction `S° → T°`. Explicitly,
`F.map f ≫ e(T).hom = e(S).hom ≫ toAlmost.map (f|S°)`
([Sch12] Definition 5.1, pp. 21–22, and Proposition 5.5, p. 23).
`fieldIntegralEquivalence` and `almostReductionEquivalence` state
the two equivalences of [Sch12] Theorem 5.2. The definitions of both almost-perfectoid predicates
have base, zero and dual-number Checks: dual numbers fail the Frobenius condition even though
their underlying module is flat.

In characteristic p, `almostInverseLimitRing` is the actual `Perfection` of the almost-elements
ring. Its coefficient ring maps use the sequence coordinates; `coeff_algebraMap` fixes the
inverse-Frobenius scalar action. `complete`, `integral` and `inverseComparison` identify its
completion and almost algebra with the lifting inverse. The zero algebra has zero inverse
limit. These statements use [Sch12] Remark 5.18 and the lifting references above.

<a id="p1-29"></a>

### P1.29 completed filtered colimits

**Completed filtered colimits of perfectoid Tate rings are perfectoid, and tilting commutes with
them.** (*completed-colimit-of-perfectoid-tate-rings*) Let `J` be a small filtered category (hence
nonempty) and `j ↦ R_j` a functor from `J` to perfectoid Tate rings (each in the sense of P1.1)
with continuous ring homomorphisms `φ_a : R_i → R_j` (no injectivity, flatness or density assumed;
characteristic `0`, `p` or neither; no perfectoid base field). Fix `j₀ ∈ J` and a
pseudo-uniformizer `ϖ♭ ∈ R_{j₀}♭` with compatible roots, `ϖ = ϖ♭♯`, which exists by
*exists-pseudouniformizer-with-compatible-roots*; on the tail `J_{j₀} = Under j₀` (final in `J`,
Mathlib `CategoryTheory.Under.final_forget`) the images of `ϖ` and of its roots are
pseudo-uniformizers with compatible roots and `ϖ^p | p` in every `R_j°` (a compatible
pseudo-uniformizer). The ring `R` with its maps `ι_j : R_j → R` is formed here by
`A := colim_j R_j°`, `R_0 := AdicCompletion (ϖ) A`, and `R := R_0[1/ϖ]`, with the image of
`R_0` open and `ϖ`-adic. P5.1 equips this ring with the completion of the chosen plus rings.
Its properties are: (a) `R` is a
complete Hausdorff Tate ring with pseudo-uniformizer `ϖ` and ring of definition `R_0`, `ϖ` a
nonzerodivisor on `R_0` with `R_0/ϖ^nR_0 = colim_j R_j°/ϖ^n`, the `ι_j` continuous and compatible
with dense image of `colim R_j`, the universal property for complete Hausdorff Tate rings `S` with
bounded `S⁺` (for `S` uniform: every compatible family `ψ_j : R_j → S` factors uniquely through
`R`), independence of `j₀`, of `ϖ` and of the choice of rings of integral elements `R_j⁺` in place
of `R_j°`; (b) `R` is a perfectoid Tate ring, `R_0` is `ϖ`-adically complete and integral
perfectoid with `R_0 ⊆ R°` and `ϖ^{1/p^n}R° ⊆ R_0` for all `n`, so `R`, with the `ι_j`, is the
colimit of the `R_j` in perfectoid Tate rings and in complete Hausdorff uniform Tate rings.
In addition, (c) tilting commutes with completed filtered colimits: the `R_j♭` (P1.6) form a filtered system of
perfectoid Tate rings of characteristic `p` with compatible pseudo-uniformizer `ϖ♭`, and the
continuous homomorphism from their completed colimit `T` (as in (a), with ring of definition the
`ϖ♭`-adic completion of `colim R_j♭°`) to `R♭` induced by the `ι_j♭` is an isomorphism of
perfectoid Tate rings, compatible with `♯`; dually, for a perfectoid Tate ring `R′` and a filtered
system of perfectoid `R′♭`-algebras `T_j`, the untilt over `R′` (P1.27) of their completed colimit
is the completed colimit of the untilts `T_j♯`. The `R_j` must be perfectoid: a completed colimit
of non-perfectoid rings can be perfectoid (`ℚ_p(p^{1/p^n})`) without its tilt being the colimit of
the tilts; such limits are tilde-limits (layer P7) ([Bhatt] Remark
6.2.9, p. 48, and §6.2, proof of almost purity over a field, p. 49; [Sch12] §5, Theorem 5.2 and the
text after it, p. 22, and proof of Theorem 3.7, p. 29 ("the union of the `L♯` is dense", asserted
without proof); [KL16] Corollary 3.3.22, p. 66; [Sch13] §4, end of the proof of Lemma 4.6, p. 25).
The proposed Lean carrier `completedFilteredColimit p F j₀ ϖ` is this localization of
`filteredIntegralCompletion`, with `integralDiagram` supplying the actual restricted transition
maps. `completedFilteredColimit.isTateRing` and
`completedFilteredColimit.isPerfectoidTateRing` state the Tate and perfectoid obligations
on this localization before it is bundled in `PairCat`; the instances use these theorems
([ECD] Proposition 6.5, p. 29). `ι_integral` fixes its structural maps on `R_j°`, `hasBasis_nhds_zero` fixes the topology,
and `lift_spec` with `hom_ext` fixes the continuous universal property. Prove
`completedFilteredColimit.ringOfDefinition_almostEq`
(`R_0 ≤ R°` and `ϖ^{1/p^n} • R° ≤ R_0` for all `n`), `completedFilteredColimit.tiltEquiv`
(`tilt (completedFilteredColimit F ϖ♭) ≃+* completedFilteredColimit (F ⋙ tiltFunctor) ϖ♭.toTilt`, a
homeomorphism commuting with `sharp` and sending `tilt.map (ι j)` to `ι j`),
`completedFilteredColimit.untiltEquiv` (for `G : J ⥤` perfectoid `R′♭`-algebras: the untilt over
`R′` of `completedFilteredColimit G` is isomorphic to `completedFilteredColimit (G ⋙ untiltFunctor R′)`, naturally in `G`), `completedFilteredColimit.equivOfChoice` (changing `(j₀, ϖ♭)` or
restricting along a final functor `J′ ⥤ J` gives canonically isomorphic rings, compatibly with `ι`
and `lift`; compatible rings of integral elements give the same ring) and
`completedFilteredColimit.isColimit` (the cocone `(completedFilteredColimit F ϖ♭, ι)` is a colimit
in perfectoid Tate rings with continuous ring homomorphisms). *Needs:* P1.1; P1.3;
*exists-pseudouniformizer-with-compatible-roots*; P1.6; P1.27;
*perfectoid-tate-ring-from-integral-perfectoid*; Mathlib
`CommRingCat.FilteredColimits.colimitCoconeIsColimit`, `CategoryTheory.IsFiltered`,
`CategoryTheory.Under.final_forget`, `AdicCompletion`, `IsAdicComplete`, `Localization.Away`; Tau
Ceti `TauCeti.Huber.IsTateRing`, `TauCeti.Huber.IsPseudoUniformizer.hasBasis_nhds_zero`,
`TauCeti.Huber.IsRingOfIntegralElements.mem_of_isTopologicallyNilpotent`,
`TauCeti.Huber.powerBoundedSubring`.

**Checks.**

- `completedFilteredColimit.terminalEquiv` (degenerate) — for `J` with a terminal object `j` (in particular
  a one-object `J`), `completedFilteredColimit F ≃+* F.obj j` as topological rings, compatibly with
  `ι_j`.
- `completedColimit_test_profinite` (computation) — for a perfectoid field `K` and a
  profinite set `S = lim S_i` (finite `S_i`), the completed colimit of the perfectoid rings
  `K^{S_i} = C(S_i, K)` is `C^0(S, K)` with the sup norm, and its ring of definition is `C^0(S, K°)`.
- `completedFilteredColimit_tilt_algebraicClosure` (compatibility) — for a perfectoid field `K`,
  `tiltEquiv` identifies the completion of an algebraic closure of `K♭` with the tilt of the
  completed colimit of the untilts `L♯` of the finite extensions `L` of `K♭` inside it; the union
  of the `L♯` is dense in the untilt (layer P3; [Sch12], proof of Theorem 3.7, p. 29).
  More precisely, choose a normed algebraic closure `Ω` of `tiltField p K` with its extended
  ultrametric norm, and a small filtered diagram `G` of perfectoid pairs under the tilted
  maximal-plus pair of `K`. Continuous embeddings `c_j` must identify the stage topologies
  and base maps with finite intermediate fields `L_j ⊆ Ω`, commute with transition maps,
  and have union `Ω`. Put `F = G ⋙ sliceUntiltFunctor ⋙ Under.forget`. For any stage and
  compatible pseudouniformizer roots `η` there, the theorem gives a topological ring
  isomorphism from `completedFilteredColimit (F ⋙ tiltFunctor)` to `Completion Ω`.
  Its composite with `completedFilteredColimit.tiltEquiv` takes the tilt of each structural
  map, after the slice-tilting counit identification, to `c_j` followed by completion.
  It also asserts density of the union of the structural images of the untilted stages.
- `completedFilteredColimit_nonPerfectoid_tower` (non-example) — for `p` odd, the fields
  `ℚ_p(p^{1/p^n})` are not perfectoid and their tilts in the sense of P1.7 are `𝔽_p`, while the
  completion `ℚ_p(p^{1/p^∞})^` is perfectoid with tilt `𝔽_p((t^{1/p^∞}))^`: tilting does not
  commute with completed colimits of non-perfectoid rings, so the hypothesis that every `R_j` be
  perfectoid cannot be dropped.

The perfectoidness of the Kummer field states that `Almost.kummerField p r` is
perfectoid for every prime `p` and sequence `r : ℕ → ℂ_p` with `r₀ = p` and
`rₙ₊₁^p = rₙ`, including `p = 2`. It supplies the perfectoid instance for
`fontaine_wintenberger_test_kummer_tower` ([Sch12] §1, p. 2, the completed
radical-field example preceding Definition 1.2).

`PerfectoidRingCat` is the full subcategory of perfectoid pairs with maximal plus ring.
`PairCat.maximalFunctor` supplies the continuous-ring-map diagram from any pair diagram.
`completedFilteredColimit.cocone` uses the specified structural maps, and `isColimit` states its
categorical universal property. `underObject` carries the common base map;
`untiltEquiv` compares the untilt of its completed colimit with the completed colimit after
slice untilting, with continuity in both directions. For every small filtered diagram
`G : J ⥤ Under (R′♭)`, a chosen stage `j`, compatible pseudouniformizer roots `ψ` at
`G.obj j` and `η` at its untilt, `completedFilteredColimit.untiltEquiv_ι` requires that
composing the untilt of each structural map `G.obj i → underObject G` with `untiltEquiv`
equals `ι p (G ⋙ sliceUntiltFunctor ⋙ Under.forget R′) j η i`, for every stage `i`.
This is the colimit comparison induced by the slice equivalence ([Berkeley] Theorem 6.2.7,
p. 45) and the completed-colimit universal property above; [KL16] Corollary 3.3.22,
p. 66 supplies perfectoidness in the Banach setting. `equivOfChoice_spec` fixes the cofinal
comparison on every stage.

The three Checks for `completedFilteredColimit.untiltEquiv` use these same
structural maps and [Berkeley] Theorem 6.2.7, p. 45:

- `completedFilteredColimit.untiltEquiv_zero_diagram` — when every stage of
  `G` is zero, the completed colimit of the untilted stages is zero and every
  element maps to zero under `untiltEquiv`.
- `completedFilteredColimit.untiltEquiv_two_roots` — if `j` is terminal, map
  `η.root 0 + η.root 1` from its untilt into the untilt of the colimit, apply
  `untiltEquiv`, and identify the result with the terminal untilted stage
  using `terminalEquiv`. The result is `η.root 0 + η.root 1`.
- `completedFilteredColimit.untiltEquiv_not_frobenius` — for a terminal stage
  with nonzero untilt, the image of `η.root 0` along the same composite is
  not the structural image of `(η.root 0)^p`. The terminal hypothesis makes
  the structural map injective; it is essential for this non-example.

The locally constant/profinite Check needs the functor of finite partitions of a specified
profinite space and its completed continuous-function algebra with the integral sup topology.
The algebraic-closure Check needs the finite-extension diagram and its compatible untilt
embeddings from P3. These two Checks retain those concrete carriers; the general colimit
universal property does not replace them. The non-perfectoid radical-tower Check needs the
finite radical extensions, their uniform tilts and the completion of their union, so it is a
separate comparison from the perfectoid-stage colimit.

<a id="p1-30"></a>

### P1.30 perfected Tate algebras

**Perfected Tate algebras `R⟨T_s^{1/p^∞} : s ∈ Σ⟩` in any set of variables over any perfectoid Tate
ring.** (*perfected-tate-algebra*) Let `R` be a perfectoid Tate ring (no base field) with
pseudo-uniformizer `ϖ` (`ϖ^p | p` in `R°`) and `Σ` an arbitrary set, possibly infinite (`Σ = ∅`
gives `R`; infinite `Σ` is used by layer P2 with `Σ = C♭°`, [KL16] Corollary 3.3.21, and by [AWS]
Lemma 2.8.7 with `Σ = C♭⁺`). The perfected Tate algebra `R⟨T_s^{1/p^∞} : s ∈ Σ⟩` is `A_0[1/ϖ]`,
where `A_0` is the `ϖ`-adic completion of the monoid algebra `R°[T^a : a ∈ ℤ[1/p]_{≥0}^{(Σ)}]`
(`AddMonoidAlgebra` over the finitely supported exponent vectors `Σ →₀ ℤ[1/p]_{≥0}`,
`T^a = Π_s T_s^{a(s)}`), topologised with `A_0` open and `ϖ`-adic; equivalently the completion of
the perfect polynomial ring `R[T_s^{1/p^∞} : s ∈ Σ]` for the topology in which
`R°[T_s^{1/p^∞} : s ∈ Σ]` is open with its `ϖ`-adic topology (the Gauss norm). It is a perfectoid
Tate ring with `R⟨T_s^{1/p^∞}⟩° = A_0 = {Σ_a c_aT^a : c_a ∈ R°, c_a → 0` along the cofinite filter
on exponents`}`; `R⁺⟨T_s^{1/p^∞}⟩` is a ring of integral elements for every `R⁺`. Universal
property: for every complete Hausdorff uniform Tate `R`-algebra `S` and every family, indexed by
`s ∈ Σ`, of compatible systems `(t_s^{(m)})_{m≥0}` in `S°` with `(t_s^{(m+1)})^p = t_s^{(m)}`, there
is a unique continuous `R`-algebra map `R⟨T_s^{1/p^∞}⟩ → S` with `T_s^{1/p^m} ↦ t_s^{(m)}`; for `S`
perfectoid such families are families `(x_s)` in `S♭°`, via `t_s^{(m)} = (x_s^{1/p^m})♯`. Its tilt
is `R♭⟨T_s^{1/p^∞} : s ∈ Σ⟩` with `T_s^{1/p^m} ↦ (T_s^{1/p^{m+k}})_k`, so `T_s♯ = T_s`. The
construction is functorial in `Σ`, and `R⟨T_s^{1/p^∞} : s ∈ Σ⟩` is the completed filtered colimit
(P1.29) of the algebras in finitely many of the variables. For `Σ = {1, …, n}` it is
`R⟨T_1^{1/p^∞}, …, T_n^{1/p^∞}⟩`, which contains Tau Ceti's `R⟨T_1, …, T_n⟩` as the closed subring
of series with exponents in `ℕ^n`, with a continuous `R`-linear retraction
`σ(Σ x_aT^a) = Σ_{a ∈ ℕ^n} x_aT^a`; the Laurent variant `R⟨T_s^{±1/p^∞}⟩` is perfectoid as well
([AWS] §2.1, Example 2.1.4, p. 50, Example 2.3.15, p. 56, and Lemma 2.8.7, proof, p. 71; [KL16]
Example 3.3.12, p. 61, for arbitrary index sets with plus rings, and Corollary 3.3.21, proof, p. 66,
where the source writes `A{C°}` and the images of the variables need compatible `p`-power roots,
which the elements of `C♭°` provide through `♯`; [Sch12] Proposition 5.20, proof, p. 27; [ECD]
Example 3.4(iii), p. 15). Define `perfectedTateAlgebra` (`perfectedTateAlgebra R Σ : Type` for
`Σ : Type*`, with `CommRing`, `TopologicalSpace`, `IsTopologicalRing`, `IsTateRing`, `CompleteSpace`,
`T2Space` instances and an `R`-algebra structure), `perfectedTateAlgebra.X` (`X s a = T_s^a` for
`s : Σ` and `a ∈ ℤ[1/p]_{≥0}`; `X s (a + b) = X s a * X s b`, `(X s (a/p))^p = X s a`, `X s a ∈ A°`), `perfectedTateAlgebra.lift` (for a complete Hausdorff uniform Tate `R`-algebra `S` and
`t : Σ → Perfection S° p`, the unique continuous `R`-algebra map with `X s (1/p^m) ↦ coeff m (t s)`;
`perfectedTateAlgebra.lift_spec` and `hom_ext`), `perfectedTateAlgebra.liftTilt` (for `S` perfectoid: continuous
`R`-algebra maps `perfectedTateAlgebra R Σ → S` correspond to maps `Σ → (tilt S)°`, with `sharp (tilt.map (liftTilt x) (X s 1)) = sharp (x s)`), `perfectedTateAlgebra.mapIndex` (for `g : Σ → Σ′`,
the continuous `R`-algebra map with `X s a ↦ X (g s) a`; `mapIndex id = id`, `mapIndex (g′ ∘ g) = mapIndex g′ ∘ mapIndex g`), `perfectedTateAlgebra.restrictedIncl` (for `Σ = Fin n`: the continuous
injective `R`-algebra map from `TauCeti.Huber.restrictedMvPowerSeriesCompletion n R` (`R⟨T⟩`) with
closed image, `T_i ↦ X i 1`), `perfectedTateAlgebra.retraction` (for `Σ = Fin n`: the continuous
`R`-linear `σ` onto integral exponents with `σ ∘ restrictedIncl = id`, the frame that
AdicSpacesPartIV R5 uses downstream), `perfectedTateAlgebra.plus` (for a ring of integral elements `R⁺`,
`R⁺⟨T_s^{1/p^∞}⟩` is a ring of integral elements; the resulting perfectoid pair) and
`perfectedTateAlgebra.laurent` (the Laurent variant is perfectoid); prove
`perfectedTateAlgebra.isPerfectoidTateRing`, `perfectedTateAlgebra.powerBoundedSubring_eq`
(`(perfectedTateAlgebra R Σ)° = {Σ c_a T^a | c_a ∈ R°, c_a → 0}`), `perfectedTateAlgebra.tiltEquiv`
(`tilt (perfectedTateAlgebra R Σ) ≃+* perfectedTateAlgebra (tilt R) Σ`, a homeomorphism with
`sharp (X s a) = X s a`), `perfectedTateAlgebra.equivCompletedColimit` (`perfectedTateAlgebra R Σ ≃+* completedFilteredColimit (Σ₀ ↦ perfectedTateAlgebra R Σ₀)` over finite subsets `Σ₀ ⊆ Σ`) and
`perfectedTateAlgebra.emptyEquiv` (`perfectedTateAlgebra R PEmpty ≃+* R`). *Needs:* P1.1; P1.5;
P1.6; P1.29; *rings-of-integral-elements-under-tilting*; Tau Ceti
`TauCeti.Huber.restrictedMvPowerSeriesCompletion`; Mathlib `AddMonoidAlgebra`, `AdicCompletion`,
`Localization.Away`, `Finsupp`.

**Checks.**

- `perfectedTateAlgebra.rootX_pow` (computation) — `(X s (1/p^{m+1}))^p = X s (1/p^m)`, and Frobenius
  on `A_0/ϖ` sends the class of `X s (1/p)` to that of `X s 1`; for `Σ = ℕ` the series
  `Σ_k ϖ^k X k 1` converges while, for `R ≠ 0`, `Σ_k X k 1` does not (its terms have norm one).
- `perfectedTateAlgebra.tilt_cyclotomic` (compatibility) — `tilt (ℚ_p^cycl⟨T^{1/p^∞}⟩) ≃+* (tilt ℚ_p^cycl)⟨T^{1/p^∞}⟩ = 𝔽_p((t^{1/p^∞}))^⟨T^{1/p^∞}⟩` ([ECD] Example 3.4(iii); [AWS] Example
  2.3.15).
- `perfectedTateAlgebra_fin_test` (compatibility) — for `Σ = Fin n`, `perfectedTateAlgebra R (Fin n)` is `R⟨T_1^{1/p^∞}, …, T_n^{1/p^∞}⟩` as AdicSpacesPartII uses it, `restrictedIncl` embeds
  `TauCeti.Huber.restrictedMvPowerSeriesCompletion n R` with retraction `σ`, and for `Fin n ⊆ ℕ`
  the map `perfectedTateAlgebra R (Fin n) → perfectedTateAlgebra R ℕ` is a strict inclusion
  when `R ≠ 0`. For the zero ring both algebras are zero.
- `not_isPerfectoidTateRing_restricted` (non-example) — for `n ≥ 1` and `R ≠ 0`, `R⟨T_1, …, T_n⟩`
  is not perfectoid: `T_1` is not a `p`-th power modulo `ϖ^p` (a non-example that AdicSpacesPartIV
  R5 also uses downstream).
- `perfectedTateAlgebra.lift_one` (characterisation) — `lift` with `t_s^{(m)} = 1` for all `s, m`
  is evaluation at `T = 1`, a continuous `R`-algebra map to `R`; two continuous `R`-algebra maps
  agreeing on all `X s (1/p^m)` are equal.

The proposed Lean algebra is `perfectedTateAlgebra p ϖ σ`, constructed as an actual
`AdicCompletion` of `AddMonoidAlgebra R° (σ →₀ pPowerExponents p)` followed by
`Localization.Away` at the image of `ϖ`. Here `pPowerExponents p` is the additive submonoid
of `ℚ` generated by `p⁻ⁿ`, including zero. `hasBasis_nhds_zero` specifies the completion
 topology; `X` is the image of a monomial and `rootX` specializes to exponent `p⁻ⁿ`.
`lift_spec` and `hom_ext` state the continuous universal property, including its action on
coefficients. `plus` is the closure of the subring generated by coefficients in the chosen
`R⁺` and all monomials; `pair` supplies its ring-of-integral-elements assertion.
`tiltEquiv_rootX` specifies both root indices: coordinate `m` of the inverse image of
`T_s^{1/pⁿ}` is `T_s^{1/p⁽ⁿ⁺ᵐ⁾}`. `tiltEquiv_scalar` specifies the coefficient action.

`perfectedTateAlgebra.laurent` is the analogous localization of the completed group
algebra, with exponent group `pPowerIntegers p ⊆ ℚ`. The group contains negative p-power
exponents. Its `hasBasis_nhds_zero` and `algebraMap_integral` fix the topology and coefficient
map. `restrictedIncl` uses Tau Ceti's actual `restrictedMvPowerSeriesCompletion`; `retraction`
is continuous and R-linear, is a left inverse, and kills monomials with a nonintegral exponent.
`liftTilt_spec` specifies evaluation at integral tilted elements on both base coefficients and
every root coordinate ([AWS] Example 2.3.15; [Sch12] Proposition 5.20).

`finiteDiagram` is the functor on finite subsets of the index set, using the actual
variable-inclusion maps. `equivCompletedColimit` identifies its completed filtered colimit with
the full algebra; `equivCompletedColimit_spec` fixes all structural maps and both continuities.
For a variable collision the difference of the two variables maps to zero. This tests
functoriality for noninjective index maps as well as inclusions.

### P1.31 the completed cyclotomic field

**The completed cyclotomic field `ℚ_p^cycl` is a perfectoid field; its tilt and primitive ideal.**
(*cyclotomic-perfectoid-field*) For any prime `p` and `ζ_{p^n}` a compatible system of primitive
`p^n`-th roots of unity, the completion `ℚ_p^cycl` of `ℚ_p(μ_{p^∞})` is a perfectoid field: its
valuation ring is the `p`-adic completion `ℤ_p^cycl` of `ℤ_p[μ_{p^∞}]`, its value group
`|p|^{ℤ[1/p]/(p−1)}` is nondiscrete and Frobenius on `ℤ_p^cycl/p` is surjective; the
pseudo-uniformizer `ϖ = ζ_{p²} − 1` has `ϖ^p | p` (`v_p(ϖ^p) = 1/(p−1) ≤ 1`), whereas `ζ_p − 1`
does not (`v_p((ζ_p − 1)^p) = p/(p−1) > 1`); for `p = 2`, `ϖ = ζ_4 − 1` has `ϖ² = −2ζ_4`, so
`ϖ² | 2`, while `(ζ_2 − 1)² = 4 ∤ 2` ([ECD] §3, Example
3.4(i), p. 14; [KL15] Example 3.3.8, p. 78, and Remark 3.5.12, p. 86; [BMS] Example 3.6, p. 21).
Its tilt is the completed perfection of `𝔽_p((t))` with `t = ε − 1`, `ε = (1, ζ_p, ζ_{p²}, …)`, with
`|t♯| = |p|^{p/(p−1)}` ([Berkeley] Example 6.2.4, p. 44), and it is the untilt of
`(𝔽_p((t^{1/p^∞}))^, 𝔽_p[[t^{1/p^∞}]]^)` along the primitive ideal generated by
`Σ_{i=0}^{p−1}[1 + t]^{i/p}` ([AWS] Example 2.3.8, p. 54; [KL15] Example 3.3.8, p. 78). `ℤ_p^×`
acts compatibly, by `γ(t) = (1 + t)^γ − 1` on the tilt. For a finite extension `K` of `ℚ_p`, the
completion of `K(μ_{p^∞})` is the finite extension `K·ℚ_p^cycl` of `ℚ_p^cycl`, perfectoid by the
stability of perfectoid fields under finite extension of layer P3 ([Sch12] Theorem 3.7(i); [KL15]
Theorem 3.5.6). *Needs:* P1.11; P1.15; P1.19; Mathlib `Padic`, `PadicInt`.

Lean: `padicCyclotomic` (`ℚ_p^cycl` as the closure in `ℂ_p` of the subfield generated by the
`p`-power roots of unity), `padicCyclotomic.instCompleteSpace`, `padicCyclotomic.exists_one_lt_norm`
(`‖p‖ = p⁻¹`, so the restricted norm is nontrivial), `padicCyclotomic.instNontriviallyNormedField`,
`isPerfectoidField_padicCyclotomic`.

<a id="p1-32-split-tate-rings"></a>

### P1.32 perfectoidness

[ECD] §15, Lem. 15.3, p. 90 (stated for every Tate `ℤ_p`-algebra) and [Berkeley] Lecture 10,
Lem. 10.1.6, p. 75 (stated for every Tate ring in which `p` is topologically nilpotent) give the
perfectoidness of the completed tower; here its base ring is complete, because the henselian
comparison that transfers finite étale splittings to the completion needs that hypothesis, and
the polynomial example over `ℚ_p[T]` of [*uniform-completion-etale-comparison*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.4) prevents its
removal. Throughout, `p` is a prime (`[Fact p.Prime]`, or the field `p.Prime` of the presentation
record), the case `p = 2` included; the prime enters the arithmetic only through the exponent
bound `p^{n−1} > n` of [*pseudouniformizer-dividing-p*](#p1-32-split-tate-rings) (P1.32). In the Frobenius argument
the choice of exponent also makes `p/ϖ^p` topologically nilpotent: divisibility `ϖ^p | p` alone
does not make the root algebra étale, since at `ϖ = p^{1/p}` and `f = 1 − p` the polynomial
`X^p − pX + p − 1` has a double root at `1`.

**Roots of monic polynomials with invertible derivative.** (*root-of-monic-in-split-tate-ring*)
Let `R` be a complete Hausdorff Tate ring with no nonsplit finite étale covers
(*finite-etale-split-ring*), `P ∈ R°[X]` monic of degree `d ≥ 1`, and `B := R[X]/(P)` with its
natural topology as a free `R`-module with basis `1, x, …, x^{d−1}`, `x` the class of `X`. Then
`x` is power-bounded in `B`. If moreover `P′(x) = u·(1 − τ)` in `B` with `u ∈ R^×` and `τ ∈ B`
topologically nilpotent, then `B` is complete and free, `1 − τ` is a unit (Tau Ceti
`IsTopologicallyNilpotent.isUnit_one_sub`), so `(P, 1)` is a standard étale pair (Mathlib
`StandardEtalePair`) and `B` is a faithfully flat finite étale `R`-algebra of rank `≥ 1`, which
has a section: `P` has a root in `R`, and every root of `P` in `R` lies in `R°` (a root is
integral over `R°`, which is integrally closed in `R`; `x` is power-bounded because the
subring of `R°` generated by the finitely many coefficients of `P` is bounded, also when `R` is
not uniform). Prove this ([ECD] §15, proof of Lem. 15.3, p. 90;
[Berkeley] Lecture 10, proof of Lem. 10.1.6, p. 75: the pattern "finite étale, hence a
solution"). The topological nilpotence of `τ` is what makes `P′(x)` a unit, and without it `B` need
not be étale (*frobenius-surjective-for-split-tate-ring*).
*Needs:* [*finite-etale-split-ring*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.0.0); Mathlib
`AdjoinRoot.powerBasis'`, `StandardEtalePair`, `Algebra.IsStandardEtale`,
`StandardEtalePair.homEquiv`, `Polynomial.Monic`, `Polynomial.derivative`; Tau Ceti
`IsTopologicallyNilpotent.isUnit_one_sub`, `Huber.IsPowerBounded`, `Huber.powerBoundedSubring`,
`Huber.IsTateRing`. README only (no Lean declaration).


**Checks.**

- For `P = X − a` with `a` power-bounded, the root is `a` and the derivative is one.
- For `P = X² − X` the roots 0 and 1 are power-bounded and the derivative is a unit in the quotient.
- For `P = X²` over a nonzero ring, the derivative-error unit condition fails at the zero root.

**A pseudouniformizer whose `p`-th power divides `p`.** (*pseudouniformizer-dividing-p*) Let `R`
be a complete Hausdorff Tate ring in which `p` is topologically nilpotent and which has no nonsplit
finite étale covers, and `ϖ_0 ∈ R` a pseudouniformizer (Tau Ceti `IsPseudoUniformizer`). Choose
`n ≥ 1` with `p^n ∈ ϖ_0R°` and `p^{n−1} > n` (possible, since `p^n → 0` and `ϖ_0R°` is open). Then
`X^{p^n} − ϖ_0X − ϖ_0` has a root in `R`, and every root `ϖ` is a pseudouniformizer lying in `R°`
with `ϖ^{p^n} = ϖ_0(1 + ϖ)`, `1 + ϖ ∈ (R°)^×` (so `ϖ^{p^n}` and `ϖ_0` divide each other in `R°`),
`ϖ^p | p` in `R°` and `p/ϖ^p` topologically nilpotent. Here `τ = (p^n/ϖ_0)x^{p^n−1}` is
topologically nilpotent because `x^{p^n} ∈ ϖ_0·B°`, and from `p^n = ϖ^{p^n}w` one gets
`(p/ϖ^p)^n = ϖ^{p^n−pn}w`. Prove this as `Huber.exists_isPseudoUniformizer_pow_dvd_p` ([ECD] §15,
proof of Lem. 15.3, p. 90 (the root of `x^{p^n} − ϖ_0x − ϖ_0`); [Berkeley] Lecture 10, proof of
Lem. 10.1.6, p. 75 (the strengthening `p/ϖ^p` topologically nilpotent, by taking the exponent
larger)). The strict inequality `p^{n−1} > n`, rather than [ECD]'s `p^{n−1} ≥ n`, is what makes
`p/ϖ^p` topologically nilpotent, which the Frobenius step needs. *Needs:*
[*root-of-monic-in-split-tate-ring*](#p1-32-split-tate-rings) (P1.32);
[*finite-etale-split-ring*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.0.0); Tau Ceti `Huber.IsPseudoUniformizer`,
`Huber.IsPowerBounded.isTopologicallyNilpotent_mul`, `IsTopologicallyNilpotent.isUnit_one_add`,
`Huber.powerBoundedSubring`; Mathlib `IsTopologicallyNilpotent`.

Lean: `Huber.exists_isPseudoUniformizer_pow_dvd_p` (a pseudouniformiser `ϖ ∈ R°` with `p = ϖ^p y`,
`y ∈ R°` topologically nilpotent); the rest is README only.

**Checks.**

- *test_exponent_table* (computation) — `p^{n−1} > n` excludes `n = 1` for every `p` (`p⁰ = 1`),
  and for `p = 2` also `n = 2` (`2 > 2` fails): the admissible exponents are `n ≥ 2` for `p ≥ 3`
  and `n ≥ 3` for `p = 2`; with `ϖ_0 = p` one may take `n = 2` (`p ≥ 3`) or `n = 3` (`p = 2`).
  When formalised, the computation uses a root of `X⁸ − 8X − 8` in `C₂`: it is a pseudouniformiser `v`
  with `2/v²` integral and topologically nilpotent. It also checks that the pseudouniformiser
  chosen by `Huber.exists_isPseudoUniformizer_pow_dvd_p` at `p = 2` is neither 2 nor 1.
- *test_tau_nilpotent* (computation) — in a complete split Tate ring, if `x² = ϖ(1+x)`, `ϖ` is topologically nilpotent,
  and `x,c` are power-bounded, then `cx` is topologically nilpotent: its square is a
  topologically nilpotent element times a power-bounded element. This is the quadratic
  instance of the derivative-error argument.
- *test_equality_not_nilpotent* (non-example for [ECD]'s `p^{n−1} ≥ n`) — `R = ℂ_2`, `p = 2`,
  `n = 2` (`2^{n−1} = n`), `ϖ_0 = 4` (so `p^n = 4 ∈ ϖ_0R°`): a root `ϖ` of `X⁴ − 4X − 4` has
  `|ϖ|⁴ = |4|`, so `|ϖ²| = |2|` and `p/ϖ^p = 2/ϖ²` is a unit of `R°`, not topologically
  nilpotent; `ϖ^p | p` holds.

**Surjectivity of Frobenius modulo `ϖ^p`.** (*frobenius-surjective-for-split-tate-ring*) Let `R`
be a complete Hausdorff Tate ring with no nonsplit finite étale covers and `ϖ ∈ R` a
pseudouniformizer with `ϖ^p | p` in `R°` and `p/ϖ^p` topologically nilpotent. Then for every
`f ∈ R°` there is `x ∈ R°` with `x^p − ϖ^px = f`; consequently `Φ : R°/ϖ → R°/ϖ^p`, `x ↦ x^p`
(a ring homomorphism since `p ∈ ϖ^pR°`, [ECD] §3, Def. 3.1, p. 14), is surjective. The second
hypothesis is what makes `P′ = −ϖ^p(1 − (p/ϖ^p)x^{p−1})` invertible; without it the algebra need
not be étale. Prove this ([ECD] §15, proof
of Lem. 15.3, p. 90). Over the perfectoid field `K`, the completion of `ℚ_p(p^{1/p^∞})`, with
`ϖ = p^{1/p}` (so `p/ϖ^p = 1`) and `f = 1 − p`, the polynomial `X^p − ϖ^pX − f = X^p − pX + p − 1`
has the double root `1` (`P(1) = 0 = P′(1)`), so `K[X]/(P)` is not étale; [ECD]'s printed proof
asserts étaleness without this hypothesis. *Needs:* [*root-of-monic-in-split-tate-ring*](#p1-32-split-tate-rings) (P1.32);
[*pseudouniformizer-dividing-p*](#p1-32-split-tate-rings) (P1.32); [*finite-etale-split-ring*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.0.0); Tau Ceti
`Huber.IsPowerBounded.isTopologicallyNilpotent_mul`, `Huber.powerBoundedSubring`; Mathlib
`frobenius`. README only (no Lean declaration).

**Checks.**

- *test_double_root* (non-example) — `P = X^p − pX + p − 1` over `K = ℚ_p(p^{1/p^∞})^∧`,
  `ϖ = p^{1/p}`, `f = 1 − p`: `P(1) = 1 − p + p − 1 = 0` and `P′(1) = p − p = 0`, and
  `τ = (p/ϖ^p)x^{p−1} = x^{p−1}` is not topologically nilpotent; the root algebra is not étale.
- *test_frobenius_mod* (computation) — a root `x ∈ R°` of `X^p − ϖ^pX − f` has
  `x^p − f = ϖ^p·x ∈ ϖ^pR°`, so `Φ(x mod ϖ) = f mod ϖ^p`.
- *test_frobenius_injective* (characterisation) — an element whose `p`-th power is power-bounded
  is power-bounded; applied to `x/ϖ` when `x^p ∈ ϖ^pR°`, this makes `Φ` injective
  ([ECD] §3, Rem. 3.2, p. 14).

**A complete uniform Tate ring without nonsplit covers is perfectoid.**
(*split-tate-ring-is-perfectoid*) Let `R` be a complete Hausdorff uniform Tate ring (`R°`
bounded) in which `p` is topologically nilpotent. If `R` has no nonsplit finite étale covers, then
`R` is a perfectoid Tate ring in the sense of layer P1 ([ECD] §3, Def. 3.1,
p. 14): there is a pseudouniformizer `ϖ` with `ϖ^p | p` in `R°` (and `p/ϖ^p` topologically
nilpotent) such that `Φ : R°/ϖ → R°/ϖ^p` is an isomorphism; injectivity of `Φ` is automatic
([ECD] §3, Rem. 3.2, p. 14; layer P1,
*perfectoid-condition-independent-of-pseudouniformizer* (a), with the surjectivity bookkeeping of
its *frobenius-surjectivity-criterion*; the split-ring argument is the content here). Prove this
as `Huber.isPerfectoid_of_isFiniteEtaleSplit`, whose conclusion is the perfectoid Tate-ring
predicate that PerfectoidSpaces owns and every roadmap shares, `Perfectoid.IsPerfectoidTateRing p R`
(Frobenius surjective on `R°/p`), equivalent to the form above by [ECD] §3, Rem. 3.2, p. 14
([Berkeley] Lecture 10, proof of Lem. 10.1.6, p. 75: the perfectoidness argument uses only
completeness, uniformity and splitness of the completed ring). Completeness and the splitting hypothesis matter: `ℚ̄_p` has no
nonsplit finite étale
covers but is not complete, and `ℂ_p⟨T⟩` is complete and uniform but not perfectoid, hence by this
lemma has a nonsplit finite étale cover. *Needs:* [*pseudouniformizer-dividing-p*](#p1-32-split-tate-rings) (P1.32);
[*frobenius-surjective-for-split-tate-ring*](#p1-32-split-tate-rings) (P1.32); [*finite-etale-split-ring*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.0.0); layer P1.

Lean: `Huber.isPerfectoid_of_isFiniteEtaleSplit` (`R°` bounded as the uniformity hypothesis; its
conclusion is PerfectoidSpaces' `Perfectoid.IsPerfectoidTateRing`); the rest is README only.


**Checks.**

- The complete algebraically closed field `ℂ_p` satisfies the splitness hypothesis and is perfectoid.
- The complete uniform ring `ℂ_p⟨T⟩` fails perfectoidness and therefore cannot satisfy finite-étale splitness.
- The uncompleted algebraic closure of `ℚ_p` fails the completeness hypothesis.

**Perfectoid uniform completion.** (*perfectoid-uniform-completion*) Let `(A, A⁺)` be a Huber pair
with `A` a complete Hausdorff Tate ring in which `p` is topologically nilpotent (a complete Tate
`ℤ_p`-algebra), `A⁺` any ring of integral elements, `T` any finite étale torsor tower over `A`
(*finite-etale-torsor-tower*) and `(Â_∞, Â_∞⁺)` its completed tower pair (*completed-tower-pair*).
Then `Â_∞` is a perfectoid Tate ring (layer P1: `Perfectoid.IsPerfectoidTateRing p Â_∞`,
the predicate PerfectoidSpaces owns); explicitly, `Â_∞` contains a
pseudouniformizer `ϖ`, a root of `X^{p^n} − ϖ_0X − ϖ_0` for any pseudouniformizer `ϖ_0` of `A` and
`n` with `p^n ∈ ϖ_0Â_∞°` and `p^{n−1} > n`, such that `ϖ^p | p` in `Â_∞°`, `p/ϖ^p` is
topologically nilpotent and `Φ : Â_∞°/ϖ → Â_∞°/ϖ^p` is bijective. So `(Â_∞, Â_∞⁺)` is a perfectoid
Huber pair with a continuous action of the profinite group `G`. Prove this ([ECD] §15, Lem. 15.3,
p. 90, the statement,
with completeness of `A` added; [ECD] §15, proof of Lem. 15.3, p. 90, pseudouniformizer then
Frobenius; [Berkeley] Lecture 10, Lem. 10.1.6, p. 75, the completion of the colimit with
`colim R_i°` open and bounded; [ECD] §3, Def. 3.1, p. 14, the definition verified). Completeness
of `A` cannot be omitted: for the non-complete Tate `ℤ_p`-algebra `ℚ_p[T]` (ring of definition
`ℤ_p[T]`, `p`-adic topology) every tower has colimit `B[T]` with `B` a colimit of finite étale
`ℚ_p`-algebras, ind-finite-étale over `ℚ_p`, and the uniform completion `B̂⟨T⟩` is not perfectoid
(its power-bounded subring is `B̂°⟨T⟩`, and modulo `p` the class of `T` is not a `p`-th power in
`(B̂°/p)[T]`), although [ECD] Lem. 15.3 is stated for all Tate `ℤ_p`-algebras. *Needs:*
[*completed-tower-pair*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.5); [*finite-etale-invariance-along-tower*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.5); [*split-tate-ring-is-perfectoid*](#p1-32-split-tate-rings) (P1.32);
[*finite-etale-torsor-tower*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.3); layer P1; Tau Ceti `Huber.IsPseudoUniformizer`. README only (no Lean declaration).

**Checks.**

- *completion_test_Qp* (computation) — for the field tower over `ℚ_p` the perfectoid ring
  produced is `ℂ_p`, for `p = 2` as for odd `p`.


**Checks.**

- The trivial-group tower over a perfectoid split base has that original perfectoid ring as its completion.

- For the zero Tate base, all stage and completed rings are zero and the perfectoid conditions are vacuous.

### P1.32a examples and non-examples

**Examples and non-examples of perfectoid Tate rings.** (*perfectoid-examples*) The following are
perfectoid Tate rings, each a declaration of the library, checked against P1.1 through the criteria
of this layer: (1) `ℂ_p` (Mathlib `PadicComplex p`), a perfectoid field whose tilt contains
`p♭ = (p, p^{1/p}, …)` with `(p♭)♯ = p` ([Bhatt] Example 3.1.2(2), p. 9); (2) the completion `K_1`
of `ℚ_p(p^{1/p^∞})`, a perfectoid field with valuation ring the `p`-adic completion of
`ℤ_p[p^{1/p^∞}]` and tilt `𝔽_p((t^{1/p^∞}))^` with `t♯ = p`, the untilt along `(p − [t])` ([AWS]
Example 2.3.7, p. 54; [Bhatt] Example 3.1.2(1), p. 9); (3) the completed perfection
`𝔽_p((t^{1/p^∞}))^`, a perfectoid field of characteristic `p` equal to its own tilt; (4) `ℚ_p^cycl`
(P1.31); (5) `ℚ_p^cycl⟨T^{1/p^∞}⟩` (P1.30); (6) the ring
`R = ℤ_p^cycl[[T^{1/p^∞}]]⟨(p/T)^{1/p^∞}⟩[1/T]` with `ϖ = T^{1/p}` (`ϖ^p = T` divides `p` in `R°`),
which does not contain a nonarchimedean field ([ECD] §3, Example 3.4(iv), p. 15; [Berkeley] Example
6.1.5(4), p. 42); (7) products such as `K × K♭` for a characteristic-zero perfectoid field `K`, where `p` is a nonzero
nonunit (the product of two Tate rings is Tate, with pair of definition the product,
`Perfectoid.IsTateRing.instProd`, this roadmap's declaration in the `Perfectoid` namespace:
the library does not have a product instance). Non-examples: `ℚ_p` and every discretely valued nonarchimedean field ([ECD] Remark 3.3,
p. 14; [Berkeley] Remark 6.1.4, p. 42), `𝔽_p((t))` (not perfect), `ℂ_p[ε]/(ε²)` (not uniform),
`R⟨T_1, …, T_n⟩` for `n ≥ 1` and `R ≠ 0` (`T_1` not a `p`-th power modulo `ϖ^p`). *Needs:* P1.1;
P1.15; P1.30; P1.31; P1.2; Mathlib `PadicComplex`, `Padic`; Tau Ceti `TauCeti.Huber.Padic.isTateRing`,
`TauCeti.Huber.PadicInt.not_isTateRing`, `TauCeti.Huber.LaurentSeries.isTateRing`.

### P1.33 compatibility with AdicSpaces' perfect field `F` and `A_inf`

**Compatibility with AdicSpaces' perfect field `F` and `A_inf = W(𝒪_F)`.**
(*adicspaces-layer-6-compatibility*) Let `F` be a complete rank-one nonarchimedean perfect field of
characteristic `p` with a pseudo-uniformizer `ϖ`, as in the Foundations of adic spaces roadmap
(AdicSpaces), layer 6.1 (no algebraic closedness). Tau Ceti's Fargues–Fontaine input is more
general than a field: a perfect ring `R` (`[CharP R p] [PerfectRing R p]`) with an element `ϖ`
and the `(p, [ϖ])`-adic topology on `W(R)` (`TauCeti.FarguesFontaine.spaY`, `frobeniusHomeomorph`,
`windowU`, `windowV`, `spaX`); the statements below are for `R = 𝒪_F`. Then (a) `F` is a perfectoid
field of characteristic `p` (P1.2, P1.11), its tilt is `F` itself with `♯` the identity (P1.4), and
`(F, 𝒪_F)` is a perfectoid Tate pair of characteristic `p` ([ECD] §3, Proposition 3.5, p. 15;
[Sch12] Lemma 3.4(iv), p. 16, and §3, p. 15); (b) AdicSpaces' `A_inf = W(𝒪_F)`, with pair of
definition `(A_inf, (p, [ϖ]))`, is the ring `W(F♭⁺) = W(𝒪_F)` of this layer, its `(p, [ϖ])`-adic
completeness, separatedness and Huber-ring structure being the library's
`WittVector.isAdicComplete_span_p_teichmuller`, `isHausdorff_span_p_teichmuller` and
`TauCeti.Huber.WittVector.isHuberRing_adicTopology_span_p_teichmuller` (for the integers of a
normed field, `Complete.lean` has the instance directly), cited in
*witt-vectors-of-perfect-plus-ring* ([Bhatt] Example 6.1.6, p. 43); (c) the untilts of `F` are the
pairs `(W(𝒪_F)[1/[ϖ]]/(ξ), W(𝒪_F)/(ξ))` for primitive `ξ ∈ A_inf` (P1.21). AdicSpaces' curve
`𝒴 ⊆ Spa(A_inf, A_inf)` and its Frobenius windows stay with AdicSpaces and the library; this layer
only identifies the input objects. For `F` the completed perfection of `𝔽_p((t))`, the untilts include
`ℚ_p(p^{1/p^∞})^` (`ξ = p − [t]`) and `ℚ_p^cycl` (`ξ = Σ_{i<p}[1 + t]^{i/p}`). *Needs:* P1.2; P1.11;
P1.4; *witt-vectors-of-perfect-plus-ring*; P1.21; the Foundations of adic spaces roadmap
(AdicSpaces), layer 6.1; Tau Ceti `TauCeti.FarguesFontaine.spaY`,
`WittVector.isAdicComplete_span_p_teichmuller`,
`TauCeti.Huber.WittVector.isHuberRing_adicTopology_span_p_teichmuller`.

For a complete nontrivially normed ultrametric field `F` with `CharP F p` and
`IsPerfectoidField p F`, `tilt.ainfEquiv` identifies
`WittVector p (Pair.tilt p (Pair.powerBounded F)).plus` with
`WittVector p (NormedField.valuation (K := F)).integer`. Here the latter integer
subring is `𝒪_F = {x : F | ‖x‖ ≤ 1}`. The perfect-field hypothesis above supplies
`IsPerfectoidField p F` by `IsPerfectoidField.iff_perfectRing`.
`tilt.ainfEquiv_coeff` specifies the map on every Witt coordinate as
`tilt.equivOfCharP`, so it is the Witt functor applied to the integral tilt
identification ([ECD] Lemmas 3.10–3.11, pp. 16–17).
For integral parameters `t ∈ F♭⁺` and `ϖ ∈ 𝒪_F` with
`tilt.equivOfCharP t = ϖ`, `tilt.ainfEquiv_map_ideal` identifies
`(p,[t])^n` with `(p,[ϖ])^n` for every `n : ℕ`, and
`tilt.ainfEquiv_continuous` states continuity in both directions for these
adic topologies. These are consequences of Witt functoriality and the
power-ideal neighbourhood bases; for a pseudouniformizer `ϖ` they identify
precisely the inputs to `TauCeti.FarguesFontaine.spaY` and `windowU`.
The two statements also apply to zero and unit parameters; they do not
assert that those parameters are pseudouniformizers.

**Checks.**

- `ainfEquiv_teichmuller` — for corresponding integral parameters, the comparison
  sends `[t]` to `[ϖ]`, fixing the second generator of the adic ideal.
- `ainfEquiv_prime_nonzero` — the comparison fixes `p`; this Witt vector is
  nonzero although its zeroth coordinate is zero, distinguishing it from
  the characteristic-`p` coefficient ring (also when `p = 2`).
- `ainfEquiv_zero_parameter` — setting the second parameter to zero gives
  continuity in both directions for the `p`-adic topologies.

### Examples

The worked examples of this layer are the checks above together with P1.31 and P1.32a: `ℚ_p` as the
complete uniform Tate ring with bijective Frobenius on `𝔽_p` that is not perfectoid, and
`ℂ_p[ε]/(ε²)` as the non-uniform ring a definition without uniformity would accept; `ℂ_p` with
`ϖ = p^{1/p}` and its gauge `2^{-p v_p(x)}`; the completed perfection of `𝔽_p((t))` equal to its own
tilt; `ε = (ζ_{p^n})_n` in `(ℚ_p^cycl)♭` with `ε♯ = 1`, `(ε − 1)♯ ≠ 0` and
`|(ε − 1)♯| = |p|^{p/(p−1)}`; `uniformTilt ℚ_p ≅ 𝔽_p` with the discrete topology; `𝒪/m` almost zero
and `𝒪/p^{1/p}𝒪` not, over `𝒪_{ℂ_p}`; the primitive elements `p − [p♭]` and `Σ_{i<p}[ε]^{i/p}` of
`W(𝒪_{ℂ_p♭})` against the non-primitive `p²`, `[ϖ]` and `[ε] − 1`; the untilts of
`𝔽_p((t^{1/p^∞}))^` along `(p)`, `(p − [t])` and `(Σ_{i<p}[1 + t]^{i/p})`, namely the field itself,
`ℚ_p(p^{1/p^∞})^` and `ℚ_p^cycl`, with the Frobenius-twisted marking of `ℂ_p` as the non-isomorphic
marked untilt; `K⟨T⟩` failing the metric criterion at `T`; `ℤ_p` against the integral perfectoid
`ℤ_p[p^{1/p^∞}]^`; `C^0(S, K)` as a completed colimit and the tower `ℚ_p(p^{1/p^n})` for which
tilting does not commute with the completed colimit; `ℚ_p^cycl⟨T^{1/p^∞}⟩` with its tilt
`𝔽_p((t^{1/p^∞}))^⟨T^{1/p^∞}⟩` against the non-perfectoid `R⟨T_1, …, T_n⟩`; and the non-field
perfectoid ring `ℤ_p^cycl[[T^{1/p^∞}]]⟨(p/T)^{1/p^∞}⟩[1/T]`.


**Additional Checks.** The labels below name further computed Checks, each to become a Lean
`example`; the ones already present in `Suggested.lean` are listed after the table.

| Check | Computed result |
| --- | --- |
| `perfectoid_padic_negative` | `ℚ_p` fails the pseudo-uniformizer divisibility despite surjective Frobenius on its residue field. |
| `perfectoid_dual_numbers` | `ℂ_p[ε]/(ε²)` has all three Frobenius ingredients except bounded `R°`; its nilpotent line is unbounded. |
| `perfectoid_zero` | Every quotient in the defining criterion is zero for the zero Tate ring. |
| `perfectoid_uniformity_required` | The entire right side of the mod-`p` criterion holds for the dual numbers, showing why boundedness is a separate hypothesis. |
| `integral_padic_negative` | In `ℤ_p`, admissible divisibility forces a unit `π`; completion at `(π)` would instead give zero. |
| `cyclotomic_two_roots` | For a primitive fourth root, `(ζ₄−1)²=−2ζ₄` divides `2` integrally, whereas `(ζ₂−1)²=4` does not. |
| `gauge_padic_formula` | With `ϖ^p=p` in `ℂ_p`, the formula is `gauge x=2^{-p v_p(x)}` for nonzero `x`. |
| `gauge_wrong_root` | The zeroth root in `ℂ_p` cannot equal `p`: its `p`-th power would not divide `p` integrally. |
| `gauge_normalisation` | On `ℂ_p`, the values at `ϖ`, `ϖ²`, and `0` are `1/2`, `1/4`, and `0`. |
| `gauge_product` | The two product gauges give `(1/2,1/2)` at `(a,0)` and `(1/4,1/2)` at `(0,a²)`, ruling out one exponent relating them. |
| `roots_padic_empty` | No compatible pseudo-uniformizer satisfying the divisibility exists in `ℚ_p`. |
| `roots_charP_unique` | In a characteristic-`p` perfectoid Tate ring, fixing the zeroth coordinate fixes all roots. |
| `tilt_cyclotomic_twice` | Tilting the cyclotomic tilt again gives it back through a continuous zeroth-coordinate ring equivalence with continuous inverse. |
| `tilt_cyclotomic_epsilon` | The compatible primitive roots have zeroth coordinate `1`, but form a tilted element distinct from `1`. |
| `tilt_coordinate_nonadditive` | Zeroth coordinates fail to preserve `ε−1` in the cyclotomic example. |
| `pair_padicComplex_unique` | Every ring of integral elements of `ℂ_p` is `𝒪_{ℂ_p}`; a putative second plus ring is not a valid test. |
| `field_padic_negative` | The discretely valued field `ℚ_p` is not perfectoid, although its mod-`p` Frobenius is surjective. |
| `field_padicComplex` | `ℂ_p` is perfectoid, its nonzero norms are rational powers of `p`, and integral elements have integral `p`-th roots. |
| `field_discrete_charP` | A complete characteristic-`p` field with a nontrivial discrete value group is neither perfect nor perfectoid. |
| `primitive_mixed_generators` | The radical generator `p−[p♭]` and the shifted cyclotomic sum are primitive over `𝒪_{ℂ_p♭}`; for odd `p`, the radical generator has first two Witt coordinates `−p♭` and `1`. |
| `primitive_p_squared_and_teichmuller` | In a nonzero characteristic-`p` perfectoid plus ring, `p²` and a single Teichmüller lift have zero first Witt coordinate and are not primitive. |
| `primitive_p` | The Witt vector `p` and its principal ideal are primitive. |
| `primitive_two_coordinates` | At `p=2`, the first two Witt coordinates of `2−[x]` are `x` and `1+x²`, fixing the sign and coordinate convention. |
| `untilt_p_recovery` | The untilt along `(p)` recovers the original topological ring and its specified plus ring. |
| `untilt_p_minus_t` | Along `(p−[t])`, the sharp image of the pseudo-uniformizer `t` is `p`. |
| `untilt_cyclotomic_recovery` | The shifted cyclotomic sum recovers `ℚ_p^cycl`, its plus ring, and the canonical tilt comparison. |
| `untilt_p_squared_nonreduced` | Along `(p²)` over a nonzero perfect base, `p` is nonzero and has square zero; this quotient is not a perfectoid untilt. |
| `untilt_zero_base` | The same algebraic quotient over the zero base is zero. |
| `untilt_small_quotient` | The radical untilt identifies the integral quotients modulo `p` and modulo `p♭`. |
| `untilt_large_quotient_negative` | `𝒪_{ℂ_p}/p²` does not have characteristic `p`, excluding an arbitrary-scale quotient comparison. |
| `frobenius_two_terms` | On `𝔽₂/(0)`, both quotient-Frobenius constructions give `Φ(1+1)=0` and `Φ(1)=1`, fixing the power and quotient conventions. |
| `frobenius_unit_ideal` | For `I=J=A`, and for `ϖ=1`, both quotient maps have zero target. |
| `frobenius_polynomial` | Frobenius on `𝔽₂[T]/(0)` misses `T`; both quotient-map constructions fail surjectivity. |
| `cyclotomic_root_membership` | A `p^n`-th root of unity lies in `ℚ_p^cycl` and has norm one, including `n=0`. |
| `cyclotomic_p_norm` | In `ℚ_p^cycl`, `p≠0` and `‖p‖=p⁻¹`. |
| `cyclotomic_nondiscrete` | The norms in `ℚ_p^cycl` meet every interval `(1,ε)` with `ε>1`. |
| `roots_first_two` | In `ℂ_p`, the first two recurrences hold in `R` and `R°`; the zeroth root is nonzero. |
| `roots_exponent_convention` | `zpow 0 1=1`, `zpow p 1=zpow 1 0`, `zpow 1 0 · zpow (-1) 0=1`, and `zpow 2 0=ϖ²`. |
| `roots_nonintegral_inverse` | In `ℂ_p`, `zpow 1 0` is the integral zeroth root and its inverse is not power-bounded. |
| `roots_zero_gauge` | On the zero ring all roots vanish and the gauge is zero; `gauge(0^0)=0≠1=(gauge 0)^0`. |
| `tilt_zero_characteristic` | The zero tilt has characteristic one, not the prime `p`; sharp and coordinates vanish and the tilted plus ring is the whole zero ring. |
| `tilt_roots_and_plus` | In `ℂ_p`, the first two coordinates of `ϖ.toTilt` are its roots, sharp is the integral zeroth root, and the nonzero element lies in the tilted plus ring. |
| `sharp_cyclotomic_two_terms` | For `ε=(ζ_{p^n})`, `ε♯=(ε²)♯=1`, but `(ε−1)♯≠ε♯−1`. |
| `tilt_charP_two_terms` | In characteristic `p`, the zeroth-coordinate equivalence sends `ϖ♭+1` to `ϖ+1` and `(ϖ♭)²` to `ϖ²`; sharp is additive here. |
| `tilt_charP_plus` | The characteristic-`p` equivalence takes the tilted plus ring back to the specified plus ring, including a nonmaximal choice. |
| `tilt_map_identity_roots` | Tilting the identity of `ℂ_p` fixes `ϖ♭+1`, its pair map fixes that element, and the transported first root is unchanged. |
| `tilt_map_projection` | The two projections from `ℂ_p×ℂ_p` select the corresponding coordinates of the tilted element and compatible roots. |
| `tilt_map_swap` | Swapping the two factors swaps the zeroth tilted coordinate and the first compatible root. |
| `almost_residue` | For every perfectoid field `K`, including `ℂ_p`, `K°/m_K` is almost zero and every compatible root of a pseudo-uniformizer annihilates it. |
| `almost_principal_quotient` | `𝒪_{ℂ_p}/ϖ` is not almost zero: the first root of `ϖ` does not belong to `(ϖ)`. |
| `almost_zero_base` | Over the zero ring every module is almost zero. |
| `primitive_zero_ring` | Every Witt vector and every ideal over the zero ring is primitive; negative examples require a nonzero base. |
| `primitive_ideal_p_squared` | The ideal `(p²)` in `W(𝒪_{ℂ_p♭})` has no primitive generator. |
| `primitive_ideal_top` | The unit ideal in `W(𝒪_{ℂ_p♭})` has no primitive generator. |
| `untilt_unit_ideal` | The algebraic localisation for `J=⊤` is zero, with whole plus ring; primitivity is needed for a nonzero topological untilt. |
| `boundedWitt_zero_base` | Over the zero ring the multiplicative set contains zero and the localisation is zero. |
| `boundedWitt_inverts_roots` | A Teichmüller lift of a pseudo-uniformizer belongs to the multiplicative set and becomes a unit in the bounded Witt ring. |
| `boundedWitt_does_not_invert_p` | Over `𝒪_{ℂ_p♭}`, `p` is outside that multiplicative set and remains a nonunit in the bounded Witt ring. |
| `untilt_charP_recovery` | For `J=(p)`, the comparison recovers the chosen plus ring, sharp is the inverse ring map, and `tiltEquiv` is the zeroth-coordinate comparison. |
| `untilt_charP_two_terms` | For `J=(p)`, sharp carries `x+y` to the sum of the images, unlike the mixed-characteristic cyclotomic witness. |
| `untilt_map_characteristic_p` | The identity pair map induces the identity on every algebraic quotient, including nonprimitive ideals. |
| `untilt_map_to_zero` | Enlarging any ideal to the unit ideal makes the induced untilt map zero. |
| `untilt_mixed_root` | For `J=(p−[p♭])` over `ℂ_p♭`, the topological comparison takes sharp of `p♭` to `p`, recovers `𝒪_{ℂ_p}`, and agrees with `tiltEquiv`. |
| `norm_rescaling_two_terms` | On `ℂ₂`, the absolute value gives `2,4` the values `1/2,1/4`; its square gives `1/4,1/16`. The identity is a continuous embedding between these normed presentations, but is not isometric. |
| `pair_tilt_projection` | The first product projection on perfectoid pairs selects the first zeroth coordinate after tilting. |
| `pair_tilt_swap` | The factor swap on perfectoid pairs swaps both entries of the first tilted coordinate. |
| `untilt_map_projection` | Along `(p)`, untilting the first projection of `ℂ_p♭×ℂ_p♭` recovers that projection and both specified plus rings; the projection has a nonzero kernel. |
| `integral_perfect_charP` | Every perfect characteristic-`p` ring is integral perfectoid, taking the adic element `0`; no topology is required. |
| `integral_zero` | The zero ring satisfies the integral predicate; its Fontaine condition uses the unique map. |
| `integral_padic_negative_predicate` | `ℤ_p` fails the integral predicate, since every `π` with `π^p ∣ p` is a unit. |
| `integral_padicComplex` | `𝒪_{ℂ_p}` is integral perfectoid, with an admissible `p`-th root of `p`. |
| `integral_adic_element_matters` | A nonzero perfect characteristic-`p` ring (for example `𝔽₂`) is integral perfectoid and complete for `(0)`, but is not complete for `(1)`; divisibility by `p=0` alone does not select an admissible topology. |


The following Checks specify the categorical, normed and completed constructions. Each label
names a Check specified in this README only.

| Check | Computation or boundary case |
| --- | --- |
| `uniformTilt_cyclotomic` | For the compatible primitive roots of unity, sharp of `ε` is 1 but sharp of `ε−1` is nonzero; the perfectoid comparison preserves coordinate 1. |
| `uniform_zero_integral` | Over the zero base the uniform tilt and its plus ring are zero, and `θ` is the zero map. |
| `uniform_charP_theta` | Over a perfect characteristic-p base, the uniform Fontaine kernel is exactly `(p)`. |
| `uniform_cyclotomic_theta` | The cyclotomic element `[ε]−1` lies in `ker θ` although `ε−1` is nonzero in the uniform tilt. |
| `uniform_map_two_coordinates` | Applying a continuous ring map to the uniform tilt applies that map to coordinates 0 and 1 separately. |
| `uniform_map_identity` | The identity map on the base fixes `x+1` and `x²` in the uniform tilt. |
| `uniform_plus_small_inverse` | The inverse of a compatible small unit is outside every tilted plus ring when the base is nonzero. |
| `exponents_first_two` | The exponent monoid contains `1` and `1/p`, and `p·(1/p)=1`. |
| `exponents_zero` | Zero is a nonnegative exponent; `−1` is excluded. |
| `exponents_not_thirds` | At `p=2`, the exponent `1/3` is excluded. |
| `perfected_empty` | The empty-variable perfected Tate algebra is topologically `R`, and its comparison fixes coefficients. |
| `perfected_two_roots` | The first two root monomials satisfy `T^(1/p)` raised to p equals T, and `T^(1/p²)` raised to p equals `T^(1/p)`. |
| `perfected_infinite_series` | Over a nonzero base, the weighted series `∑ϖⁿT_n` converges and `∑T_n` does not; the proposed Lean form uses a universe lift of `ℕ` as the variable set. |
| `perfected_zero` | Over the zero ring both the integral completion and its localized perfected algebra are zero. |
| `perfected_plus_roots` | The first and second root monomials belong to the plus ring for any chosen base plus ring. |
| `perfected_plus_zero` | The perfected plus ring of the zero algebra is the whole ring. |
| `perfected_plus_inverse` | The inverse base pseudo-uniformizer is outside the perfected plus ring over a nonzero base. |
| `perfected_tilt_two_roots` | The inverse tilting comparison sends `T` to a sequence with coordinates `T` and `T^(1/p)`. |
| `laurent_exponents_signed` | The Laurent exponent group contains both `−1` and 1. |
| `laurent_exponents_root` | The Laurent exponent group contains `1/p` and `p·(1/p)=1`. |
| `laurent_exponents_not_thirds` | The Laurent exponent group at `p=2` still excludes `1/3`. |
| `laurent_inverse_monomial` | In the Laurent algebra `T·T⁻¹=1` and `T⁻¹` is power-bounded. |
| `laurent_empty` | The Laurent algebra with no variables is topologically `R`. |
| `laurent_zero` | Both Laurent completion and localization are zero over a zero base. |
| `laurent_two_roots` | The Laurent root exponents `1/p` and `1/p²` satisfy the two successive p-th-power equations. |
| `restricted_two_monomials` | The retraction preserves the ordinary polynomial `T+T²`. |
| `restricted_fractional_root` | The retraction kills `T^(1/p)`, which has no integral exponent. |
| `restricted_empty` | With no variables, the ordinary inclusion and the retraction are both bijective. |
| `restricted_not_surjective` | Over a nonzero base `T^(1/p)` is outside the ordinary Tate algebra image. |
| `liftTilt_ones` | Evaluation at the constant root family 1 sends `T+T^(1/p)` to 2. |
| `liftTilt_zeroes` | Evaluation at the constant root family 0 sends `T+1` to 1. |
| `liftTilt_small_unit` | Evaluation at the compatible small unit sends `T` and `T^(1/p)` to `ϖ₀` and `ϖ₁`. |
| `marked_self_kernel` | The characteristic-p self untilt has primitive ideal `(p)`. |
| `marked_automorphism` | A marking-preserving automorphism is the identity. |
| `marked_distinct_ideals` | Untilts from distinct primitive ideals admit no marked isomorphism. |
| `marked_zero` | Every marked untilt of a zero pair has zero underlying ring and unit primitive ideal. |
| `marked_padicComplex_kernel` | For the canonical marking of `ℂ_p` and `sharp(t)=p`, the marked ideal is `(p−[t])`. |
| `marked_frobenius_twist` | Composing the marking of `ℂ_p` with Frobenius changes its primitive ideal and destroys marked isomorphism. |
| `pair_zero` | The zero pair admits exactly one morphism from every perfectoid pair. |
| `pair_tilt_two_coordinates` | The categorical tilted pair map agrees with the base map in coordinates 0 and 1. |
| `triple_characteristic_p` | Classifying a characteristic-p pair gives the Witt ideal `(p)`. |
| `slice_base` | Tilting the identity algebra over the base gives the identity algebra over its tilt. |
| `slice_characteristic_p_ideal` | For a characteristic-p base, extending the base Fontaine ideal gives `(p)` in the target Witt ring. |
| `slice_zero_target` | The slice untilting functor sends a zero target to a zero target. |
| `tiltField_two_powers` | An element with sharp `p` has norm `‖p‖`, and its square has norm `‖p‖²`. |
| `tiltField_charP_isometry` | The characteristic-p zeroth-coordinate comparison preserves distances. |
| `tiltField_cyclotomic_norm` | For the cyclotomic root sequence, `‖ε‖=1` and `‖ε−1‖=‖p‖^(p/(p−1))`. |
| `tiltField_zero_and_one` | The field comparison sends zero to zero and the tilted norms of zero and one are 0 and 1. |
| `tiltField_map_two_terms` | The field map induced by the identity fixes `x+1` and `x²`. |
| `tiltField_residue_small` | An integral tilted element of norm less than one has zero image under the residue-field comparison. |
| `filtered_terminal` | If the indexing category has a terminal stage, its element `x+1` is recovered by the completed-colimit comparison. |
| `filtered_zero_terminal` | A filtered diagram with a zero terminal stage has zero completed colimit, allowing noninjective transition maps. |
| `filtered_two_roots` | The structural map preserves the first two compatible-root equations after completion. |
| `tilted_roots_first_two` | Sharp of the first two tilted roots gives the first two original roots. |
| `tilted_roots_zero` | Over the zero ring the zeroth tilted root is zero. |
| `tilted_roots_nonintegral_inverse` | Over a nonzero base the inverse tilted pseudo-uniformizer is not power-bounded. |

| `almost_frobenius_classical` | For the unit almost ideal, almost bijectivity is ordinary bijectivity. |
| `almost_frobenius_zero_ideal` | For the zero almost ideal every ring map is almost bijective. |
| `almost_frobenius_identity` | The identity ring map is almost bijective for every almost ideal. |
| `almost_integral_base` | The integral base almost algebra is perfectoid. |
| `almost_integral_zero` | The zero integral almost algebra is perfectoid. |
| `almost_integral_dual_numbers` | Dual numbers over `K°` fail the integral Frobenius condition. |
| `almost_mod_base` | The reduced base almost algebra is perfectoid over `K°/ϖ`. |
| `almost_mod_zero` | The zero reduced almost algebra satisfies the mod-`ϖ` predicate. |
| `almost_mod_dual_numbers` | Dual numbers over `K°/ϖ` fail the reduced Frobenius condition. |
| `colimit_constant` | Any filtered constant diagram recovers its value, continuously and with every structural map identified. |
| `colimit_cofinal_coordinates` | The cofinal comparison fixes the images of the first two chosen roots. |
| `finite_colimit_two_roots` | The finite-variable colimit identifies T and `T^(1/p)` at every finite stage. |
| `finite_colimit_zero` | The finite-variable diagram over the zero ring has zero completed colimit. |
| `variable_collision` | Over a nonzero base, identifying two indices sends their nonzero variable difference to zero. |
| `variable_index_empty` | The map from no variables sends the element corresponding to `r` to the constant coefficient `r`. |
| `variable_index_zero` | Every index map over the zero base sends every element to zero. |
| `sharp_quotient_generator` | The chosen tilted generator maps to zero in the sharp quotient. |
| `sharp_quotient_two_terms` | The quotient comparison sends `x+y` and `xy` to the sum and product of their sharp residues. |
| `sharp_quotient_zero` | For a zero Tate ring the quotient is zero and the comparison sends one to zero. |
| `adic_presentation_zero` | The zero ring has an admissible presentation with `π=0`. |
| `adic_presentation_perfect_charP` | A perfect characteristic-p ring has an admissible presentation with `π=0`. |
| `adic_presentation_padic_negative` | No element of `ℤ_p` gives an integral perfectoid adic presentation. |
| `adic_presentation_unit_negative` | A nonzero ring admits no presentation with `π=1`. |
| `almost_limit_scalar_coordinates` | The first two scalar coordinates are `a·x₀` and `a^(1/p)·x₁`. |
| `almost_limit_zero` | A zero almost algebra has zero inverse-limit ring. |
| `almost_limit_base` | The inverse limit for the reduced base recovers `K°` as a `K°`-algebra. |
| `uniform_quotient_padic` | Over `ℚ_p` the unit has coordinates 1 and 1 after integral quotient comparison; `1−1` maps to zero. |
| `uniform_quotient_cyclotomic` | The image of `ε−1` modulo `p` has coordinate zero equal to zero and coordinate two nonzero. |
| `uniform_quotient_small_root` | For a characteristic-p small root system modulo its zeroth root, coordinate zero vanishes and coordinate one does not. |

| `value_group_unit` | The tilted value group contains a p-th root of the unit value. |
| `value_group_two_roots` | The first two chosen root norms satisfy their successive p-th-power equations. |
| `field_product_control` | The product of two copies of a perfectoid field is not a field, and neither is the product of their normed tilts. |

`triple_unit_counit` states the triangle identity for the specified classification functor and
its primitive-untilting inverse.

The three proved Lean examples for this layer show that `p` is distinguished and that `ℂ_p`
and the completed cyclotomic field are perfectoid fields. The `ℚ_p` and zero-ring boundaries
are named targets in P1.1; the other Checks above are specified in prose.

### Dependencies

Layer P0 of this roadmap (P0.1, P0.2, P0.4, P0.5, P0.6, P0.8, P0.16, P0.20, P0.22); Tau Ceti's
Huber-ring layer (`TauCeti.Huber.IsTateRing`, `IsHuberRing`, `IsPseudoUniformizer`,
`powerBoundedSubring`, `IsBounded`, `Pair`, `Pair.Hom`, `PairOfDefinition`,
`IsRingOfIntegralElements`, `topologicallyNilpotentIdeal`, `restrictedMvPowerSeriesCompletion`,
`Padic.isTateRing`, `LaurentSeries.isTateRing`, `ValuationSpectrum.cont`); Mathlib's `Perfection`,
`PreTilt`, `Tilt`, `WittVector` and `WittVector.fontaineTheta`, `IsAdicComplete`, `AdicCompletion`,
`Localization.Away`, `PadicComplex`, `AddMonoidAlgebra`, the Serre-class and `CategoryTheory`
material listed in the subsections; Tau Ceti's Witt-vector completeness
(`WittVector.isAdicComplete_span_p_teichmuller`, `isHausdorff_span_p_teichmuller`,
`TauCeti.Huber.WittVector.isHuberRing_adicTopology_span_p_teichmuller`) and Fargues–Fontaine input
(`TauCeti.FarguesFontaine.spaY`). Other roadmaps, by subsection: P1.3 uses AdicSpacesPartII R0.4
(Tate norm data and the spectral seminorm); P1.12, and through it P1.13, uses AdicSpacesPartIII R3.9
(the Gel'fand spectrum); P1.33 uses AdicSpaces 6.1 (the perfect field `F` and `A_inf`). The
uniformity of P1.1 is Tau Ceti's `IsBounded` condition, the notion AdicSpaces 4.2 names. None of
these subsections cites this roadmap. P1.2 derives the `ϖ`-adic completeness of integral
rings from AdicEtaleGeometry A1.0.2(i), using uniformity to make them open bounded rings of
definition; P1.32 uses AdicEtaleGeometry A1.0.0, A1.0.3a and A4.2–A4.5's
split-ring, torsor-tower and completion theory. Layers P2, P3, P4, P5 and P7 of this roadmap
consume the tilting, approximation and colimit statements; the consumers in other roadmaps are
listed under Downstream consumers.

## Layer P2: rational localisation, the sheaf theorem and perfectoid spaces

The layer passes from perfectoid Tate pairs to perfectoid spaces. It constructs the tilting map
`x ↦ x♭` on adic spectra and proves that it is a homeomorphism `Spa(R, R⁺) ≅ Spa(R♭, R♭⁺)`
identifying rational subsets; shows that rational localisations of a perfectoid Tate pair are
perfectoid with tilt the rational localisation of the tilt, that completed residue fields are
perfectoid, and that perfectoid Tate pairs are stably uniform and sheafy, with `O_X` acyclic and
`O_X⁺` almost acyclic on rational subsets (the sheaf theorem); defines affinoid perfectoid spaces
and the full subcategory `Perfd` of perfectoid spaces inside the adic spaces of AdicSpaces, with
`Perf` in characteristic `p`; glues the tilt `X ↦ X♭` and proves the tilting equivalence
`Perfd/X ≃ Perf/X♭`; and builds completed tensor products of perfectoid pairs, fibre products of
perfectoid spaces (over perfectoid and over analytic bases), absolute products in `Perf`, and the
comparison with Scholze's perfectoid spaces over a perfectoid field. Throughout, `(R, R⁺)` is a
perfectoid Tate pair in the sense of ECD Definition 3.1, of characteristic `0`, `p` or mixed, with no
perfectoid base field; `ϖ = (ϖ♭)♯` is a pseudouniformiser with `ϖ^p | p` in `R°` and compatible
`p`-power roots, as in the standing conventions of this roadmap; "almost zero" refers to the
ideal of topologically nilpotent elements of `R⁺` (layer P1); and `𝔽_p((t^{1/p^∞}))` abbreviates
the completed perfection `𝔽_p((t^{1/p^∞}))^∧` of `𝔽_p((t))` (the tilt of `ℚ_p^cycl` and of
`ℚ_p(p^{1/p^∞})^`, layer P1), the uncompleted union `⋃_n 𝔽_p((t^{1/p^n}))` not being complete.

Global objects are Tau Ceti's pre-adic spaces (`TauCeti.PreAdicSpace`): the perfectoid
subcategory `Perfd` (P2.10) is cut out by sheafiness together with open covers by affinoid
perfectoid opens, the adic spectra `presentationLimitPreAdicSpace` of perfectoid Tate pairs (the
spectra of `Perfectoid.PairCat p` objects). Its charts are sheafy, so this is the usual category of
perfectoid adic spaces. AdicEtaleGeometry's analytic Yoneda carrier `YonedaAdicSpace` (rational
sheaves locally represented by complete Huber pairs, with its supplied underlying valuation-space
functor) receives perfectoid spaces, as adic spaces, through its fully faithful
`YonedaAdicSpace.ofAdicSpace`; the statements below that are phrased on that carrier, notably the
comparisons with AdicEtaleGeometry's predicates and sites, apply to a perfectoid space through this
embedding, and the comparison of the two presentations belongs to AdicEtaleGeometry. Rings and pairs
remain the pinned Tau Ceti rings and pairs. Characteristic `p` means `p = 0` on every affinoid
chart, including the zero chart; in Lean, `InCharP` asks for `p = 0` in every ring of sections. The bundled global category here uses small affinoid rings
and spaces covered by a set of such charts.

### P2.1 the tilting map on adic spectra

**The tilting map `x ↦ x♭` from `Spa(R, R⁺)` to `Spa(R♭, R♭⁺)`.** (*tilting-map-of-spectra*) Let
`(R, R⁺)` be a perfectoid Tate pair with tilt `(R♭, R♭⁺)` and sharp map `♯ : R♭ → R` (continuous
and multiplicative, not additive), `X = Spa(R, R⁺)`, `X♭ = Spa(R♭, R♭⁺)`; no base field. For
`x ∈ X` with valuation `v_x : R → Γ_x ∪ {0}`, the composite `v_x ∘ ♯ : R♭ → Γ_x ∪ {0}` is a
continuous valuation of `R♭` with `v_x(f♯) ≤ 1` for all `f ∈ R♭⁺`; its class is a point `x♭ ∈ X♭`,
characterised by `|f(x♭)| ≤ |g(x♭)| ⟺ |f♯(x)| ≤ |g♯(x)|` for all `f, g ∈ R♭` (written
`|f(x♭)| = |f♯(x)|`). The tilting map `τ_R : X → X♭`, `x ↦ x♭`, satisfies: (a)
`supp(x♭) = {f ∈ R♭ : f♯ ∈ supp(x)}`; (b) `τ_R` is continuous: if `f₁, …, f_n, g ∈ R♭` and some
`f_i` is a unit of `R♭` (for instance `f_n = (ϖ♭)^N`; the unit makes `f_i♯` a unit of `R` because
`♯` is multiplicative with `1♯ = 1`, and without such a normalisation the unit-ideal condition on
the `f♯` is not automatic before the homeomorphism is known), then `f₁♯, …, f_n♯` generate the unit
ideal of `R` and `τ_R⁻¹(U(f₁, …, f_n / g)) = U(f₁♯, …, f_n♯ / g♯)`; every rational subset of `X♭`
has a presentation with `f_i, g ∈ R♭⁺` and `f_n = (ϖ♭)^N` for some `N`; (c) naturality: for a
morphism of perfectoid Tate pairs `φ : (R, R⁺) → (S, S⁺)` (continuous, `φ(R⁺) ⊆ S⁺`) with tilt
`φ♭` (layer P1, P1.6), so that `φ ∘ ♯_R = ♯_S ∘ φ♭`, one has `τ_R ∘ Spa(φ) = Spa(φ♭) ∘ τ_S`; (d)
if `pR = 0` then `(R♭, R♭⁺) = (R, R⁺)` with `♯ = id` and `τ_R = id`. That `τ_R` is a homeomorphism
identifying rational subsets is P2.2; here only the map and its first properties are built
([ECD] §3, Thm 3.12, p. 17; [AWS] Lecture 2, §2.6, Lem. 2.6.12 with proof, pp. 64–65, for the
well-definedness of `v ↦ v ∘ ♯` without a base field; [Sch12] §6, proof of Thm 6.3, p. 31, for the
valuation check and for continuity through the preimage formula (b)). In `PerfectoidSpace`, define
`spaTilt` as the map `spa R⁺ → spa R♭⁺`, `x ↦ x♭`, sending `v_x` to `v_x ∘ ♯`; prove
`spaTilt_vle_iff` (`(spaTilt x).vle f g ↔ x.vle f♯ g♯` for `f g : R♭`), `supp_spaTilt` (the support
of `spaTilt x` is the preimage under `♯` of the support of `x`), `continuous_spaTilt`,
`spaTilt_preimage_rationalSubset` (if some `f_i ∈ T` is a unit of `R♭`, then
`spaTilt ⁻¹' rationalSubset R♭⁺ T s = rationalSubset R⁺ (T.image ♯) s♯`),
`exists_rationalSubset_eq_insert_pseudoUniformizer_pow` (every rational subset of `Spa(R♭, R♭⁺)`
equals `rationalSubset R♭⁺ (insert ((ϖ♭)^N) T) s` with `T ⊆ R♭⁺`, `s ∈ R♭⁺`, for some `N`),
`spaTilt_comp_spaComap` (`spaTilt ∘ φ.spaComap = φ♭.spaComap ∘ spaTilt`) and
`spaTilt_eq_id_of_charP` (simp: `spaTilt = id` under the identification of `R` with its tilt when
`pR = 0`). *Needs:* layer P1 (P1.4, P1.5, P1.6); Tau Ceti `TauCeti.ValuationSpectrum.spa`,
`TauCeti.ValuationSpectrum.rationalSubset`,
`TauCeti.ValuationSpectrum.rationalSubset_insert_of_forall_vle`,
`TauCeti.ValuationSpectrum.rationalSubset_image_mul_right`,
`TauCeti.ValuationSpectrum.isTopologicalBasis_spaRationalFamily`,
`TauCeti.Huber.Pair.Hom.spaComap`; Mathlib `Valuation`.

**Checks.**

- `spaTilt_test_perfectoidDisc_point` (computation) — for `R = ℚ_p^cycl⟨T^{1/p^∞}⟩` and `x` the
  point `T ↦ ϖ` (evaluation `T^{1/p^n} ↦ ϖ^{1/p^n}`), `spaTilt x` is the point `T ↦ ϖ♭` of
  `ℚ_p^cycl♭⟨T^{1/p^∞}⟩`: `|T(x♭)| = |T♯(x)| = |ϖ(x)|`.
- `spaTilt_test_charP_eq_id` (degenerate) — for `R = 𝔽_p((t^{1/p^∞}))^∧⟨T^{1/p^∞}⟩`
  (characteristic `p`), `spaTilt` is the identity of `Spa(R, R⁺)`.
- `spaTilt_test_preimage_disc` (compatibility) — on the perfectoid disc over `ℚ_p^cycl`,
  `spaTilt ⁻¹' U({T, ϖ♭}/ϖ♭) = U({T, ϖ}/ϖ)`, the instance of `spaTilt_preimage_rationalSubset`
  with `f₂ = ϖ♭`; so `τ⁻¹{|T| ≤ |ϖ♭|} = {|T| ≤ |ϖ|}`.
- `spaTilt_test_no_ringHom` (non-example) — there is no ring homomorphism `ℚ_p^cycl♭ → ℚ_p^cycl`
  (characteristic `p` to characteristic `0`), so `spaTilt` is not `Pair.Hom.spaComap` of any
  morphism of Huber pairs; a definition of `x♭` as the pullback of `x` along a ring map cannot
  typecheck in mixed characteristic.
- `spaTilt_test_vle_characterisation` (characterisation) — `x♭` is the unique point `y` of
  `Spa(R♭, R♭⁺)` with `y.vle f g ↔ x.vle f♯ g♯` for all `f g`; two points with the same relation
  coincide.

**Approximation of homogeneous functions on a perfectoid polydisc.**
(*approximation-on-perfectoid-polydisc*) Let `K` be a complete, nontrivially normed
ultrametric field with `IsPerfectoidField p K`, with tilt `K♭`, `ϖ♭ ∈ K♭` a pseudouniformiser
with `ϖ = (ϖ♭)♯` and `ϖ^p | p` in `K°`, and let `P = K⟨T₀^{1/p^∞}, …, T_n^{1/p^∞}⟩` be the
perfected Tate algebra, with `P° = K°⟨T^{1/p^∞}⟩` and monomials of total `T`-degree in `ℤ[1/p]_{≥0}`
(`K°` in degree `0`; a series is homogeneous when all its nonzero monomials have one degree;
the completed ring itself is not a direct-sum graded ring) and tilt `P♭ = K♭⟨T^{1/p^∞}⟩`, `T_i♯ = T_i` (layer P1; [KL16] §3.3, Ex. 3.3.12,
p. 61). Prove that for `f ∈ P°` homogeneous of degree `d`, every `c ∈ ℤ[1/p]` with `c ≥ 0` and every
`ε ∈ ℤ[1/p]` with `0 < ε < 1` there is `g ∈ P♭°` homogeneous of degree `d` with
`|f(x) − g♯(x)| ≤ |ϖ^{1−ε}(x)| · max(|f(x)|, |ϖ^c(x)|)` for all `x ∈ Spa(P, P°)`; in particular
`max(|f(x)|, |ϖ^c(x)|) = max(|g♯(x)|, |ϖ^c(x)|)` for all `x` ([Sch12] §6, Lem. 6.5 with proof,
pp. 33–34, the induction on `c` with loss `ε(c)`; [Bhatt] Ch. 9, Lem. 9.2.6 and the paragraph
before it, p. 101). For `f = T₀` one may take `g = T₀` for every `c`; for
`f = T₀ + ϖT₁`, `g = T₀ + ϖ♭T₁` satisfies `f ≡ g♯ mod ϖP°` because `♯` is additive modulo `ϖ`,
which gives the estimate for `c = 0`, while `c > 0` needs the inductive corrections. *Needs:*
*tilting-map-of-spectra*; layer P1 (P1.5, the sharp map; P1.30, the perfected Tate algebra and its tilt); Tau Ceti
`TauCeti.ValuationSpectrum.rationalSubset`.

`Perfectoid.homogeneousPart p ϖ n d` is the closed span of monomials of total degree `d`,
not a grading by a direct sum. `homogeneous_sharp_approximation` records the quantitative
estimate over `K` with `c = a/p^m` and `ε = b/p^k`, `0 < b < p^k`,
and with `P.plus = P°`. The definition `homogeneousPart` and its component Checks apply
more generally to a perfectoid Tate base `R₀`.

**Checks for homogeneous components.** `homogeneous_degree_zero_constant`: every scalar has
degree zero. `homogeneous_two_variables`: `T₀ + ϖT₁` has degree one.
`homogeneous_wrong_degree`: `T₀` is not degree zero over a nonzero base.
`homogeneous_zero`: zero belongs to every component. These use the completed algebra itself.
The degree-preserving estimate is [Sch12] Lemma 6.5 and Remark 6.6, pp. 33–34,
for the perfectoid field `K`.

### P2.2 the tilting homeomorphism

The tilting map of P2.1 is a homeomorphism matching rational subsets; its two inputs are the
characteristic-`p` rational localisations and the approximation lemma, proved here.

**The tilting homeomorphism `Spa(R, R⁺) ≅ Spa(R♭, R♭⁺)` identifies rational subsets.**
(*tilting-homeomorphism-and-rational-subsets*) Let `(R, R⁺)` be a perfectoid Tate pair (ECD
Definition 3.1; characteristic `0`, `p` or mixed, as in ECD Example 3.4(iv); no perfectoid base
field) with tilt `(R♭, R♭⁺)`. Then the tilting map `τ_R : X = Spa(R, R⁺) → X♭ = Spa(R♭, R♭⁺)`,
`|f(x♭)| = |f♯(x)|` (see *tilting-map-of-spectra*), is a homeomorphism, natural in morphisms of
perfectoid Tate pairs. A subset `U ⊆ X` is rational if and only if `τ_R(U) ⊆ X♭` is rational;
explicitly, for `f₁, …, f_n, g ∈ R♭⁺` with `f_n = (ϖ♭)^N`,
`τ_R(U(f₁♯, …, f_n♯ / g♯)) = U(f₁, …, f_n / g)`, and every rational subset of `X` can be written
as `U(f₁♯, …, f_n♯ / g♯)` for such `f_i, g`. Consequently `τ_R` preserves the specialisation
order (Mathlib `Topology.IsInducing.specializes_iff` applied to the homeomorphism),
quasi-compact opens and rank-one (maximal) points, and restricts, for every rational
`U ⊆ X`, to the homeomorphism `Homeomorph.image τ_R U`, identifying rational subsets of `U` and of `τ_R(U)`
([ECD] §3, Thm 3.12, p. 17, and proof of Thms 3.12–3.13, p. 17, the route followed; [Sch12] §6,
Cor. 6.7(ii)–(iii) with proof, p. 35, injectivity from the approximation lemma and surjectivity by
untilting the residue field over `R`; [KL15] Thm 3.6.14(b) and [KL16] §3.3, Thm 3.3.16, p. 63, the
explicit form of the rational subsets and an independent proof for Fontaine perfectoid Tate rings,
including those in which `p` is neither `0` nor invertible; [AWS] Lecture 2, §2.5, Thm 2.5.1,
pp. 59–60, the analytic-ring generalisation). Sch12 Theorem 6.3(i) (perfectoid affinoid
`K`-algebras over a perfectoid field) is a special case. For `R = ℚ_p^cycl⟨T^{1/p^∞}⟩`,
`τ({|T| ≤ |ϖ|}) = {|T| ≤ |ϖ♭|}`, the Gauss point goes to the Gauss point and the classical point
`T ↦ ϖ` to `T ↦ ϖ♭`. The theorem applies to ECD Example 3.4(iv),
`R = ℤ_p^cycl[[T^{1/p^∞}]]⟨(p/T)^{1/p^∞}⟩[1/T]` with `ϖ = T^{1/p}`, which lives over no field: its
spectrum has points with `|p(x)| = 0` and points with `|p(x)| ≠ 0`, both matched with points of the
characteristic-`p` space `X♭`. Without the approximation lemma one only obtains that rational
subsets of `X` are preimages of quasi-compact opens of `X♭` ([Bhatt] Rem. 9.2.8); that images of
rational subsets are rational needs *approximation-lemma*. *Needs:* *tilting-map-of-spectra*;
*rational-localization-in-characteristic-p*; *approximation-lemma*; layer P1 (P1.6, P1.18 (*rings-of-integral-elements-under-tilting* (a)),
P1.27); Tau Ceti `TauCeti.ValuationSpectrum.rationalSubset_inter`,
`TauCeti.ValuationSpectrum.rationalSubset_insert_of_forall_vle`,
`TauCeti.ValuationSpectrum.rationalSubset_image_mul_right`,
`TauCeti.ValuationSpectrum.isTopologicalBasis_spaRationalFamily`,
`TauCeti.ValuationSpectrum.spectralSpace_spa_of_pairOfDefinition`.

**Rational localisations in characteristic p are perfect, with an explicit almost integral
model.** (*rational-localization-in-characteristic-p*) Let `(R, R⁺)` be a perfectoid Tate pair with
`pR = 0` (so `R` is a perfect complete Tate ring and `R⁺` is perfect, [ECD] §3, Prop. 3.5, p. 15),
`ϖ ∈ R⁺` a pseudouniformiser, `X = Spa(R, R⁺)`, `f₁, …, f_n, g ∈ R⁺` with `f_n = ϖ^N` (every
rational subset has such a presentation, by (b) of *tilting-map-of-spectra* applied to `R = R♭`),
and `U = U(f₁, …, f_n / g)`. Prove: (i) the rational localisation `(O_X(U), O_X⁺(U))` (AdicSpaces
layer 3.1) is a perfectoid Tate pair of characteristic `p`: Frobenius is bijective on `O_X(U)` and
on `O_X⁺(U)`; (ii) for `A_U` the `ϖ`-adic completion of the subring
`R⁺[(f_i/g)^{1/p^m} : i ≤ n, m ≥ 0]` of `R[1/g]`, `A_U` is perfect and `ϖ`-torsion free,
`A_U ⊆ O_X⁺(U) ⊆ O_X(U)°` inside `O_X(U) = A_U[1/ϖ]`, and both `O_X(U)°/A_U` and `O_X⁺(U)/A_U`
are almost zero (killed by `ϖ^{1/p^k}` for all `k`, the almost setup being the ideal of
topologically nilpotent elements of `R⁺`, generated in a perfect `R⁺` by the `ϖ^{1/p^k}`); (iii)
if `(R, R⁺)` is an algebra over a characteristic-`p` perfectoid field `K`, with continuous
scalar map carrying `K°` into `R⁺` and with the parameter and its roots coming from `K`,
the surjection `ψ : R⁺[T₁^{1/p^∞}, …, T_n^{1/p^∞}] → R⁺[(f_i/g)^{1/p^∞}]`,
`T_i^{1/p^m} ↦ (f_i/g)^{1/p^m}`, has kernel containing the ideal `I` generated by the
`g^{1/p^m} T_i^{1/p^m} − f_i^{1/p^m}` (`i ≤ n`, `m ≥ 0`), and `ker ψ / I` is almost zero; (iv)
every perfectoid Tate pair of characteristic `p` is stably uniform ([KL15] §3.1,
Prop. 3.1.7 with proof, pp. 67–68, for (i) and (iv) without a base field; [AWS] Lecture 2, §2.5,
proof of Thm 2.5.3, first paragraph, p. 60, the Frobenius universal-property argument of (i);
[Sch12] §6, proof of Lem. 6.4(i) for `K` of characteristic `p`, p. 31, the presentation (iii);
[Bhatt] Ch. 9, Lem. 9.2.3 with proof, pp. 97–98, almost vanishing of the kernel of `ψ` by
perfectness). For `K = 𝔽_p((t^{1/p^∞}))^∧` and `R = K⟨T^{1/p^∞}⟩`, `U = {|T| ≤ |t|}`:
`O_X(U) = K⟨(T/t)^{1/p^∞}⟩` and `A_U = K°⟨(T/t)^{1/p^∞}⟩ = O_X(U)°` exactly; for the annulus
`{|t| ≤ |T| ≤ 1}`, `O_X(U) = K⟨T^{1/p^∞}, (t/T)^{1/p^∞}⟩` is perfect. Perfectness in (i) fails
for non-perfect rings: for the Tate algebra `𝔽_p((t))⟨T⟩` (not perfectoid) the localisation at
`{|T| ≤ |t|}` is `𝔽_p((t))⟨T/t⟩`, which is not perfect. *Needs:* layer P1 (P1.1); layer P0; Tau
Ceti `TauCeti.ValuationSpectrum.existsUnique_continuous_ringHom_of_forall_comap_mem_rationalSubset`,
`TauCeti.Huber.PairOfDefinition.completionLocObj`,
`TauCeti.Huber.PairOfDefinition.completedPlusSubring`,
`TauCeti.ValuationSpectrum.mem_iff_forall_vle_one`; Mathlib `PerfectRing`, `frobeniusEquiv`.

**Approximation lemma: every function on a perfectoid affinoid is approximated by a sharp.**
(*approximation-lemma*) Let `(R, R⁺)` be a perfectoid Tate pair with tilt `(R♭, R♭⁺)`,
`ϖ = (ϖ♭)♯`, `X = Spa(R, R⁺)`; no base field; estimates are pointwise in `Γ_x ∪ {0}` with
`|ϖ^r(x)| := |((ϖ♭)^r)♯(x)|` for `r ∈ ℤ[1/p]`. Prove: (i) for every `f ∈ R`, every `c ∈ ℤ[1/p]` with
`c ≥ 0` and every `ε ∈ ℤ[1/p]` with `0 < ε < 1` there is `g ∈ R♭` with
`|f(x) − g♯(x)| ≤ |ϖ^{1−ε}(x)| · max(|f(x)|, |ϖ^c(x)|)` for all `x ∈ X`, so that
`max(|f(x)|, |ϖ^c(x)|) = max(|g♯(x)|, |ϖ^c(x)|)`; (ii) for `f, h ∈ R` and an integer `c ≥ 0` there
are `a, b ∈ R♭` with `{x : |f(x)| ≤ |h(x)|, |ϖ^c(x)| ≤ |h(x)|} = {x : |a♯(x)| ≤ |b♯(x)|, |ϖ^c(x)| ≤ |b♯(x)|}` ([Sch12] §6, Cor. 6.7(i) with proof, pp. 34–35, the reduction to rank-one
points and to the polydisc; [KL16] §3.2, Lem. 3.2.7 (preamble), p. 57, approximation by Euclidean
division with a Fontaine primitive element, and §3.3, proof of Thm 3.3.16, p. 63, how it moves
rational subsets across tilting without a base field; [AWS] Lecture 2, §2.6, Lem. 2.6.17 with
proof, p. 66; [Bhatt] Ch. 9, Prop. 9.2.7(1)–(2) with proof, pp. 101–103, the field case with
consequence (ii)). Sch12 Corollary 6.7(i) and Bhatt Proposition 9.2.7(1) are the case of a
perfectoid affinoid algebra over a perfectoid field. For `R = ℚ_p^cycl⟨T^{1/p^∞}⟩` and `f = T + ϖ`,
`g = T + ϖ♭` satisfies `f ≡ g♯ mod ϖR°`, the estimate for `c = 0`; for `c ≥ 1` the lemma gives `g`
with `{|f| ≤ |ϖ^c|} = {|g♯| ≤ |ϖ^c|}` ([Sch12] Rem. 6.6), a rational subset of `X` that is a
preimage from `X♭`; hence every rational subset of `X` is the preimage of a rational subset of
`X♭`, the step consumed by *tilting-homeomorphism-and-rational-subsets*. *Needs:*
*tilting-map-of-spectra*; *approximation-on-perfectoid-polydisc*; layer P1 (P1.6); Tau Ceti
`TauCeti.ValuationSpectrum.spa`, `TauCeti.Huber.powerBoundedSubring`.

The quantitative target is the sharp approximation on the polydisc; its two conclusions are
the error estimate and equality of the truncated valuations. The powers are
`ϖ.root m ^ a` and `ϖ.root n ^ (p^n-b)`, so no extra roots in a valuation group are implicit.
**Checks.** `sharp_approximation_zero` uses `f = g = 0`; `sharp_approximation_exact` computes the
error of an exact sharp as zero. The existing rational-set Check also tests the empty denominator.

<a id="p2-3"></a>

### P2.3 rational localisation of perfectoid affinoids

Rational localisation: the rational localisations of a perfectoid Tate pair are perfectoid, their
tilts are the rational localisations of the tilt, they form a functor on rational subsets with
tilting as a natural isomorphism, completed residue fields are perfectoid, and the structure
presheaf is a sheaf with `O⁺` almost Čech-acyclic.

**Rational localisations of a perfectoid Tate pair are perfectoid, with tilt the rational
localisation of the tilt.** (*rational-localization-of-perfectoid-affinoids*) Let `(R, R⁺)` be a
perfectoid Tate pair (ECD Definition 3.1; characteristic `0`, `p` or mixed; no perfectoid base
field, which is the form P3.17a uses for sousperfectoid rings),
`X = Spa(R, R⁺)`, `X♭ = Spa(R♭, R♭⁺)` and `τ_R : X ≅ X♭` the tilting homeomorphism. For every
rational subset `U ⊆ X`, the rational localisation `(O_X(U), O_X⁺(U))` of AdicSpaces layer 3.1
(`O_X(U) = R⟨T/s⟩` for any presentation `U = U(T/s)`, `O_X⁺(U)` the closure in the completion of the image of the integral closure
of `R⁺[T/s]` in `R[1/s]`; Tau Ceti `PairOfDefinition.completionLocObj`, `completedPlusSubring`; the presheaf
on rational subsets is presentation independent, the comparison isomorphisms between presentations
being Tau Ceti's) is a perfectoid Tate pair, and there is a unique isomorphism of perfectoid Tate
pairs `(O_X(U), O_X⁺(U))♭ ≅ (O_{X♭}(τ_R U), O_{X♭}⁺(τ_R U))` compatible with the structure maps
from `(R♭, R♭⁺)` and with `♯`; equivalently `(O_X(U), O_X⁺(U))` is the untilt over `(R, R⁺)` of
the rational localisation of the tilt at `τ_R(U)`. For a presentation
`U = U(f₁♯, …, f_n♯ / g♯)` with `f_i, g ∈ R♭⁺` and `f_n = (ϖ♭)^N` (which exists for every `U`),
`O_X⁺(U)ᵃ = (R⁺⟨(f♯/g♯)^{1/p^∞}⟩)ᵃ`. The construction `U ↦ (O_X(U), O_X⁺(U))` is the rational
localisation in the category of perfectoid Tate pairs: every morphism `(R, R⁺) → (S, S⁺)` to a
perfectoid Tate pair whose `Spa` lands in `U` factors uniquely through it (AdicSpaces' universal
property over all complete Hausdorff Huber pairs, restricted to perfectoid targets; the tilt of the
localisation is identified only after perfectoidness is known, through the explicit model, since
the universal property quantifies over all Huber pairs, [AWS] Lecture 2, §2.5, Thm 2.5.3(b) and
proof, p. 60); and tilting gives an equivalence between the rational localisations of `(R, R⁺)`
and those of `(R♭, R♭⁺)`. In characteristic `p` the tilt identification is the identity ([ECD] §3,
Thm 3.18, p. 18, perfectoidness and the tilt identification; [HK] §3, Prop. 3.8 with proof, p. 8;
[KL16] §3.3, Thm 3.3.18, proof of case (i), p. 64, an independent Witt-vector proof;
[Berkeley] Lecture 6, Thm 6.2.6, p. 45; [Sch12] §6, Cor. 6.8 with proof, p. 35, the field-case
universal property). Sch12 Corollary 6.8, Berkeley Theorem 6.2.6, Hansen–Kedlaya Proposition 3.8
and Kedlaya–Liu II Theorem 3.3.18(i) are special cases or restatements. In `Perfectoid`, define
`rationalLocalization` as the perfectoid Tate pair `(O_X(U), O_X⁺(U))` of a rational subset `U`
of `Spa(R, R⁺)`, bundled with its structure morphism of Huber pairs from `(R, R⁺)`; prove
`rationalLocalization.isPerfectoidTateRing_affinoid` (instance: `O_X(U)` is a perfectoid Tate ring),
`rationalLocalization.tiltEquiv_affinoid` (the isomorphism `(O_X(U), O_X⁺(U))♭ ≅ (O_{X♭}(τ U), O_{X♭}⁺(τ U))`),
`rationalLocalization.tiltEquiv_affinoid_spec` (`tiltEquiv` composed with the tilt of
`R → O_X(U)` is `R♭ → O_{X♭}(τ U)`). Use Tau Ceti's supplied continuous extension,
structure-map identity and uniqueness for the rational universal property.
The additional theorem `rationalLocalization.lift_mem_plus` has Tate source and complete
Hausdorff Tate target pairs `P`, `Q`, a presentation `d`, and a pair morphism `f : P → Q`.
Assume `f(d.den)` is a unit and every `f(t) / f(d.den)`, for `t ∈ d.num`, belongs to `Q.plus`.
For any continuous ring homomorphism `g : RationalLoc d →+* S` whose composite with
`PairOfDefinition.toCompletionLoc` is `f.toRingHom`, it states
`g((RationalLoc.pair d P).plus) ⊆ Q.plus`. Together with the supplier, this gives the unique
factorization as a morphism of pairs; no perfectoid hypothesis is needed for this integral
assertion ([Wedhorn] §8.1, p. 74, the universal property immediately before Lemma 8.1;
[Sch12] Proposition 2.14, p. 9, over a nonarchimedean field).
Prove `rationalLocalization.image_spaTilt`
(`τ(U(f♯/g♯)) = U(f/g)` for `f_i, g ∈ R♭⁺`, `f_n = (ϖ♭)^N`, and every rational `U` has such a
presentation), `rootIntegralModel_almost`
(`O_X⁺(U)ᵃ ≅ (R⁺⟨(f♯/g♯)^{1/p^∞}⟩)ᵃ` as `R⁺ᵃ`-algebras), `rationalOrderIso`
(tilting is an equivalence from the rational localisations of `(R, R⁺)` to those of `(R♭, R♭⁺)`,
`U ↦ τ U`), `rationalLocalization.isUniform` (instance: `O_X(U)°` is bounded) and
`rationalLocalization.tiltEquiv_of_charP_affinoid` (simp: `tiltEquiv` is the identity when `pR = 0`).
*Needs:* *tilting-homeomorphism-and-rational-subsets*; *tilting-map-of-spectra*;
*almost-integral-model-of-untilted-rational-localization*; layer P1 (P1.1); Tau Ceti
`TauCeti.Huber.PairOfDefinition.completionLocObj`,
`TauCeti.Huber.PairOfDefinition.completedPlusSubring`,
`TauCeti.ValuationSpectrum.existsUnique_continuous_ringHom_of_forall_comap_mem_rationalSubset`,
`TauCeti.ValuationSpectrum.presentationRingEquivOfEq_coe_comp_toCompletionLoc`,
`TauCeti.Huber.Pair`.

**Checks.**

- `rationalLocalization_test_perfectoidDisc` (computation) — for `R = ℚ_p^cycl⟨T^{1/p^∞}⟩` and
  `U = {|T| ≤ |ϖ|}`, `rationalLocalization R⁺ U ≅ ℚ_p^cycl⟨(T/ϖ)^{1/p^∞}⟩` (`≅ R` via `T ↦ ϖT`) and
  `tiltEquiv` identifies its tilt with `ℚ_p^cycl♭⟨(T/ϖ♭)^{1/p^∞}⟩ = O_{X♭}({|T| ≤ |ϖ♭|})`.
- `rationalLocalization_test_noBaseField` (compatibility) — for `R` of ECD Example 3.4(iv)
  (`R⁺ = R°`) and `U = U(T, p / p) = {|T| ≤ |p|}`: on `X` one has `|p| ≤ |T|` (`p/T ∈ R°`), so
  `|p| = |T| ≠ 0` on `U`; `isPerfectoidTateRing` holds for `O_X(U)` and `p` is a unit there, a
  rational localisation of a ring in which `p` is not invertible; the statement must not require an
  algebra structure over a perfectoid field. (For the product
  `ℂ_p × ℂ_p♭` of layer P1, in which `p = (p, 0)` is a nonzero nonunit, with `U = U({ϖ, p}/p)` for
  the pseudouniformiser `ϖ = (p, t)`, `t` a pseudouniformiser of `ℂ_p♭` (so that `U` is the `ℂ_p`
  factor, on which `p` is a unit; a `ϖ` with `ϖ^p | p` would make `U` empty).
- `rationalLocalization_test_charP` (degenerate) — if `pR = 0` then `tiltEquiv` is the identity of
  `(O_X(U), O_X⁺(U))` and the almost integral model is `R⁺⟨(f/g)^{1/p^∞}⟩` itself.
- `naiveModel_smaller_root_nonmember` (non-example) — the naive model `R°⟨f♯/g♯⟩` without
  `p`-power roots is not almost equal to `O_X(U)°`: for `R = ℚ_p^cycl⟨T^{1/p^∞}⟩` and
  `U = {|T| ≤ |ϖ|}`, `O_X(U)° = K°⟨(T/ϖ)^{1/p^∞}⟩` contains `(T/ϖ)^{1/p}`, while
  `R°⟨T/ϖ⟩ = {Σ_{q ∈ ℤ[1/p]_{≥0}} c_qT^q : |c_q| ≤ |ϖ|^{-⌊q⌋}, |c_q| |ϖ|^{⌊q⌋} → 0}` (the weighted coefficients tend to zero outside finite sets) contains `c·T^{1/p}` only
  for `|c| ≤ 1`; so `ϖ^{1/p²}(T/ϖ)^{1/p} = ϖ^{1/p² − 1/p}T^{1/p} ∉ R°⟨T/ϖ⟩` (whereas
  `ϖ^{1/p}(T/ϖ)^{1/p} = T^{1/p} ∈ R° ⊆ R°⟨T/ϖ⟩`, so the exponent `1/p` is not a witness), and for
  each `k ≥ 1` the element `(T/ϖ)^{1 − 1/p^n}` with `1 − 1/p^n > 1/p^k` is not carried into `R°⟨T/ϖ⟩`
  by `ϖ^{1/p^k}`.
- `rationalLoc_test_whole` (characterisation) — for `U = X = U({1}/1)`,
  `rationalLocalization` is `(R, R⁺)` itself (`O_X(X) ≅ R`) and `tiltEquiv` is the identity of
  `(R♭, R♭⁺)`.

**Untilted rational localisations: explicit almost integral model, perfectoidness and tilt.**
(*almost-integral-model-of-untilted-rational-localization*) Let `(R, R⁺)` be a perfectoid Tate pair
with tilt `(R♭, R♭⁺)` and `ϖ = (ϖ♭)♯`, `ϖ^p | p` in `R°`; parts (i), (ii), and (iv)
require no base field. Let `f₁, …, f_n, g ∈ R♭⁺`
with `f_n = (ϖ♭)^N`, `U = U(f₁, …, f_n / g) ⊆ X♭` and `U♯ := τ_R⁻¹(U) = U(f₁♯, …, f_n♯ / g♯) ⊆ X`
(the normalisation makes `f₁♯, …, f_n♯` generate the unit ideal, (b) of *tilting-map-of-spectra*).
Prove: (i) the `ϖ`-adic completion `A` of the subring
`R⁺[(f_i^{1/p^m})♯/(g^{1/p^m})♯ : i ≤ n, m ≥ 0]` of `R[1/g♯]` is `ϖ`-torsion free and Frobenius
induces an almost isomorphism `A/ϖ^{1/p} → A/ϖ`; (ii)
`O_X(U♯) = A[1/ϖ]` is a perfectoid Tate ring and `A ⊆ O_X⁺(U♯) ⊆ O_X(U♯)°` with almost zero
quotients; (iii) if `(R, R⁺)` is an algebra over a perfectoid field `K`, with continuous
scalar map carrying `K°` into `R⁺` and with `ϖ` and all its roots coming from `K`,
`A^a` is a perfectoid `K°a`-algebra and the map
`R⁺[T₁^{1/p^∞}, …, T_n^{1/p^∞}] → A`,
`T_i^{1/p^m} ↦ (f_i^{1/p^m})♯/(g^{1/p^m})♯`, is surjective modulo `ϖ` with kernel modulo `ϖ`
almost equal to the image of the ideal generated by `(g^{1/p^m})♯ T_i^{1/p^m} − (f_i^{1/p^m})♯`;
(iv) `(O_X(U♯), O_X⁺(U♯))` is the untilt over `(R, R⁺)` of `(O_{X♭}(U), O_{X♭}⁺(U))`, which
exists by the tilting equivalence over the perfectoid base `R` (ECD Theorem 3.13; layer P1,
P1.27): there is a unique isomorphism of perfectoid Tate pairs
`(O_X(U♯), O_X⁺(U♯))♭ ≅ (O_{X♭}(U), O_{X♭}⁺(U))` compatible with the structure maps from
`(R♭, R♭⁺)` and with `♯` ([Sch12] §6, Lem. 6.4 (statement), p. 31, and proof of Lem. 6.4(i),
(iii) for general `K`, pp. 32–33, the comparison through the untilt `S`; [Bhatt] Ch. 9, Lem. 9.2.5
with proof, pp. 99–101, written with `R⁺` in place of `R°` and with the universal-property proof
of (iv); [AWS] Lecture 2, §2.8, Lem. 2.8.8, p. 72, the tilt statement (iv) without a base field;
[ECD] §3, proof of Thms 3.12–3.13, p. 17). For a perfectoid Tate ring `R₀` and
`R = R₀⟨T₀^{1/p^∞}, …, T_n^{1/p^∞}⟩`, `g ∈ R♭⁺` homogeneous and `U_c = {|g| ≤ |ϖ♭|^c}`
(`c ∈ ℤ[1/p]_{≥0}`), the lemma gives `O_X(U_c♯)°ᵃ = R°⟨(g♯/ϖ^c)^{1/p^∞}⟩ᵃ`. Over a perfectoid
field `K` this is Sch12 Lemma 6.4(i)–(iii) and Bhatt Lemma 9.2.5; for `R = ℚ_p^cycl⟨T^{1/p^∞}⟩`
and `U = {|T| ≤ |ϖ♭|}`, `O_X(U♯) = ℚ_p^cycl⟨(T/ϖ)^{1/p^∞}⟩` with tilt
`ℚ_p^cycl♭⟨(T/ϖ♭)^{1/p^∞}⟩`. The model is almost, not exactly, integral: for `R⁺ ≠ R°` (e.g.
`R⁺ = K° + K°°⟨T^{1/p^∞}⟩`) and `U = X`, `A = R⁺` while `O_X(X)° = R°`. *Needs:*
*tilting-map-of-spectra*; layer P1 (P1.6, P1.27); Tau Ceti
`TauCeti.ValuationSpectrum.existsUnique_continuous_ringHom_of_forall_comap_mem_rationalSubset`,
`TauCeti.Huber.PairOfDefinition.completionLocObj`,
`TauCeti.Huber.PairOfDefinition.completedPlusSubring`.

On the field disc put `S = T/ϖ`. `Perfectoid.discScale` sends `T^(1/p^n)` to
`ϖ^(1/p^n) S^(1/p^n)` and fixes the base field. `naiveDiscModel` is the closed subring generated
by the image of the original unit ball and `S`; `rootedDiscModel` also adjoins every
`S^(1/p^n)`. Closure is in the normalized disc's Tate topology. `rootedDiscModel_eq` identifies
the root-adjoined model with the unit ball, and `naiveDiscModel_not_almost` quantifies over
all `k ≥ 1`, with the monomials `ϖ^(1/p^k) S^(1-p^(-n))` witnessing failure of containment.

**Checks for the actual integral models.**
- `discScale_constants` fixes each scalar; `discScale_two_roots` computes the images of `T`
  and `T^(1/p)`; `discScale_not_surjective_unit_ball` detects the smaller radius.
- `naiveModel_fractional_member` puts `ϖ^(1/p)S^(1/p)` in the naive model;
  `naiveModel_smaller_root_nonmember` excludes `ϖ^(1/p²)S^(1/p)`;
  `naiveModel_integral_generator` includes `S` but excludes `S^(1/p)`.
- `rootedModel_fractional_generator` includes `S^(1/p)`;
  `rootedModel_unbounded_constant` excludes a scalar of norm greater than one;
  `rootedModel_two_term_sum` includes `S + S^(1/p)`.
- Integral base flatness fails on a scaled characteristic-p disc: let
  `K = completed perfection of 𝔽_p((ϖ))`, `C = K°⟨T^{1/p^∞}⟩`, and
  `B = K°⟨S^{1/p^∞}⟩`, with `T^{1/p^n} ↦ ϖ^{1/p^n}S^{1/p^n}`. This is the root model
  for `|T| ≤ |ϖ|`. The sequence `(ϖ,T)` is regular in `C`, since
  `C/ϖC = (K°/ϖ)[T^{1/p^∞}]` and multiplication by `T` is injective.
  Its Koszul complex after tensoring with `B` has maps
  `d₁(a,b) = ϖ(a+Sb)` and `d₂(c) = ϖc(−S,1)`, hence
  `Tor₁^C(C/(ϖ,T),B) ≅ B/ϖB`. Multiplication by `ϖ^{1/p}` does not kill the class of `1`,
  so `B^a` is not flat over `C^a`, for every prime `p`, including `2`.
These statements test the constructed subrings and distinguish parameter torsion-freeness
from almost flatness over the original plus ring.

The general model `rawRootModel` is the subring of the actual rational localization
generated by the base plus ring and all `fᵢ,m/gₘ`. Here `fᵢ,m+1^p = fᵢ,m`,
`gₘ₊₁^p = gₘ`, the zero-level numerators generate the unit ideal, and their rational subset
is the specified `U`. `rootIntegralModel` is its closure. `rootIntegralModel_completion`
identifies this closure with the separated adic completion, fixing the algebraic model;
`rootIntegralModel_invert` recovers the rational ring after inversion and states parameter
torsion-freeness. `rootIntegralModel_almost` gives root-scaled containment, and
`rootIntegralModel_frobenius` states the almost kernel and cokernel conditions modulo the
two parameters. `rootModelEval` evaluates a polynomial in variables `(i,m)` at these ratios;
`rootModelRelations` includes both `Xᵢ,m+1^p-Xᵢ,m` and `gₘXᵢ,m-fᵢ,m`.
`rootIntegralModel_presentation` specifies its almost kernel modulo the parameter for an
algebra over a complete nontrivially normed ultrametric field `K` with
`IsPerfectoidField p K`. Its scalar map is continuous and carries `K°` into `R⁺`; a
`PseudoUniformizerRoots p K` named `ϖK` satisfies
`algebraMap K R (ϖK.root k) = ϖ.root k` for every `k`.
`rawRootModel_kernel_charP` states the algebraic kernel assertion before completion under
the same scalar hypotheses and `CharP K p`, with `pR = 0`.
`rootIntegralModel_charP` includes perfectness of the chosen plus ring, including the zero
ring. The presentation is [Sch12] Lemma 6.4(i), proof on pp. 31–33: the algebraic
characteristic-`p` kernel is treated on p. 31, and the kernel modulo the parameter in
general characteristic on pp. 32–33. Its `R°` formulation gives the `R⁺` formulation
by the almost equality of `R⁺` and `R°` over `K°`. [ECD] Theorem 3.18, p. 18,
supplies the rational-localization and tilt assertions without a base field.

**Checks for the general root model.** The empty presentation gives the zero localization;
numerator and denominator one recover the base plus ring; the inverse parameter is excluded in a
nonzero localization. Evaluation fixes base coefficients, computes the sum of two consecutive root
ratios, and gives zero on the empty localization. For the relation ideal: the unit presentation
contains `X₀,₀-1`, the denominator `0` (the presentation `1/0`) gives the unit ideal, and `1` is
excluded in the nonzero unit presentation.

Lean: the theorem `isPerfectoidTateRing_rationalLocalization` (the completed rational localisation
`R⟨T/s⟩` of Tau Ceti, for an admissible presentation, is a perfectoid Tate ring: the first clause).
The bundled instance form `rationalLocalization.isPerfectoidTateRing_affinoid`, the tilt
identification and the remaining clauses are specified in this README only.

### P2.4 the presheaf of perfectoid pairs on rational subsets

**The presheaf of perfectoid pairs on rational subsets: restriction maps, plus rings, and tilting
as a natural isomorphism.** (*perfectoid-pair-rational-restriction*) Let `(R, R⁺)` be a perfectoid
Tate pair, `ϖ = (ϖ♭)♯`, `X = Spa(R, R⁺)`, `τ = τ_R`. On the poset `Rat(X)` of rational subsets of
`X`, the assignment `U ↦ (O_X(U), O_X⁺(U))` with AdicSpaces' restriction maps `ρ_{U,V}` (`V ⊆ U`;
identity and composition laws from AdicSpaces layer 3.1, Tau Ceti
`PairOfDefinition.restrictionRingHomOfSubset` for a numerator enlargement and presentation changes
for general inclusions) is a functor `Rat(X)^op → {perfectoid Tate pairs}`
(see *rational-localization-of-perfectoid-affinoids*), with: (a) transitivity: for `V ⊆ U` rational,
`V` is rational in `U ≅ Spa(O_X(U), O_X⁺(U))` and `ρ_{U,V}` is the rational localisation of
`(O_X(U), O_X⁺(U))` at `V` — the functor on `Rat(X)^op` with its transitivity and restriction
maps is library for any Huber pair (Tau Ceti `pairHomOfRationalSubsetSubset`,
`spaCompletedLocalizationHomeomorph_spaComap_pairHomOfRationalSubsetSubset`,
`presentationLimitRationalIso`, `isIso_restrictionHom_of_rationalSubset_eq`,
`completedPlusSubring_eq_completionPlus`, `mem_completedPlusSubring_iff_forall_mem_rationalSubset_vle_one`),
and what is proved here is the perfectoidness of its values and the naturality of tilting; (b) plus rings: `O_X⁺(U) = {f ∈ O_X(U) : |f(x)| ≤ 1 for all x ∈ U}`,
`O_X(U)°° ⊆ O_X⁺(U) ⊆ O_X(U)°`, and `O_X(U)°/O_X⁺(U)` is almost zero (for the ideal of
topologically nilpotent elements of `R⁺`; its extension to `O_X⁺(U)` is the ideal of
topologically nilpotent elements there), so
`O_X⁺(U)ᵃ = O_X(U)°ᵃ`; (c) tilting is a natural isomorphism: the isomorphisms `rationalTiltIso` satisfy
`tiltEquiv_V ∘ (ρ_{U,V})♭ = ρ_{τU,τV} ∘ tiltEquiv_U`, i.e. `U ↦ (O_X(U), O_X⁺(U))♭` is isomorphic
to `U ↦ (O_{X♭}(τU), O_{X♭}⁺(τU))`; (d) consequently the ring isomorphisms
`O_{X♭}⁺(τU)/ϖ♭ ≅ O_X⁺(U)/ϖ` induced by `♯` are natural in `U` and `R♭⁺/ϖ♭ ≅ R⁺/ϖ`-semilinear;
(e) the homeomorphisms `Spa(O_X(U), O_X⁺(U)) ≅ U` and `Spa(O_{X♭}(τU), O_{X♭}⁺(τU)) ≅ τU`
(AdicSpaces layer 3.1) intertwine `τ_{O_X(U)}` and `τ|_U` ([ECD] §3, Ex. 3.22(ii), p. 19, clause
(b) for each `O_X(U)`; [ECD] §3, Lem. 3.11, p. 17, clause (d) for `U = X`; [Sch12] §6, remark after
Def. 6.1, p. 30, the sandwich `O°° ⊆ O⁺ ⊆ O°`; [Bhatt] Ch. 9, proof of Lem. 9.2.5(4), p. 100,
uniqueness of maps out of rational localisations, used for naturality in (c)). In
`PerfectoidSpace`, define `rationalPairFunctor` as the functor from rational subsets of
`Spa(R, R⁺)` ordered by `⊇` to perfectoid Tate pairs; prove `CategoryTheory.Functor.map_comp`
(`ρ_{V,W} ∘ ρ_{U,V} = ρ_{U,W}`, `ρ_{U,U} = id`), `rationalPairFunctor.isRationalLocalization_map`
(transitivity (a)), `rationalPairFunctor.mem_plus_iff` (`f ∈ O_X⁺(U) ↔ ∀ x ∈ U, |f(x)| ≤ 1`),
`rationalPairFunctor.isAlmostZero_powerBounded_quot_plus` (`O_X(U)°/O_X⁺(U)` almost zero),
`rationalTiltIso` (the natural isomorphism (c)), `tilt.sharpQuotEquiv`
(the ring isomorphisms (d), natural by `tilt.mapQuot_comm`) and `rationalPairFunctor.spaTilt_comm` (under
`Spa(O_X(U), O_X⁺(U)) ≅ U` and its tilt, `spaTilt` of `O_X(U)` is the restriction of `spaTilt` of
`R`). *Needs:* *rational-localization-of-perfectoid-affinoids*; *tilting-map-of-spectra*;
*tilting-homeomorphism-and-rational-subsets*; layer P1; Tau Ceti
`TauCeti.Huber.PairOfDefinition.restrictionRingHomOfSubset`,
`TauCeti.ValuationSpectrum.mem_iff_forall_vle_one`, `TauCeti.Huber.IsRingOfIntegralElements`,
`TauCeti.ValuationSpectrum.existsUnique_continuous_ringHom_of_forall_comap_mem_rationalSubset`.

For a perfectoid pair `A` with `(p : A) = 0`,
`tilt.exists_sharpEquiv_of_prime_eq_zero` supplies a ring equivalence `A♭ ≃ A`
given by sharp, carrying `A♭⁺` onto `A⁺` and identifying the small-quotient comparison
with reduction of sharp. This includes the zero pair: its tilt, plus ring and
quotients are zero. For `A = O_X(U)`, the rational structure map carries `pR = 0`
to `pA = 0`, even when `U` is empty. Exact `CharP A p` is used only when `A` is
nonzero ([ECD] §3, Prop. 3.5, p. 15; Lem. 3.10, p. 16; Lem. 3.11, p. 17;
Thm. 3.18, p. 18).

**Checks.**

- `rationalPairFunctor_test_restriction_disc` (computation) — for `R = ℚ_p^cycl⟨T^{1/p^∞}⟩`,
  `V = {|T| ≤ |ϖ|²} ⊆ U = {|T| ≤ |ϖ|}`, the restriction
  `ℚ_p^cycl⟨(T/ϖ)^{1/p^∞}⟩ → ℚ_p^cycl⟨(T/ϖ²)^{1/p^∞}⟩` sends `(T/ϖ)^{1/p^m}` to
  `ϖ^{1/p^m}(T/ϖ²)^{1/p^m}`, and `tiltIso` transports it to the same formula over `ℚ_p^cycl♭` with
  `ϖ♭`.
- `rationalPairFunctor_test_plusModEquiv_whole` (compatibility) — for `U = X`, `plusModEquiv` is
  the ring isomorphism `R♭⁺/ϖ♭ ≅ R⁺/ϖ` induced by `♯` (ECD Lemma 3.11), and naturality for `V ⊆ X`
  is the commutation with `R⁺/ϖ → O_X⁺(V)/ϖ`.
- `rationalPairFunctor_test_plus_ne_powerBounded` (non-example) — for `K = ℚ_p^cycl`,
  `R = K⟨T^{1/p^∞}⟩` and `R⁺ = K° + K°°⟨T^{1/p^∞}⟩` (open, integrally closed), `T ∈ R° \ R⁺` while
  `ϖ^{1/p^n}T ∈ R⁺` for all `n`; a plus presheaf defined by `O_X(U)°` would violate `mem_plus_iff`
  at the points where `|T| > 1`.
- `rationalPairFunctor_test_charP` (degenerate) — if `pR = 0`, sharp identifies
  the tilted rational section pair with the original pair and `plusModEquiv`
  becomes reduction to `O_X⁺(U)/ϖ`, including for the empty rational open.
- `rationalPairFunctor_charP_empty` (non-example) — for `U = ∅`, `pA = 0`
  and the tilt equivalence sends every element to zero, while `CharP A p` is false.
  This applies in particular over the completed perfection of `𝔽_p((t))`
  ([ECD] §3, Ex. 3.4(ii), p. 15).

`Perfectoid.RationalOpen p R` is a rational subset of the supplied valuation spectrum.
`rationalPair`, `rationalMap` and `rationalPair_universal` give its completed perfectoid pair
and the factorization property for maps whose spectra land in that subset.
`rationalPair_presentation` fixes the topology, plus ring and base map by comparison with
`PairOfDefinition`'s completed presentation. `rationalRes` is the unique map commuting with
the two base maps. `tiltRationalOpen` takes the image under `spaTilt`;
`rationalTiltIso_base` and `rationalTiltIso_restriction` fix the comparison on base elements
and on nested rational subsets.

**Checks for rational pairs and restrictions.** `rational_subset_whole` makes the base map
an isomorphism; `rational_subset_empty` gives the zero ring; `rational_subset_proper` rules
out an isomorphism for a proper subset. `rational_restriction_identity`,
`rational_restriction_composition` and `rational_restriction_empty` compute identity,
composition and the zero-target map. `restriction_disc_two_radii` computes the fractional
coordinate at radii `ϖ` and `ϖ²` through the normalized disc map.
`rational_tilt_empty`, `rational_tilt_whole` and `rational_tilt_injective` test the subset
transport. `rational_tilt_zero_one`, `rational_tilt_empty_ring` and
`rational_tilt_base_coordinate` test the pair comparison on constants, the empty ring and
an arbitrary base coordinate.

`rationalOrderIso` is the order equivalence of rational subsets, with forward map
`tiltRationalOpen`; `rationalPairFunctor` uses the actual restriction morphisms,
as specified by `rationalPairFunctor_map`. The identity on base coordinates is
`rationalRes_base` evaluated at each element of `R`.
**Checks.** `rational_order_empty`, `rational_order_whole` and `rational_order_disjoint`
test the empty subset, the whole spectrum and preservation of disjointness.

### P2.5 completed residue fields

**Completed residue fields of perfectoid affinoids are perfectoid and tilt to those of the tilt.**
(*completed-residue-fields*) Let `(R, R⁺)` be a perfectoid Tate pair (no base field),
`X = Spa(R, R⁺)`, `x ∈ X`, and `(k(x)^, k(x)^⁺)` the completed residue affinoid field of `x` (the
completion of the residue field of the stalk `O_{X,x}` for the topology of `v_x`, with the
completion of the valuation ring of `v_x`; the uncompleted residue field with its valuation and
`Valued` structure and the stalk material are library, Tau Ceti
`TauCeti.ValuationSpectrum.residueFieldValuation`, `valued_algebraMap_residueFieldValuation`,
`continuous_algebraMap_residueFieldValuation`,
`isLocalRing_stalk_presentationLimitPresheafInCommRingCat`, `supp_presentationLimitStalkValuation`,
`presentationLimitStalkResidueValuation`, and AdicSpaces layer 3.3 supplies the completed residue
affinoid field; the completion and the perfectoidness are what this target adds; `k(x)^` is a complete nonarchimedean field whose
topology is that of the rank-one generisation of `x`, `x` being analytic because `R` is Tate), with
its canonical morphism `(R, R⁺) → (k(x)^, k(x)^⁺)`. Then `k(x)^` is a perfectoid field and
`(k(x)^, k(x)^⁺)` is a perfectoid Tate pair. Its tilt is canonically `(k(x♭)^, k(x♭)^⁺)`,
compatibly with the canonical morphisms from `(R, R⁺)` and `(R♭, R♭⁺)` and with `♯`; in particular
the value groups `Γ_x` and `Γ_{x♭}` coincide and `x` has rank one iff `x♭` has. Explicitly,
`k(x)^⁺` is the `ϖ`-adic completion of the image in `k(x)` of `colim_{U ∋ x} O_X⁺(U)` over the
rational neighbourhoods of `x` (an element `h` of the stalk with `|h(x)| ≤ 1` lies in `O_X⁺` of the
rational neighbourhood `{|h| ≤ 1}`) ([Sch12] §6, Cor. 6.7(ii) with proof and end of proof, p. 35,
the field case, with the characteristic-`0` case deduced after the homeomorphism; [AWS] Lecture 2,
§2.5, Cor. 2.5.7 with proof, pp. 60–61, the general statement and its last step; [KL16] §3.3,
Thm 3.3.18, class (ii), p. 64, the dense-image criterion used for `k(x)^` and its tilt). For
`R = ℚ_p^cycl⟨T^{1/p^∞}⟩`: at the Gauss point `k(x)^` is the completion of `ℚ_p^cycl(T^{1/p^∞})`
for the Gauss norm, whose tilt is the analogous completion of `ℚ_p^cycl♭(T^{1/p^∞})`; at the
evaluation point given by a compatible system `a_n ∈ ℤ_p^cycl`,
`a_{n+1}^p = a_n`, `T^{1/p^n} ↦ a_n`, one has `k(x)^ = ℚ_p^cycl`.
A value `a_0` alone does not specify such a point over `ℚ_p^cycl`. The uncompleted residue field is not
perfectoid in general: `ℚ_p^cycl(T^{1/p^∞})` is not complete. *Needs:*
*rational-localization-of-perfectoid-affinoids*; *perfectoid-pair-rational-restriction*;
*tilting-homeomorphism-and-rational-subsets*; *tilting-map-of-spectra*; layer P1 (P1.15, P1.27);
AdicSpaces layer 3.3; Tau Ceti `TauCeti.ValuationSpectrum.residueFieldValuation`,
`TauCeti.ValuationSpectrum.presentationLimitStalkResidueValuation`.

`Perfectoid.CompletedResidue` is `Completion (WithVal (residueFieldValuation x))`.
`residueMap` composes the supplier's residue map with the completion map. `residuePlus` is
the closure of the original valuation ring's image; it retains higher-rank integral data.
`residueTiltEquiv_spec` fixes both topologies, the transported plus subring, and the map
from the base tilt.

**Checks for residue fields.** `residue_zero_one` computes the two constants;
`residue_support` identifies the kernel with the point's support;
`residue_field_recovery` recovers a complete base field. `residue_integral_image` and
`residue_integral_two_terms` include integral base elements and their sums, while
`residue_inverse_parameter_nonmember` excludes `ϖ⁻¹`.
`residue_tilt_zero_one`, `residue_tilt_injective` and `residue_tilt_inverse_parameter`
test constants, the zero fibre and the chosen tilted parameter. The last Check uses
`ϖ♭` itself in the base-map equation.

### P2.6 stable uniformity and sheafiness

**Perfectoid Tate pairs are stably uniform and sheafy, with Čech-acyclic structure sheaf.**
(*perfectoid-stably-uniform-and-sheafy*) Let `(R, R⁺)` be a perfectoid Tate pair (ECD
Definition 3.1; no base field), `X = Spa(R, R⁺)`. Then (a) `(R, R⁺)` is stably uniform: `O_X(U)` is
uniform for every rational `U ⊆ X` ([HK] Def. 3.13; the predicate is Tau Ceti
`TauCeti.Huber.IsStablyUniform`, with `TauCeti.Huber.isStablyUniform_iff_forall_isUniform_presentationLimit` and
`PairOfDefinition.isStablyUniform_completion_locTopology`; the perfectoid instance is what is
proved here); (b) `(R, R⁺)` is sheafy: the structure presheaf `O_X` (AdicSpaces layers 3.3–3.5,
presentation independent) is a sheaf of complete topological rings on `X` (Tau Ceti
`Huber.IsSheafyRing`, `IsSheafyForEveryPresentation`), and `O_X⁺` is a
sheaf; the same holds for `(O_X(U), O_X⁺(U))` for every rational `U` — the implication "stably
uniform implies sheafy" is the Buzzard–Verberkmoes theorem, AdicSpaces layer 4.2, a target of that
roadmap which this one applies (Tau Ceti has the two halves for the Laurent cover,
`injective_presentationLimitMap_inf_laurentCoverOpen_of_isStablyUniform` and
`exists_presentationLimitMap_eq_of_inf_laurentCoverOpen_of_isStablyUniform`, and sheafiness for
strongly noetherian rings, `TauCeti.Huber.isSheafyRing_of_isStronglyNoetherian`, which does not apply to
perfectoid rings); (c) for every rational `U ⊆ X` and every finite covering `𝔘` of `U` by rational
subsets the augmented Čech complex `0 → O_X(U) → ∏ O_X(U_i) → ∏ O_X(U_i ∩ U_j) → ⋯` is exact in
every degree, and `H^i(U, O_X) = 0` for `i > 0`, `H⁰(U, O_X) = O_X(U)` (sheaf cohomology of the
underlying sheaf of abelian groups, Mathlib `CategoryTheory.Sheaf.H`) — this is AdicSpacesPartIII
R3.1 (*sheafy-cech-acyclicity*, for every sheafy complete Tate pair), applied to the pair
made sheafy by (b); (d) (a)–(c) depend on `R` only, not on the choice of `R⁺` ([KL15] §3.6,
Thm 3.6.15 with proof, p. 94, perfectoid `ℚ_p`-algebras; [KL16] §3.3, Cor. 3.3.19, p. 65, the
general case via Buzzard–Verberkmoes; [Berkeley] Lecture 6, Thm 6.1.10 and the following paragraph,
p. 43, and remark after Thm 6.2.6, p. 45, the characteristic-`p` case of (a); [HK] §3, Cor. 3.15 with
proof, p. 9). Kedlaya–Liu I Theorem 3.6.15, Kedlaya–Liu II Corollary 3.3.19, Berkeley
Theorems 6.1.10/7.1.1, Hansen–Kedlaya Corollary 3.15 and Kedlaya AWS Corollary 2.5.4 are this
theorem. In characteristic `p` only perfectness is used: `𝔽_p((t^{1/p^∞}))^∧⟨T^{1/p^∞}⟩` is stably
uniform because each rational localisation is perfect and complete (ECD Proposition 3.5). The uniform
non-sheafy Huber rings of Buzzard–Verberkmoes and Mihara are not perfectoid, and perfectoid rings
are never noetherian unless finite products of perfectoid fields ([AWS] Cor. 2.9.3), so Huber's
strongly noetherian sheafiness theorem does not prove the general perfectoid statement;
it does apply to perfectoid fields and their finite products. *Needs:*
*rational-localization-of-perfectoid-affinoids*; *rational-localization-in-characteristic-p*;
*perfectoid-pair-rational-restriction*; AdicSpaces 4.2 (Buzzard–Verberkmoes);
AdicSpacesPartIII R3.1 (*sheafy-cech-acyclicity*); Tau Ceti
`TauCeti.Huber.IsStablyUniform`, `TauCeti.Huber.IsSheafyRing`,
`TauCeti.ValuationSpectrum.isSheaf_iff_isSheafFor_rationalCover`,
`TauCeti.Huber.powerBoundedSubring`; Mathlib `CategoryTheory.Sheaf.H`.

`Perfectoid.isStablyUniform` has conclusion
`TauCeti.Huber.IsStablyUniform R` for a complete Hausdorff perfectoid Tate ring at a prime
`p`; its characteristic-`p` case, under `CharP R p`, uses only perfectness.
`Perfectoid.PairCat.isStablyUniform` registers the instance for the ring of a bundled
perfectoid pair, whose type determines `p`.
`Perfectoid.isSheafyRing` concludes `TauCeti.Huber.IsSheafyRing R` under the same
perfectoid hypotheses, without a characteristic or plus-ring restriction
([KL16] Cor. 3.3.19, p. 65; [ECD] Thm 3.18, p. 18). Applying it to each
`rationalPair p R U` supplies sheafiness on rational localizations. The ordinary and Čech
acyclicity conclusions are applications of AdicSpacesPartIII R3.1
(*sheafy-cech-acyclicity*; [KL15] Thm 2.4.23, p. 46),
on Part II `AdicSpace.Chart`, using `c.affinoid.ring`, `c.affinoid.spa` and
the presheaf comparison below.

Lean: `Perfectoid.isStablyUniform`, `Perfectoid.isSheafyRing`.

### P2.7 almost acyclicity of `O⁺` on rational coverings

**Almost exactness of the integral Čech complex of `O⁺` on rational coverings of a perfectoid
affinoid.** (*integral-almost-acyclicity*) Let `(R, R⁺)` be a perfectoid Tate pair (ECD
Definition 3.1; no base field), `X = Spa(R, R⁺)`. For every rational `U ⊆ X` and every finite
covering `𝔘 = (U₁, …, U_m)` of `U` by rational subsets (all finite intersections are then rational),
the augmented alternating Čech complex, with `(d⁰s)_{ij} = s_j|_{U_i∩U_j} − s_i|_{U_i∩U_j}` for `i < j`,
`C⁺(𝔘): 0 → O_X⁺(U) → ∏_i O_X⁺(U_i) → ∏_{i<j} O_X⁺(U_i ∩ U_j) → ⋯` is almost exact: each of its
cohomology modules is killed by every topologically nilpotent element of `R⁺` (equivalently by
`ϖ^{1/p^n}` for all `n`; the ideal is idempotent, layer P1). It is exact at its first two terms. The
same holds for the complex of the `O_X(V)°`, which is almost equal to `C⁺(𝔘)` termwise; in the
almost category `C⁺(𝔘)ᵃ` is exact. Consequently `H^i(U, O_X⁺)ᵃ = 0`, i.e. `H^i(U, O_X⁺)` is almost
zero, for every rational `U ⊆ X` and every `i > 0` (see *cech-to-sheaf-cohomology-on-rational-basis*)
([Berkeley] Lecture 7, proof of Thm 7.4.4, p. 53, bounded torsion plus the Frobenius step in
characteristic `p`; [Sch12] §6, proof of Prop. 6.14, p. 38, the tilting reduction of the general
case; [KL16] §3.3, Cor. 3.3.20(c), p. 65, and Rem. 3.3.11, p. 61, the same conclusion for Fontaine
perfectoid rings through almost optimal exactness). Over a perfectoid field this is Sch12
Proposition 6.14, proved there through `p`-finite algebras and Tate's acyclicity theorem
(see *p-finite-acyclicity-from-tate*), a route this layer keeps. For `R = ℚ_p^cycl⟨T^{1/p^∞}⟩` and the
Laurent covering `{|T| ≤ |ϖ|}`, `{|ϖ| ≤ |T|}`, `C⁺` is exact: the Laurent decomposition of
`Σ_{k ∈ ℤ[1/p]} d_k (T/ϖ)^k` splits exactly. The conclusion is about `O⁺` up to almost zero
modules: after inverting `ϖ` it recovers the exactness of the `O`-Čech complex, but `C⁺` itself need
not be exact when `R⁺ ≠ R°`. *Needs:* *perfectoid-stably-uniform-and-sheafy*;
*integral-cech-cohomology-bounded-torsion*; *almost-exactness-from-reduction-mod-pseudouniformizer*;
*tilting-homeomorphism-and-rational-subsets*; *perfectoid-pair-rational-restriction*;
*rational-localization-in-characteristic-p*; layer P1; layer P0; Mathlib `frobeniusEquiv`,
`HomologicalComplex.ExactAt`, `CategoryTheory.Sheaf.H`.

**Almost exactness lifts from reduction modulo `ϖ` for torsion-free complete complexes.**
(*almost-exactness-from-reduction-mod-pseudouniformizer*) Let `V` be a ring, `m ⊆ V` an ideal with
`m² = m` (the basic setup of layer P0, P0.1; in the application `V = R⁺` and `m` its ideal of
topologically nilpotent elements, with `ϖ ∈ m`), `ϖ ∈ V`, and call a `V`-module almost zero if `m`
kills it (a Serre class closed under products and filtered colimits; [ECD] §3, after Ex. 3.22,
p. 19). Prove: (a) for a bounded complex `C` of `V`-modules whose terms are `ϖ`-torsion free and
`ϖ`-adically complete and separated (`C = lim_n C/ϖ^n C` with surjective transition maps), if every
cohomology module of `C/ϖC` is almost zero, so is every cohomology module of `C`; (b) conversely, if
the terms of `C` are `ϖ`-torsion free and every cohomology module of `C` is almost zero, so is every
cohomology module of `C/ϖC`; (c) for `α : M → N` `V`-linear with `M` `ϖ`-adically separated and
`N` `ϖ`-torsion free, if `α ⊗ V/ϖ` has almost zero kernel then `ker α` is almost zero; if moreover
`M` is `ϖ`-adically complete, `N` is `ϖ`-adically separated and `α ⊗ V/ϖ` has almost zero
cokernel, then `coker α` is almost zero ([Sch12] §6, proof of Prop. 6.14 (characteristic `0`), p. 38,
statement (a) in the almost category; [Bhatt] Ch. 9, footnote 1 in the proof of Lem. 9.2.5, p. 100,
the kernel statement of (c)). Completeness cannot be dropped: for `V = ℤ_p`, `m = V` (so that
"almost zero" means zero; `m = 0` would make every module almost zero and the example empty) and
`C = [ℚ_p → 0]`, `C/pC = 0` is exact while `H⁰(C) = ℚ_p ≠ 0`, `ℚ_p` being `p`-torsion free but not
`p`-adically separated. In
*integral-almost-acyclicity*, (b) is applied to the characteristic-`p` integral Čech complex and (a)
to the characteristic-`0` one after identifying both modulo `ϖ`. *Needs:* layer P0 (P0.1, P0.16);
Mathlib `CategoryTheory.ObjectProperty.IsSerreClass`, `IsAdicComplete`,
`HomologicalComplex.ExactAt`.

**Integral Čech cohomology of a stably uniform sheafy Tate pair is killed by a bounded power of
`ϖ`.** (*integral-cech-cohomology-bounded-torsion*) Let `(A, A⁺)` be a complete Hausdorff Tate Huber
pair which is stably uniform (every `O_X(V)` uniform) and sheafy, with no perfectoid or noetherian
hypothesis, `ϖ ∈ A⁺` a pseudouniformiser, `X = Spa(A, A⁺)`, `U ⊆ X` rational and `𝔘 = (U₁, …, U_m)`
a finite covering of `U` by rational subsets. Let `C⁺` be the augmented alternating Čech complex
`0 → O_X⁺(U) → ∏_i O_X⁺(U_i) → ∏_{i<j} O_X⁺(U_i ∩ U_j) → ⋯ → O_X⁺(U₁ ∩ ⋯ ∩ U_m) → 0`. Prove that
`C⁺[1/ϖ]` is the augmented Čech complex of `O_X`, which is exact (AdicSpacesPartIII R3.1
(*sheafy-cech-acyclicity*),
Kedlaya–Liu I Theorem 2.4.23), that `C⁺` is exact at its first two terms, and that there is `N ≥ 0`
such that `ϖ^N` annihilates every cohomology module of `C⁺`; the same bound holds for the Čech
complex of `V ↦ O_X(V)°`, whose terms are also open and bounded ([Sch12] §6, proof of
Prop. 6.10(iii), p. 36, the open-mapping step and agreement of topologies as bounded torsion;
[Berkeley] Lecture 7, proof of Thm 7.4.4, p. 53, the same step for a perfectoid pair in
characteristic `p`). For `S = K⟨T⟩` over a complete nonarchimedean field `K` and the Laurent covering
`{|T| ≤ |ϖ|}`, `{|ϖ| ≤ |T|}` of the closed disc,
`C⁺ = [K°⟨T⟩ → K°⟨T/ϖ⟩ × K°⟨T, ϖ/T⟩ → K°⟨T/ϖ, ϖ/T⟩]` is exact (`N = 0`): every Laurent series
`Σ_{k∈ℤ} d_k(T/ϖ)^k` with `d_k → 0` in `K°` splits into its parts of non-negative and negative
degree. The lemma gives only a bounded power of `ϖ`; almost vanishing needs Frobenius
(see *integral-almost-acyclicity*). *Needs:* AdicSpacesPartIII R3.1 (*sheafy-cech-acyclicity*);
Tau Ceti
`TauCeti.Huber.IsTateRing.isOpenMap`, `TauCeti.Huber.IsTateRing.isStrictMap_of_isClosed_range`,
`TauCeti.ValuationSpectrum.mem_iff_forall_vle_one`, `TauCeti.Huber.powerBoundedSubring`; Mathlib
`Topology.IsStrictMap`, `HomologicalComplex.ExactAt`.

The general Čech carrier is `Perfectoid.RationalTateSheaf`: an additive sheaf with the
base integral scalar action, additive coordinates in each supplied completed rational
localization, and restriction equations for the continuous localization maps. A Boolean
selects the chosen plus ring or the power-bounded ring in those coordinates.
`rationalTateSheaf` constructs both from stable uniformity.
`integral_cech_bounded_torsion` quantifies over every finite rational covering of a rational
open and supplies one exponent for both integral conventions and all positive degrees.
The positive-degree Čech complex is Mathlib’s. Exactness at the two augmentation terms
is the sheaf equalizer condition; localization at the parameter identifies its rational
terms. The bounded-torsion deduction uses strictness of the complete Tate-module
complex: each image is closed by rational acyclicity, the open-mapping theorem compares
its quotient topology with the kernel topology, and a finite alternating complex gives
a common exponent. The explicit printed calculation in Sch12 6.10(iii) is over a field;
the Tate-module strict-map input is `TauCeti.Huber.IsTateRing.isStrictMap_of_isClosed_range`.

**Checks.** `general_integral_sections_empty` gives zero sections;
`general_integral_sections_whole` recovers the supplied plus ring;
`general_integral_parameter_torsionfree` rules out parameter torsion in sections on every open.
`bounded_torsion_single_open` gives zero positive cohomology, hence exponent zero,
for a one-member covering.

**From almost Čech acyclicity on the rational basis to almost vanishing of sheaf cohomology.**
(*cech-to-sheaf-cohomology-on-rational-basis*) Let `V` be a ring and `m ⊆ V` an ideal with `m² = m`
(so almost zero modules form a Serre class closed under products and filtered colimits, layer P0).
Let `X` be a topological space with a basis `ℬ` of quasi-compact open subsets closed under finite
intersections (so every open covering of `U ∈ ℬ` is refined by a finite covering by members of
`ℬ`), and `F` a sheaf of `V`-modules on `X`. Prove: if for every `U ∈ ℬ` and every finite covering
`𝔘` of `U` by members of `ℬ` the Čech cohomology `Ȟ^q(𝔘, F)` is almost zero for all `q ≥ 1`, then
the sheaf cohomology `H^q(U, F)` (derived-functor cohomology of the underlying sheaf of abelian
groups, Mathlib `CategoryTheory.Sheaf.H`) is almost zero for every `U ∈ ℬ` and `q ≥ 1`;
equivalently `H^q(U, Fᵃ) = 0` for the associated sheaf of almost modules (layer P0, P0.29). For
`m = V` this is Cartan's criterion ([Stacks] Tag 01EW (Lemma 20.11.9)), also used in
DiamondsAndVStacks D0.19 and for `O_X` in AdicSpacesPartIII R3.1 (*sheafy-cech-acyclicity*), clause (2). It
applies to `X = Spa(A, A⁺)` for a Huber
pair with `ℬ` the rational subsets, a basis of quasi-compact opens closed under finite intersections
with finite rational refinements of every covering of a rational subset ([KL15] §2.4, proof of
Prop. 2.4.21, p. 46, the passage from Čech to sheaf cohomology on the rational basis; [Sch12] §6,
proof of Prop. 6.14, p. 38, for the perfectoid application; [HK] §3, Def. 3.9, p. 8, the convention for sheaf cohomology on `Spa`). With `V = R⁺`, `m`
the topologically nilpotent elements and `F = O_X⁺` on a perfectoid affinoid,
*integral-almost-acyclicity* supplies the hypothesis and the conclusion is ECD Theorem 3.24. The
hypothesis must hold for all `U ∈ ℬ` and all finite coverings: Čech acyclicity for the coverings of
`X` alone does not control `H^q(X, F)`. *Needs:* layer P0 (P0.1, P0.29); Mathlib `CategoryTheory.Sheaf.H`, `CategoryTheory.cechComplexFunctor`,
`CategoryTheory.ObjectProperty.IsSerreClass`; Tau Ceti
`TauCeti.ValuationSpectrum.isTopologicalBasis_spaRationalFamily`,
`TauCeti.ValuationSpectrum.isCompact_of_mem_spaRationalFamily`,
`TauCeti.ValuationSpectrum.rationalSubset_inter`,
`TauCeti.ValuationSpectrum.exists_finite_spaRationalFamily_refinement`.

For the abstract lifting argument, apply the short exact sequences between successive
`C/ϖ^n C` to propagate almost-zero cohomology. Idempotency of `m` makes almost-zero modules
closed under extensions. The reductions form a termwise surjective tower; completeness
identifies its derived limit with `C`. The product description of `Rlim` and its cohomology
sequence then preserve almost-zero modules. The precise ordinary inverse-limit inputs are
[Stacks, Lemma 15.88.1(2), (3), (5)](https://stacks.math.columbia.edu/tag/07KV) and
[§15.89](https://stacks.math.columbia.edu/tag/0CQC), applied to `V`-modules. Reduction in the
reverse direction uses the multiplication-by-`ϖ` triangle.
`Perfectoid.complexMod` uses Mathlib extension and restriction of scalars along `V → V/(ϖ)`;
`almost_complex_reduction_iff`, `almost_lift_kernel` and `almost_lift_cokernel` state the
complex and linear-map forms with their completeness and torsion hypotheses.
For `almost_lift_cokernel`, idempotency strengthens the reduction hypothesis to
`mN ⊆ f(M) + ϖ·mN`. Successive corrections give a convergent series in the complete
source; the remaining error lies in every `ϖ^n N` and vanishes by separation of the target.
This applies to every idempotent ideal, including the zero and unit ideals.

**Checks for reduction.** `reduction_unit` gives the zero complex modulo one;
`reduction_zero` recovers the original complex modulo zero;
`reduction_characteristic_two` computes `2x = 0` after reduction modulo two.
`almost_lifting_zero_ideal` and `almost_lifting_identity` test the vacuous ideal and identity
map; `almostExact_test_nonseparated` remains the failure when separation is omitted.

For the Cartan step, [Stacks, Lemma 20.11.9](https://stacks.math.columbia.edu/tag/01EW) verifies
the ordinary statement. Its dimension-shifting proof can be run modulo the Serre class of
`m`-annihilated modules: filtered refinements compute degree one, finite products give the
Čech terms, and `m² = m` controls extensions. The additional identification with derived
cohomology in the category of almost sheaves requires an exact almostification comparison;
that comparison is a separate target of layer P0. The ordinary-cohomology almost-vanishing
claim above does not assume that comparison. The almost Cartan criterion `almost_cartan` has the
general signature: a scalar action `V → End(F)`, an idempotent ideal, a quasi-compact basis closed
under intersections, and almost vanishing for every finite basis covering. Its conclusion
uses Mathlib’s cohomology presheaf on each basis open. The computed scalar Checks
`cartan_zero_scalar`, `cartan_unit_scalar` and `cartan_two_term_additivity` give zero,
identity and the sum of two scalar actions.

### P2.8 the sheaf theorem

The sheaf theorem collects the sheaf property, the perfectoidness of the rational localisations
and the (almost) vanishing of cohomology; the `p`-finite route of Scholze is kept alongside it.

**The sheaf theorem: `O_X` is a sheaf with vanishing higher cohomology and `H^i(X, O_X⁺)` is almost
zero.** (*sheaf-theorem-and-almost-acyclicity*) Let `(R, R⁺)` be a perfectoid Tate pair (ECD
Definition 3.1: any perfectoid Tate ring, of characteristic `0`, `p` or mixed, with no perfectoid
base field) and `X = Spa(R, R⁺)`. Then: (i) `O_X` and `O_X⁺` are sheaves on `X`; (ii) for every
rational `U ⊆ X`, `(O_X(U), O_X⁺(U))` is a perfectoid Tate pair with tilt
`(O_{X♭}(U♭), O_{X♭}⁺(U♭))`, `U♭` the image of `U` in `X♭`; (iii) `H⁰(X, O_X) = R` and
`H^i(X, O_X) = 0` for `i > 0` (AdicSpacesPartIII R3.1 (*sheafy-cech-acyclicity*), for the pair
made
sheafy by *perfectoid-stably-uniform-and-sheafy*); (iv) `H⁰(X, O_X⁺) = R⁺` and, for `i > 0`, the `R⁺`-module
`H^i(X, O_X⁺)` is almost zero, i.e. killed by every topologically nilpotent element of `R⁺`
(equivalently `H^i(X, O_X⁺ᵃ) = H^i(X, O_X°ᵃ) = 0` in the almost category, [Sch12] §6, Rem. 6.12,
p. 36); (v) (i)–(iv) hold for every rational `U` in place of `X`. Sheaf cohomology is that of
sheaves of abelian groups on the topological space `X`. This is ECD Theorems 3.18 and 3.24 in their
stated generality; ECD's citations [Sch12], [KL15] cover perfectoid algebras over a perfectoid field
and over `ℚ_p` (with characteristic `p`), and the case where `p` is neither zero nor invertible is
Kedlaya–Liu II §3.3 ([ECD] §3, Thm 3.18, p. 18, for (i)–(ii), and Thm 3.24, p. 19, for (iv);
[Sch12] §6, Thm 6.3(iii)–(iv), p. 30, and Prop. 6.14, p. 37, over a perfectoid field; [KL15] §3.6,
Thm 3.6.15, p. 94, for the vanishing of `H^i(X, O)` and the almost vanishing of `H^i(X, O⁺)`;
[Berkeley] Lecture 7, Thm 7.4.4, p. 53, (iii) for perfectoid Tate–Huber pairs; [HK] §6, Prop. 6.14,
p. 19, (iv) without a base field; [AWS] Lecture 2, §2.9, Thm 2.9.5, p. 73, (iv) for analytic
perfectoid pairs). Sch12 Theorem 6.3(iii)–(iv) and Proposition 6.14, Kedlaya–Liu II
Corollary 3.3.20, Berkeley Theorem 7.4.4, Hansen–Kedlaya Proposition 6.14 and Kedlaya AWS
Theorem 2.9.5 are instances. For `K` a perfectoid field and `X = Spa(K, K°)`, `H^i(X, O_X⁺) = 0` for
`i > 0` because `X` is a point with the trivial covering only, and `H⁰ = K°`; almost zero cannot be
improved to zero in general (Sch12 Remark 6.12 states the result as `m`-torsion). For `R` of ECD
Example 3.4(iv) the theorem applies: the Tate-ring proofs of the ingredients use no base field.
*Needs:* *perfectoid-stably-uniform-and-sheafy*; *rational-localization-of-perfectoid-affinoids*;
*perfectoid-pair-rational-restriction*; *integral-almost-acyclicity*;
*cech-to-sheaf-cohomology-on-rational-basis*; layer P0 (P0.5, P0.29); Tau Ceti
`TauCeti.ValuationSpectrum.mem_iff_forall_vle_one`; Mathlib `CategoryTheory.Sheaf.H`,
`TopCat.Presheaf.IsSheaf`.

**Almost acyclicity for p-finite perfectoid affinoids via Tate's acyclicity theorem.**
(*p-finite-acyclicity-from-tate*) Let `K` be a perfectoid field of characteristic `p` with
pseudouniformiser `ϖ`. A perfectoid affinoid `K`-algebra `(R, R⁺)` is `p`-finite if it is the
completed perfection of a reduced affinoid `K`-algebra `(S, S⁺)` of topologically finite type in
the convention of Sch12 Definition 2.6(iii), which includes `S⁺ = S°` (`S` a quotient of a Tate
algebra; an arbitrary smaller plus ring is not part of Sch12 Definition 6.9): `R⁺` is the `ϖ`-adic
completion of `colim_Φ S⁺` and `R = R⁺[ϖ⁻¹]` ([Sch12] §6, Def. 6.9, p. 35). Every perfectoid Tate
pair of characteristic `p` is a perfectoid affinoid algebra over `𝔽_p((t^{1/p^∞}))^∧` via `t ↦ ϖ`,
with `R⁺` an algebra over its ring of integers (Sch12 proof of Proposition 6.14; the same statement
is clause (a) of P5.13.2's *char-p-perfectoid-base-field*, which uses this subsection
and is not an input here), so every characteristic-`p` perfectoid Tate pair is in the scope of the base-field
convention; but only those that are completed perfections of reduced affinoid `K`-algebras are
`p`-finite, and a general characteristic-`p` perfectoid Tate pair is a completed direct limit of
`p`-finite ones (*completed-direct-limits-of-p-finite-affinoids*), not itself `p`-finite
(`K⟨T_s^{1/p^∞} : s ∈ ℕ⟩` in infinitely many variables is not). For such `(S, S⁺)` and `Y = Spa(S, S⁺)` prove: (i) `S°` is open and bounded
(`S` is uniform); (ii) for every rational `V ⊆ Y`, `O_Y(V)` is again a reduced affinoid `K`-algebra
of topologically finite type; (iii) for every finite covering of a rational `V ⊆ Y` by rational
subsets, every cohomology module of the augmented Čech complex of the `O_Y(·)°` is killed by a
power of `ϖ`; and for `X = Spa(R, R⁺)`: (iv) the projection `X → Y` is a homeomorphism identifying
rational subsets; (v) for rational `U ⊆ X` corresponding to `V ⊆ Y`, `(O_X(U), O_X⁺(U))` is the
completed perfection of `(O_Y(V), O_Y⁺(V))`; (vi) for every finite rational covering of `X` the
augmented Čech complex `0 → O_X(X)°ᵃ → ∏ O_X(U_i)°ᵃ → ∏ O_X(U_i ∩ U_j)°ᵃ → ⋯` is exact, so `O_X` is a
sheaf and `H^i(X, O_X°ᵃ) = 0` for `i > 0`, equivalently `H^i(X, O_X⁺)` is almost zero (Sch12
Proposition 6.11, Remark 6.12) ([Sch12] §6, proof of Prop. 6.10, p. 36, the rigid-analytic inputs
(i)–(iii); proof of Prop. 6.11(i) and (iii), pp. 36–37, for (iv) and (vi); [Sch12] arXiv v1,
Def. 2.6(iii), p. 8, published version p. 254 (PDF p. 10), the topologically-finite-type convention
with the maximal plus ring). The route through Tate's acyclicity is an alternative to the Frobenius
route of *integral-almost-acyclicity*; it is kept because Sch12 §7 (Lemma 7.5(ii),
Propositions 7.6, 7.7, 7.13) reduces étale statements to `p`-finite algebras and hence to
noetherian adic spaces. For `S = K⟨T⟩` and the covering by `{|T| ≤ |ϖ|}` and `{|ϖ| ≤ |T|}`,
`R = K⟨T^{1/p^∞}⟩` is `p`-finite, `X ≅ Y`, and the `O°`-Čech complex is exact both for `Y` and for
`X`. *Needs:* *integral-cech-cohomology-bounded-torsion*; *completed-direct-limits-of-p-finite-affinoids*;
layer P1; AdicSpacesPartII R0.9 (*reduced-affinoid-supremum-norm*); Tau Ceti
`TauCeti.Huber.IsStronglyNoetherian`,
`TauCeti.Huber.IsTateRing.isOpenMap`.

**Perfectoid affinoid algebras in characteristic p are completed direct limits of p-finite ones.**
(*completed-direct-limits-of-p-finite-affinoids*) Let `K` be a perfectoid field of characteristic
`p` with pseudouniformiser `ϖ`, and `(R, R⁺)` a perfectoid affinoid `K`-algebra such that `R⁺` is a
`K°`-algebra (every perfectoid Tate pair of characteristic `p` satisfies this for
`K = 𝔽_p((t^{1/p^∞}))^∧` with `t ↦ ϖ`, Sch12 proof of Proposition 6.14; P5.13.2's
*char-p-perfectoid-base-field* (a) cites this for (i), downstream). Prove: (i) `(R, R⁺)` is the completion of a filtered direct limit of
`p`-finite perfectoid affinoid `K`-algebras `(R_I, R_I⁺)` indexed by the finite subsets `I ⊆ R⁺`:
`S_I` is the image of `K⟨T_i : i ∈ I⟩ → R` with the quotient topology, `S_I⁺` the integral closure
in `S_I` of the image of `K°⟨T_i : i ∈ I⟩` (this choice makes `S_I⁺ ⊆ R⁺` automatic, `R⁺` being
integrally closed and containing the images of the `T_i`; Sch12 instead takes `S_I⁺ = S_I°` and
uses Tate's theorem that power-bounded elements of a reduced affinoid algebra are integral over
`K°⟨T⟩`; `S_I⁺` is in general not the preimage of `R⁺` in `S_I`), and `(R_I, R_I⁺)` the completed
perfection of `(S_I, S_I⁺)`; precisely `R⁺/ϖ^n = colim_I R_I⁺/ϖ^n` for all `n`; (ii)
`Spa(R, R⁺) ≅ lim_I Spa(R_I, R_I⁺)`, and every rational `U ⊆ X` is the preimage of a rational
`U_I ⊆ X_I = Spa(R_I, R_I⁺)` for some `I`; (iii) `(O_X(U), O_X⁺(U))` is the completion of
`colim_{J ⊇ I} (O_{X_J}(U_J), O_{X_J}⁺(U_J))`, `U_J` the preimage of `U_I`; (iv) if `U_I ⊆ X_I` is a
quasi-compact open containing the image of `X`, then the image of `X_J` lies in `U_I` for some
`J ⊇ I` ([Sch12] §6, Lem. 6.13(i) with proof, p. 37, for (i), the colimit description modulo `ϖ^n`
and the hypothesis on `S_I⁺`; proof of Lem. 6.13(iv), p. 37, for (iv); proof of Prop. 6.14, p. 37,
the reduction that makes the `K°`-algebra hypothesis harmless). For `R = K⟨T^{1/p^∞}⟩` and
`I = {T}`, `S_I = K⟨T⟩` and `R_I = R` for this particular index; this does not assert that every transition
beyond it is an isomorphism. Part (iv) lets a
finite rational covering of `X`, and finite étale covers over it, descend to a finite `X_J` (Sch12
Propositions 6.14 and 7.6). *Needs:* layer P1; AdicSpacesPartII R0.19.5’s spectral-limit calculus; Mathlib
`UniformSpace.Completion`; Tau Ceti `TauCeti.Huber.restrictedMvPowerSeriesCompletion`,
`TauCeti.ValuationSpectrum.rationalSubset`.

`Perfectoid.PFiniteModel` records a reduced strictly finite-type affinoid, its maximal plus
ring, a continuous topological embedding into the perfected algebra, density of its
`p`-power roots, and equality of the completed root-integral model with the chosen plus
ring. `IsPFinite` means that such a model exists. `PFiniteSystem` records the finite-subset
diagram, its compatible scalar maps and its map to `R`. At `I`, the reduced source is
topologically generated by the elements of `I` and embeds algebraically in `R`; it keeps
its affinoid quotient topology. The completed-perfection map need not be an embedding.
`pfiniteSystem_isColimit`, `pfiniteSystem_mod`, `pfiniteSystem_topologicalLimit`,
`pfiniteSystem_rational_descent` and `pfiniteSystem_eventual_image` state the completed
colimit, its quotient comparisons, the valuation-space limit and finite-stage descent.
`pfiniteSystem_rational_colimit` includes the rational-localization comparison, with
transition maps and the map to the limiting localization fixed over the base stages.
The modulo-parameter comparison states both surjectivity and eventual vanishing of a
representative, so it specifies the kernel as well as the image.

**Checks for p-finite models.** `pfinite_field` recovers `K`; `pfinite_disc` uses the completed
perfection of `K⟨T⟩`; `pfinite_infinite_variables_nonexample` excludes the perfected algebra
on countably infinitely many independent variables from being a single p-finite object.
`pfinite_system_empty` recovers the scalar field at the empty index when `R` is nonzero
(the zero ring instead gives a zero stage); `pfinite_system_singleton` lifts its named
integral generator; `pfinite_system_two_generators` computes the sum and product of two
generators in their common stage. `pfiniteModel_spa_homeomorph` and
`pfiniteModel_rational_correspondence` fix the p-finite source comparison on valuations.

`rationalPairFunctor` uses opposite inclusion of rational opens; its map is exactly
`rationalRes`, as `rationalPairFunctor_map` states. **Checks.**
`rationalPairFunctor_whole` identifies the whole-space value with the original pair;
`rationalPairFunctor_empty` gives the zero ring; `rationalPairFunctor_proper` shows that
a proper rational subset cannot have a surjective map on spectra.

### P2.9 affinoid perfectoid spaces

**Affinoid perfectoid spaces: adic spectra of perfectoid Huber pairs.** (*affinoid-perfectoid-space*)
A space `X` here is an object of Tau Ceti's `PreAdicSpace`; the adic spaces among them are the
sheafy ones (`PreAdicSpace.isAdic`). AdicEtaleGeometry's analytic carrier `YonedaAdicSpace`, the full
subcategory of `J_rat`-sheaves on `CAffᵒᵖ` covered by open affinoid Yoneda spectra
([Berkeley] §3.4, Def. 3.4.1, printed p. 22), receives adic spaces through
`YonedaAdicSpace.ofAdicSpace`, and the statements below phrased on it are read through that
embedding. `X` is
*affinoid perfectoid* if there are a complete Hausdorff Huber pair `(R, R⁺)` (Tau Ceti `Huber.Pair`:
`R⁺ ⊆ R°` open and integrally closed) with `R` a perfectoid Tate ring in the sense of ECD
Definition 3.1 (layer P1, P1.1: complete, uniform, with a pseudouniformiser `ϖ` such that `ϖ^p | p`
in `R°` and Frobenius `R°/ϖ → R°/ϖ^p` bijective) and an isomorphism of adic spaces
`X ≅ Spa(R, R⁺)`. Here `Spa(R, R⁺)` means Tau Ceti's `presentationLimitPreAdicSpace` of the pair
(for a pair of definition contained in `R⁺`), for `R : Perfectoid.PairCat p`; it is an adic space,
and its image under `YonedaAdicSpace.ofAdicSpace` is the Yoneda spectrum
`YonedaAdicSpace.ofSpaY (CAff.of R.pair)`, because perfectoid Tate pairs are sheafy
(see *perfectoid-stably-uniform-and-sheafy*). For such `X` the global sections recover the pair,
`(O_X(X), O_X⁺(X)) ≅ (R, R⁺)`, by the library: `TauCeti.ValuationSpectrum.isIso_toPresentationLimit_top`
for `O_X(X) = R` and `TauCeti.ValuationSpectrum.mem_iff_forall_vle_one` (Wedhorn Prop. 7.52(1),
`IsRingOfIntegralElements.mem_of_forall_vle_one`) for `O_X⁺(X) = {f : |f(x)| ≤ 1 for all x} = R⁺`;
so `X` is affinoid perfectoid iff it is isomorphic to the Yoneda spectrum of a complete
sheafy Huber pair and `O_X(X)` is a perfectoid Tate ring, and
then `X ≅ Spa(O_X(X), O_X⁺(X))` canonically. By [Wedhorn] §8.2, Rem. 8.24 and Prop. 8.25, p. 80,
`Spa` is an anti-equivalence from complete perfectoid Huber pairs with morphisms of Huber pairs
(Tau Ceti `Huber.Pair.Hom`: continuous ring maps carrying plus ring into plus ring) to the full
subcategory of affinoid perfectoid spaces. Conventions: (1) the predicate asks for a
presentation by a perfectoid ring ([Berkeley] Lecture 7, §7.1, Rem. 7.1.3, printed p. 49); it is
not the conjunction "affinoid and a perfectoid space"; the two agree in characteristic `p`
(see *characteristic-p-affinoid-perfectoid-criterion*), and their agreement in characteristic `0` is
not asserted; (2) no perfectoid base field is fixed ([ECD] §3, Ex. 3.4(iv), p. 15; [Sch12] §6,
after the proof of Thm 6.3, p. 38, names `Spa(R, R⁺)` of a perfectoid affinoid `K`-algebra, and the
base field is dropped here), `R⁺` is an arbitrary ring of integral elements, and the zero ring is
allowed, so the empty adic space is affinoid perfectoid; (3) morphisms are morphisms of adic
spaces. In `PerfectoidSpace`, define `isAffinoidPerfectoidModel` (the object property of
`PreAdicSpace` satisfied by the spaces `presentationLimitPreAdicSpace` of perfectoid Tate pairs) and
`isAffinoidPerfectoid` as its isomorphism closure, the predicate on a pre-adic space `X`
(there exist a complete Hausdorff Huber pair `(R, R⁺)` with `R` perfectoid and an isomorphism
`X ≅ Spa(R, R⁺)`), so that closure under isomorphisms holds by construction; and `spa` as
`Spa(R, R⁺)` of a complete Hausdorff Huber pair with `R` perfectoid. The proposed lemma names
written `IsAffinoidPerfectoid.*` below refer to this object property. Prove
`isAffinoidPerfectoid_presentationLimitPreAdicSpace` (`Spa(R, R⁺)` is affinoid perfectoid), `IsAffinoidPerfectoid.isPerfectoidTateRing_sections` (if `X` is
affinoid perfectoid then `O_X(X)` is a perfectoid Tate ring, `(O_X(X), O_X⁺(X))` is a complete
Huber pair, and `X → Spa(O_X(X), O_X⁺(X))` corresponding to the identity is an isomorphism),
the characterisation of affinoid perfectoid spectra (for a complete Hausdorff Huber pair
`A : CAff` with `[IsTateRing A.A]` and `hA : A.Sheafy`, `Spa A` is affinoid perfectoid iff
`IsPerfectoidTateRing p A.A`; sheafiness identifies the global sections with `A.A`,
by [Wedhorn] §8.2, Def. 8.21 and Prop. 8.25, p. 80, and the perfectoid presentation
uses [Berkeley] §7.1, Thm 7.1.1 and Rem. 7.1.3, printed p. 49),
`spaHomEquiv` (for a perfectoid space `Y`) and `spaHomEquiv_adic` (for any
`Y : TauCeti.AdicSpace`, embedded by `YonedaAdicSpace.ofAdicSpace`):
morphisms `Y → Spa(R, R⁺)` correspond naturally to
continuous unital ring maps `R → O_Y(Y)` carrying `R⁺` into `O_Y⁺(Y)`; for a complete sheafy
affinoid `Y = Spa(S, S⁺)` these are `Huber.Pair.Hom (R, R⁺) (S, S⁺)`. The global sections of an
arbitrary `Y` need not form a Huber ring, so the general right-hand side is the subtype of ring
maps with these two properties, `affinoidPerfectoidEquiv` (`Spa` is an equivalence from the opposite of the
complete perfectoid Huber pairs to the full subcategory of affinoid perfectoid spaces),
`IsAffinoidPerfectoid.of_iso` (closed under isomorphisms, Mathlib
`ObjectProperty.IsClosedUnderIsomorphisms`), `IsAffinoidPerfectoid.spectralSpace` (the underlying
space is spectral, in particular quasi-compact and quasi-separated),
`IsAffinoidPerfectoid.isAnalyticPoint` (every point is analytic, `R` being Tate) and
`IsAffinoidPerfectoid.rationalSubset` (rational subsets are affinoid perfectoid,
*affinoid-perfectoid-basis*). *Needs:* layer P1 (P1.1); *perfectoid-stably-uniform-and-sheafy*;
*sheaf-theorem-and-almost-acyclicity*; AdicSpaces layers 3.5 and 5; Tau Ceti `TauCeti.AdicSpace`,
`TauCeti.PreAdicSpace`, `TauCeti.ValuationSpectrum.isIso_toPresentationLimit_top`,
`TauCeti.ValuationSpectrum.mem_iff_forall_vle_one`, `TauCeti.Huber.Pair`,
`TauCeti.Huber.Pair.Hom`, `TauCeti.Huber.IsTateRing`, `TauCeti.ValuationSpectrum.spa`,
`TauCeti.ValuationSpectrum.spaAnalytic_eq_spa_of_isTateRing`,
`TauCeti.ValuationSpectrum.spectralSpace_spa_of_pairOfDefinition`; Mathlib
`CategoryTheory.ObjectProperty.IsClosedUnderIsomorphisms`.

**Checks.**

- `IsAffinoidPerfectoid.spa_Cp` (computation) — `Spa(ℂ_p, O_{ℂ_p})` is affinoid perfectoid and its
  underlying space is a single point (the rank-one valuation).
- `affinoid_padic_nonexample` (non-example) — `Spa(ℚ_p, ℤ_p)` is an affinoid adic space
  (strongly noetherian, AdicSpaces layer 4.1) but not affinoid perfectoid: any isomorphism
  `Spa(ℚ_p, ℤ_p) ≅ Spa(R, R⁺)` gives `R ≅ O(Spa(ℚ_p, ℤ_p)) = ℚ_p`, which is not perfectoid (ECD
  Remark 3.3); a definition asking only for surjective Frobenius on the residue field would accept
  it.
- Degenerate: the empty adic space is affinoid perfectoid,
  presented by the zero ring with the zero plus ring (`0` is a pseudouniformiser of the zero ring
  and ECD Definition 3.1 holds vacuously); a definition requiring a nonzero ring or a base field
  fails this.
- `IsAffinoidPerfectoid.spa_ecd34iv` (computation) — for
  `R = ℤ_p^cycl[[T^{1/p^∞}]]⟨(p/T)^{1/p^∞}⟩[1/T]` with `ϖ = T^{1/p}` (ECD Example 3.4(iv)),
  `Spa(R, R°)` is affinoid perfectoid; no perfectoid base field is required.
- `spaHomEquiv_perfectoidField` (characterisation) — for a perfectoid field `K` and a complete
  perfectoid Tate pair `(S, S⁺)`, morphisms `Spa(S, S⁺) → Spa(K, K°)` are exactly the continuous
  ring maps `K → S` carrying `K°` into `S⁺`.
- Compatibility: `X` is affinoid perfectoid iff it
  is isomorphic to the Yoneda spectrum of a complete sheafy Huber pair (the image of the
  classical affinoid adic space under `ofAdicSpace`) and `O_X(X)` is a perfectoid Tate ring. For
  `A : CAff` with `[IsTateRing A.A]` and `hA : A.Sheafy`, the Check applies the characterisation
  of affinoid perfectoid spectra to identify this with `IsPerfectoidTateRing p A.A`.

`PerfectoidSpace.isAffinoidPerfectoid` asks for an isomorphism to the spectrum
`presentationLimitPreAdicSpace` of a `PairCat` object (`isAffinoidPerfectoidModel`). `ofPair`
constructs this spectrum in `Perfd`; `spaFunctor_map` specifies its action on morphisms, which on
the analytic carrier is AdicEtaleGeometry's sheafified Yoneda map. `affinoidPerfectoidEquiv`
and its comparison with `spaFunctor` give the anti-equivalence.

**Checks for the affinoid carrier and anti-equivalence.** `affinoid_spa_agreement` agrees with
the supplied spectrum; `affinoid_zero_pair` gives the empty spectrum;
`affinoid_padic_nonexample` excludes `Spa(ℚ_p, ℤ_p)`.
`affinoid_equivalence_empty`, `affinoid_equivalence_distinguishes_maps` and
`affinoid_equivalence_recovers_pair` test the empty object, two unequal morphisms and recovery
of the original complete pair.

For a perfectoid test space `Y`, `GlobalPairMap` is the type of continuous ring maps
`R → globalSections Y` carrying `R⁺` into `globalPlus Y`. `spaHomEquiv_apply` fixes the
correspondence as pullback followed by the inverse of the affinoid section equivalence.
**Checks.** `affine_mapping_empty` gives a unique map into the sections of the empty space;
`affine_mapping_recovers_pair` recovers the Huber-pair maps when `Y` is affinoid;
`affine_mapping_inverse_parameter` excludes the image of an inverse parameter from the
plus ring of any nonzero target chart.

For a classical `Y : TauCeti.AdicSpace`, `spaHomEquiv_adic` states the mapping equivalence
on AdicEtaleGeometry's Yoneda carrier, using the actual topological global sections and
`AdicEtaleGeometry.AdicSpace.Oplus Y ⊤`. Its value is the supplied
`YonedaAdicSpace.mapsFromAdicSpaceEquiv` applied to the underlying sheaf morphism
([Wedhorn] §8.2, Prop. 8.25, p. 80). This applies to every classical adic source,
without a perfectoid hypothesis. The supplier includes the topology and integral condition.

The comparison with the analytic carrier identifies affinoid perfectoidness of
`YonedaAdicSpace.ofAdicSpace.obj X`, for every
`X : TauCeti.AdicSpace`, with the existence of a perfectoid pair `R`, its sheafiness
proof `hR`, and a classical isomorphism `X ≅ (CAff.of R.pair).adicSpa hR`, that is, with
`isAffinoidPerfectoid` of the underlying pre-adic space.
It follows from the supplier's fully faithful embedding and `ofAdicSpace_spa`
([Berkeley] Prop. 3.2.2, printed p. 20, and §3.4, printed p. 22).
**Checks.** `adic_comparison_affinoid` tests a perfectoid pair;
`adic_comparison_empty` tests the empty classical source;
`adic_comparison_padic_nonexample` excludes the embedded spectrum of `ℚ_p`.

Lean: `PerfectoidSpace.isAffinoidPerfectoidModel`, `isAffinoidPerfectoid`,
`isAffinoidPerfectoid_presentationLimitPreAdicSpace`.

### P2.10 perfectoid spaces

Perfectoid spaces: the full subcategory `Perfd` of adic spaces, its characteristic-`p` part
`Perf`, and the glued tilt.

**Perfectoid spaces: the full subcategory Perfd of adic spaces, and Perf in characteristic p.**
(*perfectoid-space*) Fix a prime `p`. A *perfectoid space* is an adic space, that is, a sheafy
object `X` of Tau Ceti's `PreAdicSpace`, covered by open subspaces that are affinoid perfectoid
(see *affinoid-perfectoid-space*): every `x ∈ X` has an open neighbourhood `U` such that the open
subspace `X|_U` (`PreAdicSpace.restrict`) is isomorphic to `Spa(R, R⁺)` with `R` a perfectoid Tate
ring and `R⁺ ⊆ R°` a ring of integral elements ([ECD] §3, Def. 3.19, p. 18; [Berkeley] Lecture 7,
§7.1, Def. 7.1.2, printed p. 49). Sheafiness is part of the definition: without it the empty
pre-adic space with nonzero sections over `∅` would qualify, the chart condition being vacuous for
it. The category `Perfd` of perfectoid spaces is the full subcategory
of `PreAdicSpace` on these objects (Mathlib `ObjectProperty.FullSubcategory`):
morphisms are those of pre-adic spaces. Open immersions are `PreAdicSpace.IsOpenImmersion`; the
point carrier is the underlying space. Through AdicEtaleGeometry's `YonedaAdicSpace.ofAdicSpace`
these agree with the analytic carrier's open immersions `YonedaAdicSpace.IsOpenImmersion` and
points `YonedaAdicSpace.space` (`YonedaAdicSpace.ofAdicSpace_space`, and
*adic-spaces-in-yoneda-adic-spaces* for open immersions). `Perf ⊆ Perfd` is the full subcategory of perfectoid spaces of characteristic
`p`, those with `p = 0` in `O_X(X)` (in Lean, `InCharP` asks for `p = 0` in `O_X(U)` for every open
`U`, equivalent by `mem_perf_iff` below); equivalently, `X` admits a (necessarily unique) morphism to the
discrete adic space `Spa(𝔽_p, 𝔽_p)`, which is an adic space (AdicSpacesPartII R0.2
(*discrete-huber-pair-sheafy*)) but
not a perfectoid space, and which `Perf` does not contain ([ECD] §1, p. 3). With the
definition: every point of a perfectoid space is analytic, so every morphism of adic spaces into a
perfectoid space is adic (AdicSpacesPartII R0.6 (*adic-morphism*)); an adic space with `p = 0`
in its structure sheaf
is perfectoid iff it is covered by open subspaces `Spa(R, R⁺)` with `R` a perfect complete Tate
ring ([ECD] §3, Prop. 3.5, p. 15). In `PerfectoidSpace`, define `isPerfectoidSpace` (the object
property of `PreAdicSpace`: sheafy, and every point has an open neighbourhood `U` with `X|_U` affinoid
perfectoid), the category `Perfd` (its full subcategory, with hom sets those of pre-adic spaces),
`toAdicSpace` (the inclusion of `Perfd` into pre-adic spaces, landing in adic spaces, Mathlib
`ObjectProperty.ι`, fully faithful by `ObjectProperty.fullyFaithfulι`),
`ofSpa` (`Spa(R, R⁺)` as an object of `Perfd`), `InCharP` and `Perf` (the full subcategory on `p = 0` in
every ring of sections, fully faithful in `Perfd`); the proposed lemma names written
`IsPerfectoidSpace.*` below refer to the object property `isPerfectoidSpace`. Prove
`isAffinoidPerfectoid_le_isPerfectoidSpace` (affinoid perfectoid spaces are perfectoid spaces),
`isPerfectoidSpace_le_isAdic` (perfectoid spaces are adic spaces, their charts being sheafy by
`Perfectoid.isSheafyRing`), `IsPerfectoidSpace.of_iso` (closed under isomorphisms), `IsPerfectoidSpace.of_openCover` (covered
by perfectoid open subspaces implies perfectoid), `IsPerfectoidSpace.isAnalyticPoint` (every point
analytic, hence every `T → X` adic by `AdicSpace.isAdic_iff_image_analyticLocus`), `isPerfectoidSpace_restrict`
(open subspaces are perfectoid, *open-subspaces-of-perfectoid-spaces*),
`isBasis_affinoidPerfectoidOpens` (affinoid perfectoid opens form a basis,
*affinoid-perfectoid-basis*), `mem_perf_iff` (`X ∈ Perf` iff `p = 0` in `O_X(U)` for every open
`U` iff there is a unique morphism `X → Spa(𝔽_p, 𝔽_p)` iff every affinoid perfectoid open has a ring
of characteristic `p`) and the characteristic-`p` criterion (for `p = 0` in `O_X(X)`: perfectoid iff
covered by `Spa` of perfect complete Tate rings, *characteristic-p-affinoid-perfectoid-criterion*).
*Needs:* *affinoid-perfectoid-space*; layer P1 (P1.2); AdicSpacesPartII R0.2, R0.6
(*discrete-huber-pair-sheafy*, *adic-morphism*,
*adic-iff-analytic-locus*: adic morphisms, finiteness classes, locally noetherian spaces); Tau Ceti
`TauCeti.ValuationSpectrum.spaAnalytic_eq_spa_of_isTateRing`,
`TauCeti.ValuationSpectrum.IsAnalyticPoint`; Mathlib `CategoryTheory.ObjectProperty.FullSubcategory`,
`CategoryTheory.ObjectProperty.ι`, `CategoryTheory.ObjectProperty.fullyFaithfulι`,
`CategoryTheory.ObjectProperty.IsClosedUnderIsomorphisms`, `CharP`.

**Checks.**

- `perfectoid_empty` (degenerate) — the empty adic space is a perfectoid space (covered by
  the empty family); an empty pre-adic space is perfectoid exactly when it is sheafy, its sections
  over `∅` vanishing, the chart condition being vacuous (a Lean `example`).
- `perfectoid_padic_nonexample` (non-example) — `Spa(ℚ_p, ℤ_p)` is not a perfectoid space: its only
  nonempty open subset is the whole space, whose ring `ℚ_p` is not perfectoid (ECD Remark 3.3).
- `not_isPerfectoidSpace_closedUnitDisc_Qp` (non-example) — the closed unit disc
  `Spa(ℚ_p⟨T⟩, ℤ_p⟨T⟩)` is not a perfectoid space: the classical point `T = 0` has completed residue
  field `ℚ_p`, discretely valued and not perfectoid, whereas every completed residue field of a
  perfectoid space is a perfectoid field (*completed-residue-fields*; Kedlaya AWS
  Corollary 2.5.7).
- `not_isPerfectoidSpace_spa_Fp` (non-example) — the discrete `Spa(𝔽_p, 𝔽_p)` is an adic space of
  characteristic `p` (AdicSpacesPartII R0.2 (*discrete-huber-pair-sheafy*)) and is not
  perfectoid: its unique point has open support, so it is not
  analytic; a construction treating `Spa 𝔽_p` as a perfectoid Tate space, e.g. as a final object of
  `Perf`, is wrong.
- Computation: gluing two copies of the perfectoid
  closed disc `Spa(ℂ_p⟨T^{1/p^∞}⟩, O_{ℂ_p}⟨T^{1/p^∞}⟩)` along the open complement of `T = 0`
  (AdicSpaces layer 5.3) gives a perfectoid space which is quasi-compact but not quasi-separated:
  the two charts meet in a non-quasi-compact open.

For every `X : TauCeti.AdicSpace`, the comparison with the analytic carrier says that
`YonedaAdicSpace.ofAdicSpace.obj X` is perfectoid in the Yoneda sense exactly when each
`x : X.obj` lies in the image of a classical open immersion from the sheafy spectrum
of a perfectoid pair. This is the classical covering definition of [ECD] §3,
Def. 3.19, p. 18, transported along the fully faithful comparison of [Berkeley] §3.4,
printed p. 22; preservation of points and open immersions identifies the covering condition.
The empty covering is allowed. Thus the Yoneda predicates also express the stated
classical notions, which are the Lean definitions, without identifying the two ambient categories by
definition.

`PerfectoidSpace.isPerfectoidSpace` asks for sheafiness and open affinoid perfectoid charts
covering every point; `Perfd` is its full subcategory of `PreAdicSpace`. `InCharP` requires `p = 0`
in every ring of sections, equivalently (by `mem_perf_iff`) in every perfectoid affinoid chart, and
`Perf` is the corresponding full subcategory.
**Checks.** `perfectoid_empty`, `perfectoid_affinoid` and `perfectoid_padic_nonexample` test
empty, perfectoid affinoid and nonperfectoid field spectra. `characteristic_p_tilt`,
`characteristic_p_empty` and `characteristic_p_mixed_nonexample` test a tilted chart, the
empty space and a characteristic-zero chart. The last condition excludes the zero ring
through `CharZero`, rather than imposing nonemptiness on all spaces.

Lean: `PerfectoidSpace.isPerfectoidSpace`, `Perfd`, `InCharP`, `Perf`,
`isAffinoidPerfectoid_le_isPerfectoidSpace`, `isPerfectoidSpace_le_isAdic`,
`isPerfectoidSpace_restrict`, `isBasis_affinoidPerfectoidOpens`.

### P2.11 the glued tilt

P2.11 owns glued tilting on `PerfectoidSpace.Perfd`, including `tiltFunctor`,
`tiltHomeomorph`, the affinoid and restriction comparisons, and
`sliceTiltFunctor_isEquivalence`. AdicSpacesPartIV R5 consumes `PerfectoidSpace.tiltFunctor`
directly, downstream; P2 uses no subsection of AdicSpacesPartIV.

**The glued tilt `X ↦ X♭` of a perfectoid space.** (*perfectoid-spaces-and-glued-tilting*) For every
perfectoid space `X` (*perfectoid-space*; any characteristic, no base field) there is a perfectoid
space `X♭` of characteristic `p`, the *tilt* of `X`, with a homeomorphism `|X| ≅ |X♭|`, `x ↦ x♭`,
constructed by gluing ([ECD] §3, the sentence after Def. 3.19, p. 18; [Berkeley] Lecture 7, §7.1,
after Rem. 7.1.3, printed p. 49, the target `Perf`; [Sch12] §6, Def. 6.16 and Prop. 6.17 with
proof, p. 38, over a perfectoid field, where property (a) is "a formal consequence" of tilting of
algebras, of rational localisations and of maps into affinoid adic spaces; [Bhatt] §9.3,
Thm 9.3.10(1), printed p. 111). Affinoid case: for `X = Spa(R, R⁺)` with `R` perfectoid,
`X♭ := Spa(R♭, R♭⁺)`, where `(R♭, R♭⁺)` is the tilt of the pair (layer P1: `R♭ = lim_{x ↦ x^p} R`,
`R♭⁺ = lim_{x ↦ x^p} R⁺`), and `x ↦ x♭` is the tilting homeomorphism `|f(x♭)| = |f♯(x)|`
(*tilting-map-of-spectra*, *tilting-homeomorphism-and-rational-subsets*), which matches rational
subsets `U ↔ U♭` and satisfies `(O_{X♭}(U♭), O⁺_{X♭}(U♭)) ≅ (O_X(U), O⁺_X(U))♭` compatibly with
restrictions (see *rational-localization-of-perfectoid-affinoids*). General case: `X♭` is glued
(AdicSpaces layer 5.3) from the tilts `V♭` of all affinoid perfectoid open subspaces `V ⊆ X`, along
the tilts of their common rational refinements (*affinoid-perfectoid-basis* (iii)); the family being
canonical, `X♭` is defined up to unique isomorphism, and `|X♭| = |X|` as sets with the same open
subsets. A morphism `f : Y → X` of perfectoid spaces gives `f♭ : Y♭ → X♭`, affinoid-locally `Spa` of
the tilt of the corresponding morphism of perfectoid Tate pairs (layer P1, P1.6), glued because
morphisms of adic spaces form a sheaf; so `X ↦ X♭` is a functor `Perfd ⥤ Perf` with `|f♭| = |f|`
under the homeomorphisms. Properties: (a) for `U ⊆ X` open, `(X|_U)♭ = X♭|_{U♭}`, `U♭` being `U`
viewed in `|X♭|`; `U` is affinoid perfectoid iff `U♭` is, and then
`(O_{X♭}(U♭), O⁺_{X♭}(U♭)) ≅ (O_X(U), O⁺_X(U))♭`; (b) the sharp maps glue to a continuous
multiplicative map of sheaves of topological monoids `O_{X♭} → O_X` on `|X| = |X♭|`, `f ↦ f♯`, with
`|f♯(x)| = |f(x♭)|`, carrying `O⁺_{X♭}` into `O⁺_X`; it is not additive; (c) for `X` of
characteristic `p`, projection to the zeroth coordinate `R♭ → R` is a ring isomorphism, giving a
natural isomorphism `X♭ ≅ X`; (d) `X` is not determined by `X♭` (the tilt forgets the untilt); the
equivalence of slice categories, with the description of maps into `X♭` by untilts, is
*tilting-slice-equivalence*. In `PerfectoidSpace`, define `tilt` as the functor `Perfd ⥤ Perf`,
`X ↦ X♭`, `f ↦ f♭`; prove `tiltHomeomorph` (`|X| ≃ₜ |X♭|`, Mathlib `Homeomorph`, natural in `X`:
`|f♭| ∘ h_Y = h_X ∘ |f|`; on affinoids the tilting homeomorphism), `tiltSpaIso`
(`(Spa(R, R⁺))♭ ≅ Spa(R♭, R♭⁺)`, natural in morphisms of complete perfectoid Tate pairs),
`tiltSectionsIso` (for `U ⊆ X` affinoid perfectoid, `(O_{X♭}(U♭), O⁺_{X♭}(U♭)) ≅ (O_X(U), O⁺_X(U))♭`
as topological rings with plus rings, compatibly with restriction to affinoid perfectoid `U' ⊆ U`),
the statement that `U` is affinoid perfectoid iff `U♭` is, `tiltRestrictIso`
(`(X|_U)♭ ≅ X♭|_{U♭}`; the tilt maps open immersions to open immersions and open covers to open
covers), `globalSharp` (the map of sheaves of topological monoids (b)), `tiltCharPIso` (for
`X ∈ Perf` a natural isomorphism `X♭ ≅ X`, i.e. `tilt` restricted to `Perf` is isomorphic to the
identity), `tiltSpaIso_naturality` (simp: under `tiltSpaIso`, the tilt of `Spa(φ)` is `Spa(φ♭)`),
`tiltHomeomorph_image_rationalSubset` (for affinoid perfectoid `U = Spa(R, R⁺) ⊆ X` the
homeomorphism carries `U(f_1♯, …, f_n♯ / g♯)` to `U(f_1, …, f_n / g)`, `f_i, g ∈ R♭`, and every
rational subset of `U♭` arises so) and `sliceTiltFunctor_isEquivalence` (the equivalence `Perfd/X ≃ Perf/X♭`,
*tilting-slice-equivalence*). *Needs:* *perfectoid-space*; *affinoid-perfectoid-space*;
*affinoid-perfectoid-basis*; *open-subspaces-of-perfectoid-spaces*; *tilting-map-of-spectra*;
*tilting-homeomorphism-and-rational-subsets*; *rational-localization-of-perfectoid-affinoids*;
layer P1 (P1.5, P1.6); Tau Ceti `TauCeti.ValuationSpectrum.spa`; Mathlib
`CategoryTheory.GlueData`, `Homeomorph`.

**Checks.**

- `tilt_spa_cyclotomic` (computation) — `Spa(ℚ_p^cycl, ℤ_p^cycl)♭ ≅ Spa(𝔽_p((t^{1/p^∞})), 𝔽_p[[t^{1/p^∞}]])`, `t = (1, ζ_p, ζ_{p²}, …) − 1` (Berkeley Example 6.2.4),
  and the homeomorphism is the identity of the one-point space.
- `tilt_spa_perfectedDisc` (computation) — for a perfectoid Tate ring `R`:
  `Spa(R⟨T^{1/p^∞}⟩, R⁺⟨T^{1/p^∞}⟩)♭ ≅ Spa(R♭⟨T^{1/p^∞}⟩, R♭⁺⟨T^{1/p^∞}⟩)` with
  `T ↦ (T, T^{1/p}, …)` (Kedlaya AWS Example 2.3.15).
- `tilt_empty` (degenerate) — the tilt of the empty perfectoid space is empty.
- `tilt_characteristic_p` (compatibility) — for `X ∈ Perf` the natural isomorphism `X♭ ≅ X` holds,
  e.g. `Spa(𝔽_p((t^{1/p^∞})))♭ ≅ Spa(𝔽_p((t^{1/p^∞})))` via the zeroth coordinate.
- `tilt_not_injective_on_iso_classes` (non-example) — `Spa(ℚ_p^cycl, ℤ_p^cycl)` and
  `Spa(𝔽_p((t^{1/p^∞})), 𝔽_p[[t^{1/p^∞}]])` are not isomorphic (different characteristic) but have
  isomorphic tilts; so `tilt : Perfd ⥤ Perf` is not an equivalence, and a construction claiming to
  recover `X` from `X♭` without a chosen base is wrong.
- `tiltHomeomorph_rationalSubset` (characterisation) — for `X = Spa(R, R⁺)` perfectoid and
  `f_1, …, f_n, g ∈ R♭` generating the unit ideal, the tilting homeomorphism maps
  `U(f_1♯, …, f_n♯ / g♯)` onto `U(f_1, …, f_n / g)`.

**Rational subsets of affinoid perfectoid opens are affinoid perfectoid, and affinoid perfectoid
opens form a basis.** (*affinoid-perfectoid-basis*) Let `X` be a perfectoid space. Prove: (i) if
`V ⊆ X` is an open subspace with an isomorphism `V ≅ Spa(R, R⁺)`, `R` perfectoid, then for every
rational subset `W = R(T/s) ⊆ Spa(R, R⁺)` (Tau Ceti `ValuationSpectrum.rationalSubset`, `T·R` open;
for `R` Tate, `T` generates the unit ideal) the open subspace `W ⊆ X` is affinoid perfectoid:
`W ≅ Spa(R⟨T/s⟩, R⟨T/s⟩⁺)`, the rational localisation of AdicSpaces layer 3.1, a perfectoid Tate
pair (see *rational-localization-of-perfectoid-affinoids*), and
`(O_X(W), O_X⁺(W)) ≅ (R⟨T/s⟩, R⟨T/s⟩⁺)` compatibly with restriction from `V`
(see *perfectoid-pair-rational-restriction*); (ii) the affinoid perfectoid open subspaces of `X` form a
basis of the topology of `|X|`, and every quasi-compact open subset of `X` is a finite union of
them; (iii) for affinoid perfectoid opens `V, V' ⊆ X` and `x ∈ V ∩ V'` there is an open `W` with
`x ∈ W ⊆ V ∩ V'` that is a rational subset of `V` and of `V'`, hence affinoid perfectoid by (i);
(iv) for every affinoid perfectoid open `V ⊆ X`, `O_X(V)` is a perfectoid Tate ring and
`V ≅ Spa(O_X(V), O_X⁺(V))` canonically. The intersection of two affinoid perfectoid opens need not
be affinoid, nor quasi-compact ([Sch12] §6, proof of Prop. 6.17, p. 38, open subspaces and their
global sections, giving (iv); [Wedhorn] §8.2, Rem. 8.23(3), p. 80, part (iii); [ECD] §3, Thm 3.18,
p. 18, part (i)). For the perfectoid closed disc `X = Spa(ℂ_p⟨T^{1/p^∞}⟩, O_{ℂ_p}⟨T^{1/p^∞}⟩)`, the
annulus `{|p| ≤ |T| ≤ 1} = R({p, T}/T)` is affinoid perfectoid with ring
`ℂ_p⟨T^{1/p^∞}, (p/T)^{1/p^∞}⟩`; in the perfectoid disc with doubled origin
(the doubled-origin computation of P2.10) the two charts are affinoid perfectoid and meet in a
non-quasi-compact open. *Needs:* *perfectoid-space*; *affinoid-perfectoid-space*;
*rational-localization-of-perfectoid-affinoids*; *perfectoid-pair-rational-restriction*; Tau Ceti
`TauCeti.ValuationSpectrum.rationalSubset`,
`TauCeti.ValuationSpectrum.isTopologicalBasis_spaRationalFamily`,
`TauCeti.ValuationSpectrum.isCompact_of_mem_spaRationalFamily`,
`TauCeti.ValuationSpectrum.spaLocalizationHomeomorph`.

**Open subspaces of perfectoid spaces are perfectoid; open immersions in Perfd.**
(*open-subspaces-of-perfectoid-spaces*) Let `X` be a perfectoid space and `U ⊆ |X|` open. The
affinoid case is *rational-localization-of-perfectoid-affinoids* together with the library fact
that a rational subset is the open affinoid subspace `Spa` of its localisation
(`TauCeti.ValuationSpectrum.spaLocalizationHomeomorph`); the gluing is what is proved here. Prove
that the open adic subspace `X|_U = (U, O_X|_U, (v_x)_{x ∈ U})` of AdicSpaces layer 5.1
([Wedhorn] §8.2, Rem. 8.23(2), p. 80) is a perfectoid space, and that the inclusion
`j : X|_U → X` is an open immersion of perfectoid spaces (Tau Ceti `PreAdicSpace.IsOpenImmersion`)
with the universal property in `Perfd` and in adic spaces: a morphism `f : Y → X` from an adic space `Y` factors through `j`, uniquely, iff
`f(|Y|) ⊆ U`. Conversely: (a) the source of an open immersion of adic spaces into a perfectoid space
is perfectoid; (b) an adic space covered by open subspaces that are perfectoid spaces is
perfectoid; (c) for open `U' ⊆ U`, `(X|_U)|_{U'} = X|_{U'}`. Consequently `Perfd` is closed under
open subspaces, the open immersions in `Perfd` are AdicSpaces' open immersions between perfectoid
spaces, and `Perf` is closed under open subspaces ([Sch12] §6, proof of Prop. 6.17, p. 38). For
`X = Spa(𝔽_p((t^{1/p^∞}))⟨T^{1/p^∞}⟩)` the punctured disc `{T ≠ 0}` is an open perfectoid subspace
in `Perf`, not quasi-compact and so not affinoid; a morphism
`Spa(ℂ_p, O_{ℂ_p}) → Spa(ℂ_p⟨T^{1/p^∞}⟩, O_{ℂ_p}⟨T^{1/p^∞}⟩)`, `T^{1/p^n} ↦ a^{1/p^n}` for a
compatible system of `p`-power roots of `a ∈ O_{ℂ_p}`, factors through the annulus
`{|p| ≤ |T| ≤ 1}` iff `|p| ≤ |a|`. *Needs:* *perfectoid-space*; *affinoid-perfectoid-basis*.

`PerfectoidSpace.tiltFunctor`, `tiltSpaIso` and `tiltSpaIso_naturality` identify the glued
functor with affinoid tilting. `tiltHomeomorph_spa` identifies its underlying map with the
valuation-defined `spaTilt`, using the natural affinoid coordinates `spaSpaceEquiv`.
The restriction specification fixes the open embedding and its exact image; `tiltRestrictIso_inclusion`
fixes the comparison on open subspaces by their inclusions. `isBasis_affinoidPerfectoidOpens`
asserts the basis property without a quasi-separatedness assumption.

**Checks for global tilting and opens.** `tilt_empty`, `tilt_characteristic_p` and
`tilt_affinoid_valuation` test the empty tilt, characteristic-p agreement and the strict
valuation order `0 < 1`. `restriction_empty`, `restriction_whole` and `restriction_proper`
give the empty space, an isomorphism, and a non-isomorphism. `spa_space_empty`,
`spa_space_whole` and `spa_space_map_coordinate` test the affinoid coordinates on empty and
whole subsets and on pullback of a valuation.

The global sharp map uses `PerfectoidSpace.Chart`, an open affinoid chart with its specified
inclusion. The primary carrier of global functions is the structure sheaf of the underlying
`PreAdicSpace`: `globalSections X` is a reducible abbreviation of `Γ(X, O_X)`, the global sections
of Tau Ceti's structure presheaf of `X.obj`, and `globalPlus X` of its integral subring
`Γ(X, O⁺_X)`; the description of `Γ(X, O_X)` as the subring of the product of the chart rings cut
out by all restriction equations, with the chart plus-subring conditions for `Γ(X, O⁺_X)`, is a
comparison lemma (`globalSections_eq_compatibleFamilies`), and the family type used to prove it is
auxiliary. The topology on `Γ(X, O_X)` is the one induced by the chart restriction maps; no second
ring is introduced to carry it. `globalSectionsAffinoidEquiv_spec` recovers the affinoid ring by
evaluation on its identity chart and recovers its exact plus ring; for affinoids this comparison
fixes the supplied `toPresentationLimit` sections, and for arbitrary opens it commutes with
restriction. The sharp map, pullbacks, integral conditions and the Čech and cohomology statements
below are stated on these canonical carriers. `globalPullback_chart` fixes pullback on chart maps.
`tiltChart` uses the tilted inclusion, and `globalSharp_chart` applies sharp to the
corresponding tilted chart component. `globalSharp_naturality` includes restriction to
opens, and `globalSharp_plus` gives the integral map.

**Checks for global sections and sharp.** `global_sections_empty` gives the zero ring;
`global_sections_affinoid` recovers the affinoid ring; `global_plus_inverse_parameter`
excludes the inverse parameter while `global_plus_two_integral_terms` includes a sum of
integral elements. `global_sharp_zero_one` computes zero and one,
`global_sharp_two_factors` computes a product, and `global_sharp_nonadditive` at `p = 2`
compares sharp of `1 + 1 = 0` on the tilt with `1 + 1 ≠ 0` on a characteristic-zero chart.
`global_pullback_identity`, `global_pullback_composition` and
`global_pullback_affine_two_terms` fix the contravariant convention.

### P2.12 the tilting equivalence of perfectoid spaces

The tilting equivalence of perfectoid spaces: the slice equivalence over a perfectoid base, its
affinoid form, and the completed tensor products and affinoid fibre products that it is compatible
with.

**Tilting equivalence of perfectoid spaces: `Perfd/X ≃ Perf/X♭`.** (*tilting-slice-equivalence*) Let
`X` be a perfectoid space with tilt `X♭` (*perfectoid-spaces-and-glued-tilting*; any characteristic,
no perfectoid base field). (a) The functor `Perfd/X → Perf/X♭ = Perfd/X♭` on slice categories
(Mathlib `CategoryTheory.Over`), `(f : Y → X) ↦ (f♭ : Y♭ → X♭)`, is an equivalence of categories (a
Mathlib `CategoryTheory.Equivalence`). (b) It commutes with restriction to open subspaces of `X`, and
a perfectoid space `Y` over `X` is affinoid perfectoid iff `Y♭` is; for `X = Spa(A, A⁺)` it restricts
on affinoid perfectoid spaces to the equivalence of
*affinoid-perfectoid-pairs-tilting-equivalence*. (c) Untilts over `X`: for every perfectoid space
`T` of characteristic `p` and every morphism `g : T → X♭` there are a perfectoid space `T♯` with a
morphism `T♯ → X` and an isomorphism `ι : (T♯)♭ ≅ T` over `X♭`, and the pair `(T♯ → X, ι)` is unique
up to unique isomorphism; so `Hom_Perf(T, X♭)` is in natural bijection with isomorphism classes of
such pairs, functorially in `T`. (d) Under the homeomorphisms `|Y| ≅ |Y♭|` the structure maps
correspond. Unit and counit are isomorphisms compatible with restriction to open subspaces of `X`.
The quasi-inverse depends on `X`: there is no quasi-inverse to tilting that recovers every untilt without a chosen
base (`tilt_not_injective_on_iso_classes`) ([ECD] §3, Cor. 3.20, p. 19, statement (a) over an
arbitrary perfectoid base; [Berkeley] Lecture 7, §7.1, Thm 7.1.4, printed pp. 49–50; [Sch12] §6,
Prop. 6.17, p. 38, the field case; [Bhatt] §9.3, Thm 9.3.10(2), printed p. 111). Sch12
Proposition 6.17 is the case `X = Spa(K, K⁺₀)` for a perfectoid field `K`,
where `K⁺₀` is the smallest ring of integral elements (P2.19)
(see *perfectoid-spaces-over-a-perfectoid-field-comparison*). For `X = Spa(ℚ_p^cycl, ℤ_p^cycl)`, the
perfectoid disc over `𝔽_p((t^{1/p^∞}))` untilts to `Spa(ℚ_p^cycl⟨T^{1/p^∞}⟩, ℤ_p^cycl⟨T^{1/p^∞}⟩)`,
and `Spa(𝔽_p((t^{1/p^∞})))` itself (with the identity to `X♭`) untilts to `X`. *Needs:*
*perfectoid-spaces-and-glued-tilting*; *affinoid-perfectoid-basis*;
*open-subspaces-of-perfectoid-spaces*; *affinoid-perfectoid-pairs-tilting-equivalence*; Mathlib
`CategoryTheory.Over`, `CategoryTheory.Equivalence`.

**Tilting is an equivalence between perfectoid Tate pairs over `(A, A⁺)` and over `(A♭, A♭⁺)`,
compatible with rational localisation.** (*affinoid-perfectoid-pairs-tilting-equivalence*) Let
`(A, A⁺)` be a complete perfectoid Tate pair (`A` perfectoid Tate ring, `A⁺ ⊆ A°` a ring of integral
elements; no perfectoid base field) with tilt `(A♭, A♭⁺)`. Let `Perfd_{(A,A⁺)}` be the category of
complete perfectoid Tate pairs `(B, B⁺)` with a morphism of Huber pairs `(A, A⁺) → (B, B⁺)`,
morphisms being morphisms of Huber pairs under `(A, A⁺)`, and define `Perfd_{(A♭,A♭⁺)}` likewise.
The tilting equivalence of perfectoid `A`-algebras, with quasi-inverse the untilt along a
generator `ξ` of `ker(θ : W(A♭°) → A°)`, is P1.27 (*tilting-equivalence-and-explicit-tilt*); what
is proved here is its form for pairs under an arbitrary `(A, A⁺)` and its compatibility with
rational localisation: the tilt `(B, B⁺) ↦ (B♭, B♭⁺)` is an equivalence
`Perfd_{(A,A⁺)} ≃ Perfd_{(A♭,A♭⁺)}` (plus rings correspond by *rings-of-integral-elements-under-tilting*,
P1.18), with quasi-inverse sending `(S, S⁺)` to the untilt `(W(S⁺)[[ϖ♭]⁻¹]/ξ, W(S⁺)/ξ)` along a
generator `ξ` of `ker(θ : W(A♭⁺) → A⁺)`, `ϖ♭ ∈ A♭` a pseudouniformiser (ECD Lemma 3.14 and Theorem
3.17; layer P1); it carries rational localisations to rational localisations: for `f_1, …, f_n, g ∈ B♭`
generating the unit ideal, `(B⟨f♯/g♯⟩)♭ ≅ B♭⟨f/g⟩` over `B♭`, and every rational localisation of `B`
arises so; and that through `Spa` (see *affinoid-perfectoid-space*) it gives an equivalence between
affinoid perfectoid spaces over `Spa(A, A⁺)` and over `Spa(A♭, A♭⁺)`, compatible with the tilting
homeomorphisms ([ECD] §3, Thm 3.13 and its proof, p. 17, the general-base tilting equivalence of
rings, upgraded here to pairs under `(A, A⁺)`, and Thm 3.17, p. 18, the explicit quasi-inverse;
[AWS] §2.3, Rem. 2.3.17, p. 56, the relative equivalence over an arbitrary perfectoid base;
[Sch12] §6, Lem. 6.2, p. 30, the field case with plus rings). For `(A, A⁺) = (K, K°)` a perfectoid
field this is Scholze's Lemma 6.2 (for pairs whose plus ring contains `K°`); for
`(A, A⁺) = (ℚ_p^cycl, ℤ_p^cycl)`, `(𝔽_p((t^{1/p^∞}))⟨T^{1/p^∞}⟩, 𝔽_p[[t^{1/p^∞}]]⟨T^{1/p^∞}⟩)`
untilts to `(ℚ_p^cycl⟨T^{1/p^∞}⟩, ℤ_p^cycl⟨T^{1/p^∞}⟩)`. *Needs:* *affinoid-perfectoid-space*;
*rational-localization-of-perfectoid-affinoids*; layer P1 (P1.6, P1.19, P1.27); Tau Ceti
`TauCeti.Huber.Pair`, `TauCeti.Huber.Pair.Hom`.

`PerfectoidSpace.sliceTiltFunctor` sends an arrow and its commuting triangles to their
tilts; `sliceTiltFunctor_isEquivalence` gives the equivalence of slices in `Perfd`.
Every space mapping to the characteristic-p base has characteristic p, giving the stated
`Perf` slice. Pair-level tilting is already `Perfectoid.PairCat.sliceTiltingEquivalence`
in P1, with its specified `sliceTiltFunctor`; no second pair equivalence is defined here.
**Checks.** `slice_identity` computes the terminal arrow; `slice_affinoid_arrow` fixes the
forward arrow; `slice_fibre_product` compares a product in the slice. The P1 Checks
`slice_base`, `slice_characteristic_p_ideal` and `slice_zero_target` retain the base-marked
untilt and zero-ring tests.

### P2.13 completed tensor products of perfectoid pairs

**Completed tensor products of perfectoid Tate pairs are perfectoid, uniform, and commute with
tilting.** (*completed-tensor-of-perfectoid-is-perfectoid*) Let `(A, A⁺) → (B, B⁺)` and
`(A, A⁺) → (C, C⁺)` be morphisms of complete Hausdorff Huber pairs (Tau Ceti `Huber.Pair.Hom`) with
`A, B, C` perfectoid Tate rings (ECD Definition 3.1; any characteristic; no perfectoid base field).
Both maps are adic (AdicSpacesPartII R0.1 (*adic-ring-homomorphism*)), so the completed tensor
product
`(D, D⁺) = B ⊗̂_A C` with `inl, inr` (AdicSpacesPartII R0.3 (*completed-tensor-product*); `D⁺`
the closure of the integral closure of the image of
`B⁺ ⊗_{A⁺} C⁺`) is defined. Then: (i) `D` is a perfectoid Tate ring; in particular `D` is uniform and
the uniformisation `D → B ⊗̂ᵘ_A C` (AdicSpacesPartII R0.4 (*uniform-completed-tensor-product*))
is an isomorphism (the tensor-product norm is the
spectral norm, [KL16] §3.3, Thm 3.3.13(b), p. 62); (ii) `(D, D⁺)` with `inl, inr` is a pushout of
`(B, B⁺) ← (A, A⁺) → (C, C⁺)` in the category of complete perfectoid Tate pairs — given (i), this
is the universal property of AdicSpacesPartII R0.3
(*completed-tensor-product-universal-property*,
for complete Hausdorff targets with closed integrally closed plus ring) restricted to perfectoid
targets; (iii) tilting
commutes with it: there is a unique isomorphism of Huber pairs
`(D♭, D♭⁺) ≅ (B♭, B♭⁺) ⊗̂_{(A♭, A♭⁺)} (C♭, C♭⁺)` whose composites with `inl♭, inr♭` are the structure maps of `B♭ ⊗̂_{A♭} C♭`
([AWS] §2.4, Thm 2.4.1, p. 57; [KL16] Thm 3.3.13(a)); (iv) for a pseudouniformiser `ϖ = (ϖ♭)♯ ∈ A`
with `ϖ^p | p` in `A°` and compatible `p`-power roots (layer P1, P1.3), the `ϖ`-adic completion `F̂`
of the image `F` of `B° ⊗_{A°} C°` in `B ⊗_A C` satisfies `ϖ^{1/p^n} D° ⊆ F̂ ⊆ D°` for all `n`, and
the `ϖ`-power torsion of `B° ⊗_{A°} C°` is killed by every `ϖ^{1/p^n}`; so `F̂ → D°` is an almost
isomorphism (layer P1), which over a perfectoid field `K` (the case `A = K`) is Scholze's
statement that `B°ᵃ ⊗̂_{A°ᵃ} C°ᵃ` is a perfectoid `K°ᵃ`-algebra ([Sch12] §6, proof of Prop. 6.18,
p. 38, the affinoid formula and the plus-ring convention). The perfectoid base `A` is essential:
`ℂ_p ⊗̂_{ℚ_p} ℂ_p` is not uniform (recorded in AdicSpacesPartII R0.4
(*uniform-completed-tensor-product*)) ([KL16] §3.3, Rem. 3.3.14, p. 62: the earlier
`ℚ_p`-version [KL15] §3.6, Prop. 3.6.11, p. 92, had an incomplete proof, repaired by
Thm 3.3.13). The perfectoidness target for `B ⊗̂_A C` uses `IsAdicHom` for both structure maps,
provided by AdicSpacesPartII R0.1 (*adic-ring-homomorphism*).
`B ⊗̂_A A⟨T^{1/p^∞}⟩ ≅ B⟨T^{1/p^∞}⟩` for every perfectoid Tate pair `(B, B⁺)` under
`(A, A⁺)`, compatibly with tilting, so `A⟨T^{1/p^∞}⟩ ⊗̂_A A⟨S^{1/p^∞}⟩ ≅ A⟨T^{1/p^∞}, S^{1/p^∞}⟩`;
for a finite Galois extension `L/K` of perfectoid fields with group `G`, `L ⊗̂_K L ≅ ∏_{g ∈ G} L`
(already `L ⊗_K L ≅ ∏_G L` algebraically, and the product is complete), a perfectoid Tate ring
that is not a field; that `L` is itself perfectoid is layer P3 (P3.7), cited for the example only
and not an input to this target. *Needs:* AdicSpacesPartII layer
R0 (*adic-ring-homomorphism*, *completed-tensor-product*,
*completed-tensor-product-universal-property*, *uniformization*, *uniform-completed-tensor-product*:
the completed tensor product); *perfect-uniform-completed-tensor*;
*tilting-homeomorphism-and-rational-subsets*; layer P1 (P1.3, P1.6, P1.27); layer P0 (P0.5,
P0.16); Tau Ceti `TauCeti.Huber.Pair`, `TauCeti.Huber.Pair.Hom`,
`TauCeti.Huber.Pair.isRingOfIntegralElements_integralClosure`; Mathlib `UniformSpace.Completion`,
`integralClosure`.

**Characteristic p: the completed tensor product of perfect complete Tate rings is perfect,
uniform and perfectoid.** (*perfect-uniform-completed-tensor*) Let `(A, A⁺) → (B, B⁺)` and
`(A, A⁺) → (C, C⁺)` be morphisms of complete Hausdorff Huber pairs with `A, B, C` perfectoid Tate
rings of characteristic `p`, equivalently perfect complete Tate rings (ECD Proposition 3.5). Both
maps are adic (AdicSpacesPartII R0.1 (*adic-ring-homomorphism*)), so `(D, D⁺) = B ⊗̂_A C` of
AdicSpacesPartII R0.3 (*completed-tensor-product*) is defined. Let `ϖ ∈ A` be a
pseudouniformiser, `F ⊆ B ⊗_A C` the image of `B° ⊗_{A°} C°` (a ring of definition for R0's
topology, because `ϖB° ⊆ B⁺ ⊆ B°` and likewise for `C`) and `F̂ ⊆ D` its `ϖ`-adic completion.
Prove: (i) `D` and `F̂` are perfect rings; (ii) `D` is a perfect complete Tate ring, hence uniform
and perfectoid, and the uniformisation `D → B ⊗̂ᵘ_A C` (AdicSpacesPartII R0.4
(*uniform-completed-tensor-product*)) is an isomorphism; (iii)
`ϖ^{1/p^n} D° ⊆ F̂ ⊆ D°` for all `n ≥ 0`, so `ϖD° ⊆ F̂` and `F̂ → D°` is injective with cokernel
killed by every `ϖ^{1/p^n}`, an almost isomorphism for the almost setup of `A` (layer P1); (iv) the
`ϖ`-power torsion of `B° ⊗_{A°} C°` is killed by every `ϖ^{1/p^n}` ([KL15] §3.1, Rem. 3.1.6(c),
p. 67, (i)–(ii) for perfect uniform Banach rings, perfectness being what makes the completed tensor
product uniform; [Bhatt] §9.3, proof of Cor. 9.3.13, printed p. 111; [Sch12] §6, proof of
Prop. 6.18, p. 38, statement (iv)). For a perfectoid field `K` of characteristic `p`,
`K⟨T^{1/p^∞}⟩ ⊗̂_K K⟨S^{1/p^∞}⟩ = K⟨T^{1/p^∞}, S^{1/p^∞}⟩` with `F̂ = K°⟨T^{1/p^∞}, S^{1/p^∞}⟩`;
without perfectness the conclusion fails: `𝔽_p((t))⟨T⟩ ⊗̂_{𝔽_p((t))} 𝔽_p((t))⟨S⟩` is uniform but not
perfect. *Needs:* AdicSpacesPartII R0.1, R0.3, R0.4 (*adic-ring-homomorphism*, *completed-tensor-product*,
*uniformization*, *uniform-completed-tensor-product*); layer P1; Tau Ceti
`TauCeti.Huber.Pair.Hom`, `TauCeti.Huber.powerBoundedSubring`; Mathlib `PerfectRing`,
`Algebra.TensorProduct.includeLeft`, `UniformSpace.Completion`.

`Perfectoid.PairCat.tensorPair` uses the supplied completed tensor ring and plus pair,
with scalar structures determined by the given pair morphisms. `tensorInclusions_spec`
identifies the inclusions with the actual pure tensors. `tensorPair_isPushout` supplies
the pair-level universal property; `tensorTiltIso_inclusions` identifies both tilted
inclusions under the tensor comparison.
**Checks.** `tensor_pair_zero`, `tensor_pair_identity` and `tensor_pair_two_inclusions`
test a zero factor, an identity base change and equality of the two base images.
`tensor_tilt_zero`, `tensor_tilt_two_constants` and `tensor_tilt_left_coordinate` test the
zero tensor, zero and one, and a coordinate from the left factor.

`PairCat.tensorPlusModel` is the closed image of the tensor of the *chosen* plus rings,
before integral closure. `tensorPlusModel_almost` states both containment in the supplied
completed-tensor plus ring and annihilation of its quotient by every parameter root.
`integral_tensor_torsion` states that parameter-power torsion in the algebraic integral
tensor is killed by each root; its two algebra structures are restrictions of the given
pair maps. **Checks.** `tensor_plus_model_zero` gives the zero factor,
`tensor_plus_model_identity` recovers the other plus ring under identity base change,
and `tensor_plus_model_inverse_parameter` excludes the inverse base parameter.
`tensor_affinoid_pullback` states the universal property of the specific tensor cone
against every Yoneda-adic space.

<a id="p2-14"></a>

### P2.14 the uniform completed tensor product over an arbitrary Tate ring

**The uniform completed tensor product of perfectoid rings over an arbitrary Tate ring is
perfectoid.** (*uniform-completed-tensor-of-perfectoid-over-any-base*) Let `(A, A⁺)` be a complete
Hausdorff Huber pair with `A` a Tate ring (not necessarily perfectoid, uniform or of characteristic
`p`), and let `(A, A⁺) → (B, B⁺)`, `(A, A⁺) → (C, C⁺)` be morphisms of Huber pairs with `B, C`
perfectoid Tate rings. Then the uniform completed tensor product `B ⊗̂ᵘ_A C` (AdicSpacesPartII layer
R0 (*uniform-completed-tensor-product*)) is a perfectoid Tate ring ([KL16] §3.3, Cor. 3.3.21, p. 66;
[KL15] §3.6, Cor. 3.6.18,
p. 96, over `ℚ_p`, where Kedlaya–Liu's `D` is the uniform completion of `B ⊗_A C` in the present notation). If `A` is perfectoid it equals
`B ⊗̂_A C` (*completed-tensor-of-perfectoid-is-perfectoid* (i)); otherwise `B ⊗̂_A C` need not be
uniform (`ℂ_p ⊗̂_{ℚ_p} ℂ_p` is not), and the uniformisation cannot be omitted ([KL15] §3.1,
Rem. 3.1.6(c), p. 67). For `B = C = ℚ_p^cycl` over `ℚ_p`, `ℚ_p^cycl ⊗̂ᵘ_{ℚ_p} ℚ_p^cycl` is a
perfectoid Tate ring over the non-perfectoid base `ℚ_p`; `ℂ_p ⊗̂ᵘ_{ℚ_p} ℂ_p` is perfectoid although
`ℂ_p ⊗̂_{ℚ_p} ℂ_p` is not uniform. *Needs:* AdicSpacesPartII R0.4 (*uniformization*,
*uniform-completed-tensor-product*: the spectral
seminorm, uniformisation and the uniform completed tensor product);
*completed-tensor-of-perfectoid-is-perfectoid*; *perfectoid-dense-image-criterion*; layer P1 (P1.5,
P1.30); Tau Ceti `TauCeti.Huber.IsTateRing`, `TauCeti.Huber.Pair.Hom`.

**A uniform Tate ring receiving a dense-image map from a perfectoid ring is perfectoid.**
(*perfectoid-dense-image-criterion*) Let `f : A → B` be a continuous ring homomorphism with dense
image from a perfectoid Tate ring `A` to a complete Hausdorff uniform Tate ring `B` (`B°` bounded).
Prove that `B` is a perfectoid Tate ring and that `f♭ : A♭ → B♭` has dense image ([AWS] §2.8,
Lem. 2.8.4, p. 70, and §2.4, Thm 2.4.3(a), p. 58; [KL16] §3.3, Thm 3.3.18(ii), p. 64, the class of
morphisms along which the perfectoid property transfers; [KL15] §3.6, Thm 3.6.17(b), p. 95, over
`ℚ_p`). Uniformity of `B` is a hypothesis: `K⟨T^{1/p^∞}⟩ → K⟨T^{1/p^∞}⟩/(T)` (closure of the
ideal) is surjective, but the target is not reduced, hence not uniform and not perfectoid
([AWS] §2.4, Ex. 2.4.9, p. 59). If `B` is the completed residue field of a point of `Spa(A, A⁺)`,
a uniform nonarchimedean field receiving a dense-image map from the completed colimit of rational
localisations, `B` is perfectoid (Kedlaya AWS Corollary 2.5.7, second proof); the uniformisation of
`B ⊗̂_A C` with `B, C` perfectoid receives a dense-image map from a perfectoid ring
(see *uniform-completed-tensor-of-perfectoid-over-any-base*). *Needs:* layer P1 (P1.1, P1.19); Tau Ceti
`TauCeti.Huber.IsTateRing`, `TauCeti.Huber.powerBoundedSubring`, `TauCeti.Huber.IsBounded`.

The additional targets are the perfectoidness of the uniform completed tensor product,
`isPerfectoid_of_denseRange` and `denseRange_tiltMap`. The first allows an arbitrary Tate
base and uses the supplied `Uniformization (CompletedTensor A B C)`.
`tensorIntegralModel` is the closure of the subring generated by the two power-bounded
images under the actual completed tensor inclusions. `tensorIntegralModel_almost` bounds
it between all root-scaled copies of `D°` and `D°`; it does not replace integral closure by
an equality. `completedTensor_frobenius_bijective` uses `p = 0` in both factors and allows
the zero tensor ring.

**Checks for tensor conventions and integral models.** `tensor_inclusion_two_terms` computes
`b ⊗ c = inl(b) inr(c)`; `tensor_integral_two_inclusions` includes `b ⊗ 1 + 1 ⊗ c`;
`tensor_integral_zero` gives the whole zero ring; `tensor_integral_unbounded_scalar` excludes
`ϖ⁻¹` in a nonzero tensor. `tensor_plus_zero` computes the zero plus ring,
`uniform_tensor_zero` the zero uniform tensor, and `tensor_uniformization_perfectoid_base`
identifies ordinary and uniform completion over a perfectoid base.

### P2.15 affinoid perfectoid fibre products

**Affinoid perfectoid fibre products: `Spa(B ⊗̂_A C)` is a fibre product in adic spaces, compatible
with open subspaces.** (*perfectoid-fibre-product-universal-property*) Let `X = Spa(A, A⁺)`,
`Y = Spa(B, B⁺)`, `Z = Spa(C, C⁺)` be affinoid perfectoid spaces (any characteristic, no base field)
and `Y → X ← Z` morphisms, corresponding ([Wedhorn] Prop. 8.25) to morphisms of complete perfectoid
Tate pairs `(A, A⁺) → (B, B⁺)` and `(A, A⁺) → (C, C⁺)`. Let `(D, D⁺) = B ⊗̂_A C`, a complete
perfectoid Tate pair (see *completed-tensor-of-perfectoid-is-perfectoid*), hence sheafy
(see *perfectoid-stably-uniform-and-sheafy*). Then (a) `W = Spa(D, D⁺)` with `p = Spa(inl)` and
`q = Spa(inr)` is a fibre product `Y ×_X Z` in AdicSpaces' category of adic spaces (a Mathlib
`IsPullback` square there, against all adic spaces `T`, not only perfectoid ones), hence in `Perfd`,
and `W` is affinoid perfectoid; this is AdicSpacesPartII R0.7 (*affinoid-fibre-product*,
affinoid fibre products)
with its sheafiness hypothesis discharged; (b) for rational subsets `U' ⊆ X`, `U ⊆ Y`, `V ⊆ Z` with
`U` and `V` mapping into `U'`, the open `p⁻¹(U) ∩ q⁻¹(V) ⊆ W` is a rational subset and equals
`Spa(O_Y(U) ⊗̂_{O_X(U')} O_Z(V)) = U ×_{U'} V` (AdicSpacesPartII R0.3
(*completed-tensor-rational-localisation*)); (c) for arbitrary open subspaces `U ⊆ Y`,
`V ⊆ Z`, the open subspace `p⁻¹(U) ∩ q⁻¹(V)` with the restricted projections is a fibre product
`U ×_X V` in adic spaces; (d) the map `|W| → |Y| ×_{|X|} |Z|` is surjective (AdicEtaleGeometry
A0.1 (*tate-completed-tensor-points*)) and not
injective in general ([Sch12] §6, proof of Prop. 6.18, p. 38, the universal property over a
perfectoid field; [AWS] §2.4, before Thm 2.4.1, p. 57, why the perfectoid case discharges the
sheafiness hypothesis; [ECD] §8, Prop. 8.3, p. 41, stability of affinoid perfectoid spaces under
fibre products). `Spa(K⟨T^{1/p^∞}⟩) ×_{Spa K} Spa(K⟨S^{1/p^∞}⟩) ≅ Spa(K⟨T^{1/p^∞}, S^{1/p^∞}⟩)` for
every perfectoid Tate ring `K`, with `{|T| ≤ |ϖ|} × {|S| ≤ |ϖ|}` corresponding to
`Spa(K⟨(T/ϖ)^{1/p^∞}, (S/ϖ)^{1/p^∞}⟩)`. *Needs:* *completed-tensor-of-perfectoid-is-perfectoid*;
*affinoid-perfectoid-space*; *perfectoid-stably-uniform-and-sheafy*;
*rational-localization-of-perfectoid-affinoids*; AdicSpacesPartII R0.1, R0.3, R0.7
(*adic-ring-homomorphism*, *completed-tensor-rational-localisation*, *affinoid-fibre-product*);
AdicEtaleGeometry A0.1 (*tate-completed-tensor-points*);
Mathlib `CategoryTheory.IsPullback`; Tau Ceti
`TauCeti.Huber.Pair.Hom.spaComap`,
`TauCeti.Huber.Pair.Hom.spaComap_preimage_rationalSubset`.

### P2.16 fibre products of perfectoid spaces

Fibre products of perfectoid spaces: over a perfectoid base in all adic spaces, over an analytic
base in `Perfd`, absolute products in `Perf`, and the comparison with Scholze's category over a
perfectoid field.

**Fibre products of perfectoid spaces exist in adic spaces and are perfectoid.**
(*fibre-products-of-perfectoid-spaces*) Let `f : Y → X` and `g : Z → X` be morphisms of perfectoid
spaces (any characteristic; no perfectoid base field). The fibre product `W = Y ×_X Z` exists in
AdicSpaces' category of adic spaces and is a perfectoid space; with its projections `p : W → Y`,
`q : W → Z` it is therefore also a fibre product in `Perfd`, `Perfd` has all fibre products (Mathlib
`HasPullbacks`), and the inclusion `Perfd ⥤ AdicSpace` preserves them. Construction: `W` is glued
(AdicSpaces layer 5.3) from the affinoid perfectoid spaces
`W_{U,U',V} = Spa(O_Y(U) ⊗̂_{O_X(U')} O_Z(V)) = U ×_{U'} V`
(see *perfectoid-fibre-product-universal-property*), for affinoid perfectoid opens `U' ⊆ X`,
`U ⊆ f⁻¹(U')`, `V ⊆ g⁻¹(U')`, along their common open subspaces, which are fibre products of the
intersections (open compatibility (c) there). Conventions: the affinoid formula uses
AdicSpacesPartII R0.3 (*completed-tensor-product*)
with plus ring the closure of the integral closure of the image of `O_Y⁺(U) ⊗ O_Z⁺(V)`, which is
almost, but not in general exactly, the completed tensor product of the plus rings
(see *completed-tensor-plus-ring-almost-formula*); the ring is already uniform, so no uniformisation is
performed. Affinoid perfectoid spaces and `Perf` are closed under fibre products; tilting commutes
with them (see *tilting-commutes-with-fibre-products*); `|W| → |Y| ×_{|X|} |Z|` is surjective but not
injective in general. Over a perfectoid field this is Sch12 Proposition 6.18; fibre products over a
non-perfectoid analytic base are *fibre-products-over-analytic-base*, and absolute products in
`Perf` are *products-in-perf* ([Sch12] §6, Prop. 6.18, p. 38, and the paragraph before it, p. 38,
where fibre products of general adic spaces exist only under finiteness hypotheses (Huber's
Proposition 1.2.2; AdicSpacesPartII R0.7 (*fibre-products-existence*)); [Berkeley] Lecture 8,
Prop. 8.2.5(3), printed p. 59,
existence over no base field; [Bhatt] §9.3, Cor. 9.3.13, printed p. 111, the field case with the
reduction to characteristic `p`). In `PerfectoidSpace`, define `pullback` as the fibre product
`W = Y ×_X Z` in `Perfd`; prove `pullback.fst` and `pullback.snd` (the projections `p`, `q`),
`pullback.condition` (`p ≫ f = q ≫ g`), `pullback.lift` (for an adic space `T`, not necessarily
perfectoid, with `a : T → Y`, `b : T → Z`, `a ≫ f = b ≫ g`, the unique `T → Y ×_X Z`, with
the projection equations and uniqueness from `pullback.isPullback_toAdicSpace`; for perfectoid
`T`, use Mathlib's `CategoryTheory.Limits.pullback.lift_fst`, `.lift_snd` and `.hom_ext`),
`hasPullbacks` (instance `HasPullbacks Perfd`),
`pullback.isPullback_toAdicSpace` (the image square in adic spaces is an `IsPullback` square; the
inclusion preserves pullbacks, and `W` agrees with any fibre product of adic spaces of
AdicSpacesPartII R0.7 (*fibre-products-existence*) that
exists for the same diagram), `pullbackSpaIso`
(`Spa(B, B⁺) ×_{Spa(A, A⁺)} Spa(C, C⁺) ≅ Spa(B ⊗̂_A C)` over the projections, with R0's plus ring),
`pullback.isAffinoidPerfectoid` (instance), `pullbackRestrictIso`
(`p⁻¹(U) ∩ q⁻¹(V) ≅ U ×_X V` for open `U ⊆ Y`, `V ⊆ Z`), `pullback_points`
(`|Y ×_X Z| → |Y| ×_{|X|} |Z|` surjective, by AdicEtaleGeometry A0.1
(*tate-completed-tensor-points*) on complete Tate affinoid charts),
`Perf.hasPullbacks` (`Perf` closed under fibre
products; `HasPullbacks Perf`), binary products in `Perfd/X` (given by
fibre products, Mathlib `Over.ConstructProducts.over_binaryProduct_of_pullback`) and
`pullbackTiltIso` (`(Y ×_X Z)♭ ≅ Y♭ ×_{X♭} Z♭` over the tilted projections,
*tilting-commutes-with-fibre-products*). *Needs:* *perfectoid-fibre-product-universal-property*;
*perfectoid-space*; *affinoid-perfectoid-basis*; *completed-tensor-plus-ring-almost-formula*;
AdicSpacesPartII R0.6 (*adic-morphism*); AdicEtaleGeometry A0.1
(*tate-completed-tensor-points*); Mathlib
`CategoryTheory.Limits.HasPullbacks`,
`CategoryTheory.IsPullback`, `CategoryTheory.Over.ConstructProducts.over_binaryProduct_of_pullback`.

**Checks.**

- `pullback_perfectedDiscs` (computation) — for a perfectoid Tate ring `R`:
  `Spa(R⟨T^{1/p^∞}⟩, R⁺⟨T^{1/p^∞}⟩) ×_{Spa(R, R⁺)} Spa(R⟨S^{1/p^∞}⟩, R⁺⟨S^{1/p^∞}⟩) ≅ Spa(R⟨T^{1/p^∞}, S^{1/p^∞}⟩, R⁺⟨T^{1/p^∞}, S^{1/p^∞}⟩)` over the two projections.
- `pullback_galois_points` (computation) — for a finite Galois extension `L/K` of perfectoid fields
  with group `G`, `Spa(L, L°) ×_{Spa(K, K°)} Spa(L, L°) ≅ ⊔_{g ∈ G} Spa(L, L°)` has `|G|` points
  while `|Spa L| ×_{|Spa K|} |Spa L|` is one point: the underlying space of a fibre product is not
  the fibre product of underlying spaces.
- `pullback_id` (degenerate) — `Y ×_X X ≅ Y` via `p`, for the identity of `X`; and `Y ×_X ∅ = ∅`.
- `pullback_plus_ne_completedTensor_plus` (non-example) — for `K` the completion of
  `ℚ̆_p(μ_{p^∞})` and `L = K(p^{1/ℓ})` (`ℓ` a prime not dividing `p(p − 1)`), the plus ring of
  `Spa L ×_{Spa K} Spa L` is `(L°)^ℓ`, which strictly contains the image of `L° ⊗_{K°} L°`
  (*completed-tensor-of-perfectoid-is-perfectoid* (iv)). The example states the strict
  inclusion for every nontrivial finite Galois extension of complete normed perfectoid fields
  whose automorphisms act trivially on residue classes; `tame_cubic_perfectoid_extension`
  supplies the residue-characteristic-two instance. A construction using the completed tensor
  product of plus rings as plus ring produces a pair that is not a Huber pair.
- `pullback.isPullback_toAdicSpace` (compatibility) — the square `(W, p, q, f, g)` is a pullback
  (Mathlib `IsPullback`) in AdicSpaces' category of adic spaces, and on affinoid pieces the
  canonical map `Spa(B ⊗̂ᵘ_A C) → Spa(B ⊗̂_A C)` from AdicSpacesPartII R0.4
  (*uniform-completed-tensor-product*) is an isomorphism.
- `pullback_tilt` (characterisation) — for `Y → X ← Z` in `Perfd`, the fibre product of
  `Y♭ → X♭ ← Z♭` in `Perf` is `(Y ×_X Z)♭` with the tilted projections.

**The plus ring of a perfectoid completed tensor product: almost equal to the completed tensor
product of plus rings, but not equal.** (*completed-tensor-plus-ring-almost-formula*) In the
situation of *completed-tensor-of-perfectoid-is-perfectoid*, with `ϖ ∈ A` as there, prove: (i) the
`ϖ`-power torsion of `B⁺ ⊗_{A⁺} C⁺` is killed by every `ϖ^{1/p^n}`; (ii) the natural map from the
`ϖ`-adic completion `(B⁺ ⊗_{A⁺} C⁺)^∧` to `D⁺` has kernel and cokernel killed by every `ϖ^{1/p^n}`,
i.e. it is an almost isomorphism of `A⁺`-modules; (iii) `D⁺` is the `ϖ`-adic completion of the
integral closure in `B ⊗_A C` of the image of `B⁺ ⊗_{A⁺} C⁺`, and it contains `D°°` and
`ϖ^{1/p^n} D°` for all `n`. The map in (ii) is not an isomorphism in general (layer P3) ([Bhatt]
§9.3, Ex. 9.3.14, printed p. 112, part (ii) over a perfectoid field; [ECD] §3, Ex. 3.22(ii), p. 19,
used in (i); [AWS] §2.8, proof of Lem. 2.8.7, p. 71, whose claim holds only up to almost isomorphism
and after passing to the integral closure, as corrected here and in layer P3). Over a perfectoid
field `K` and `B = C = K⟨T^{1/p^∞}⟩` with plus rings `K°⟨T^{1/p^∞}⟩`, the map of (ii) is an
isomorphism onto `K°⟨T^{1/p^∞}, S^{1/p^∞}⟩`. *Needs:*
*completed-tensor-of-perfectoid-is-perfectoid*; AdicSpacesPartII R0.3
(*completed-tensor-product*); layer P1; layer
P0; Mathlib `integralClosure`.

**Tilting commutes with fibre products of perfectoid spaces.**
(*tilting-commutes-with-fibre-products*) This is the space-level form of
*completed-tensor-of-perfectoid-is-perfectoid* (iii): for morphisms `Y → X ← Z` of perfectoid
spaces the tilts of the projections exhibit `(Y ×_X Z)♭` as the fibre product `Y♭ ×_{X♭} Z♭` in
`Perf` and in adic spaces, the canonical morphism `(Y ×_X Z)♭ → Y♭ ×_{X♭} Z♭` being, on affinoid
pieces, the isomorphism `(B ⊗̂_A C)♭ ≅ B♭ ⊗̂_{A♭} C♭` of P2.13 (iii), glued; more generally the tilting equivalence
`Perfd/X ≃ Perf/X♭` preserves every limit that exists in `Perfd/X` ([AWS] §2.4, Thm 2.4.1, p. 57,
the affinoid case; [ECD] §3, Cor. 3.20, p. 19, the equivalence whose limit preservation gives the
lemma). For `K` perfectoid, `(Spa K⟨T^{1/p^∞}⟩ ×_{Spa K} Spa K⟨S^{1/p^∞}⟩)♭ ≅ Spa K♭⟨T^{1/p^∞}, S^{1/p^∞}⟩`; for `L/K` finite Galois of perfectoid fields,
`(Spa L ×_{Spa K} Spa L)♭ ≅ Spa L♭ ×_{Spa K♭} Spa L♭`, both `⊔_{g ∈ G}` of the tilt. *Needs:*
*tilting-slice-equivalence*; *fibre-products-of-perfectoid-spaces*;
*completed-tensor-of-perfectoid-is-perfectoid*; Mathlib
`CategoryTheory.Over.ConstructProducts.over_binaryProduct_of_pullback`,
`CategoryTheory.Adjunction.rightAdjoint_preservesLimits`.

`PerfectoidSpace.hasPullbacks` supplies Mathlib pullbacks and their projections and
universal maps. The preservation statement tests the universal property in the
ambient category of pre-adic spaces (and, through `YonedaAdicSpace.ofAdicSpace`, in the Yoneda-adic
category), and `tilt_preservesPullbacks` gives the tilted comparison.
`pullback_points` asserts surjectivity onto compatible pairs of valuation points.
**Checks.** `pullback_identity_projection` makes base change along identity an isomorphism;
`pullback_empty_factor` gives the empty space; `pullback_surjects_compatible_points` exhibits
a lift of a compatible pair, without claiming uniqueness of that point.

Lean: `PerfectoidSpace.hasPullbacks`.

### P2.17 fibre products over an analytic base

**Fibre products of perfectoid spaces over an analytic adic space, in Perfd, via uniform completed
tensor products.** (*fibre-products-over-analytic-base*) Let `S` be an analytic adic space
(AdicSpaces layer 5; every point analytic, e.g. `Spa(K, K⁺)` for any nonarchimedean field `K`, or
any rigid space) and `f : Y → S`, `g : Z → S` morphisms from perfectoid spaces. Then the fibre
product `Y ×_S Z` exists in `Perfd`: there is a perfectoid space `W` with morphisms `p : W → Y`,
`q : W → Z`, `f p = g q`, such that for every perfectoid space `T`,
`Hom(T, W) = Hom(T, Y) ×_{Hom(T, S)} Hom(T, Z)`. Affinoid-locally, for an open affinoid
`U' = Spa(A, A⁺) ⊆ S` with `A` a complete Tate ring (the analytic locus is covered by such,
AdicSpaces layer 2.3) and affinoid perfectoid `U ⊆ f⁻¹(U')`, `V ⊆ g⁻¹(U')`, the piece of `W` over
`U, V` is `Spa(O_Y(U) ⊗̂ᵘ_A O_Z(V))`, the uniform completed tensor product (AdicSpacesPartII layer
R0 (*uniform-completed-tensor-product*)), which is perfectoid (see
*uniform-completed-tensor-of-perfectoid-over-any-base*). If `S` is
perfectoid, `W` is the fibre product of *fibre-products-of-perfectoid-spaces* (in all adic spaces).
If `S` is not perfectoid, `W` is a fibre product in `Perfd` only: the test objects are uniform, the
completed tensor product itself need not be uniform (`ℂ_p ⊗̂_{ℚ_p} ℂ_p` is not) and its sheafiness
is not known. The diagonal `Y → Y ×_S Y` is defined ([KL16] §3.3, Cor. 3.3.21, p. 66, the affinoid
pieces over a non-perfectoid base are perfectoid; [KL15] §3.1, Rem. 3.1.6(c), p. 67, the
uniformisation cannot be omitted; [AWS] §2.4, before Thm 2.4.1, p. 57, why the universal property is
in `Perfd` only). For `K` a nonarchimedean field that is not perfectoid (e.g. `ℚ_p`) and perfectoid
spaces `Y, Z` over `Spa(K, K°)`, `Y ×_{Spa K} Z` is a perfectoid space, the fibre product used for
analytic separatedness in layer [P8](../PerfectoidTowers/README.md#layer-p8-finite-quotients-and-closed-perfectoid-loci-in-towers). In `PerfectoidSpace`, define `pullbackOverAnalytic` as the
perfectoid space `Y ×_S Z` in `Perfd` for `S` analytic; prove `pullbackOverAnalytic.fst` and `.snd`
(the projections), `pullbackOverAnalytic.condition` (`p ≫ f = q ≫ g` in adic spaces),
`pullbackOverAnalytic.lift` (for a perfectoid space `T` with `a : T → Y`, `b : T → Z`,
`a ≫ f = b ≫ g`, the unique `T → Y ×_S Z`, with both projection equations in
`pullbackOverAnalytic.lift_spec` and uniqueness in `pullbackOverAnalytic.isPullback`),
`pullbackOverAnalyticSpaIso` (`Spa(B) ×_{Spa A} Spa(C) ≅ Spa(B ⊗̂ᵘ_A C)` with `A` Tate),
`pullbackOverAnalyticIsoPullback` (if `S` is perfectoid, `pullbackOverAnalytic f g → pullback f g`
is an isomorphism), `pullbackOverAnalytic.isAffinoidPerfectoid` (instance: for `Y, Z` affinoid
perfectoid and `S` affinoid with Tate ring), `pullbackOverAnalyticRestrictIso` (for opens `U ⊆ Y`,
`V ⊆ Z`, `S' ⊆ S` with `f(U), g(V) ⊆ S'`: `p⁻¹(U) ∩ q⁻¹(V) ≅ U ×_{S'} V`), `diagonalOverAnalytic`
(`Δ : Y → Y ×_S Y = lift (𝟙 Y) (𝟙 Y)`, used for analytic separatedness in layer [P8](../PerfectoidTowers/README.md#layer-p8-finite-quotients-and-closed-perfectoid-loci-in-towers)) and
`pullbackOverAnalytic.surjective_points` (`|Y ×_S Z| → |Y| ×_{|S|} |Z|` is surjective: `Spa` of
the uniformisation is homeomorphic to `Spa` of the completed tensor product, AdicSpacesPartII layer
R0 (*uniformization*), property (d), and points lift to the completed tensor by
AdicEtaleGeometry A0.1 (*tate-completed-tensor-points*)). *Needs:*
*uniform-completed-tensor-of-perfectoid-over-any-base*;
*fibre-products-of-perfectoid-spaces*; *completed-tensor-of-perfectoid-is-perfectoid*;
*perfectoid-fibre-product-universal-property*; *perfectoid-space*; AdicSpacesPartII R0.3, R0.4, R0.6
(*completed-tensor-rational-localisation*, *uniformization*, *uniform-completed-tensor-product*,
*adic-iff-analytic-locus*); AdicEtaleGeometry A0.1 (*tate-completed-tensor-points*).

**Checks.**

- `pullbackOverAnalytic_empty`, `pullbackOverAnalytic.fst_empty`, `pullbackOverAnalytic.snd_empty`, `pullbackOverAnalytic.lift_empty`: an empty factor gives an empty pullback.
- `pullbackOverAnalytic_perfectoid`, `pullbackOverAnalytic.fst_perfectoid`, `pullbackOverAnalytic.snd_perfectoid`, `pullbackOverAnalytic.lift_perfectoid`: over a perfectoid base the comparison with the perfectoid pullback preserves both projections.
- `pullbackOverAnalytic_identity`, `pullbackOverAnalytic.fst_identity`, `pullbackOverAnalytic.snd_identity`, `pullbackOverAnalytic.lift_identity`: the identity cospan has an invertible first projection and its diagonal lift composes to the identity.

- `diagonalOverAnalytic_identity`: the identity map has invertible diagonal.
- `diagonalOverAnalytic_empty`: an empty source has invertible diagonal.
- `diagonalOverAnalytic_two`: the fold from two copies of a nonempty affinoid has a noninvertible diagonal.

- `pullbackOverAnalytic_Cp_Qp` (non-example) — for `S = Spa(ℚ_p, ℤ_p)` and
  `Y = Z = Spa(ℂ_p, O_{ℂ_p})`, the affinoid piece is `Spa(ℂ_p ⊗̂ᵘ_{ℚ_p} ℂ_p)` and not
  `Spa(ℂ_p ⊗̂_{ℚ_p} ℂ_p)`: the completed tensor product is not uniform (Kedlaya–Liu I
  Remark 2.8.5), so a construction without uniformisation does not produce a perfectoid ring.
- `pullbackOverAnalytic_perfectoid` (compatibility) — if `S` is perfectoid,
  `pullbackOverAnalytic f g ≅ PerfectoidSpace.pullback f g` over `Y` and `Z`.
- `pullbackOverAnalytic_empty` (degenerate) — if `Y = ∅` then `Y ×_S Z = ∅`.
- `pullbackOverAnalytic_cyclotomic` (computation) — for `S = Spa(ℚ_p, ℤ_p)` and
  `Y = Z = Spa(ℚ_p^cycl, ℤ_p^cycl)`, `Y ×_S Z = Spa(ℚ_p^cycl ⊗̂ᵘ_{ℚ_p} ℚ_p^cycl)`, a perfectoid space
  mapping to `Y` by `p`; it is nonempty and its points surject onto `|Y| ×_{|S|} |Z|` (a point).

Analyticity of the base means locally the spectrum of a Tate pair ([Berkeley] Definition 4.3.2,
printed p. 33): AdicSpacesPartII R0.6's `AdicSpace.IsAnalytic` on adic spaces, which on
AdicEtaleGeometry's analytic carrier is `YonedaAdicSpace.IsAnalytic`.
`pullback_over_analytic` produces a perfectoid cone that is a limit against every
perfectoid test space, including a nonperfectoid analytic base.
**Checks.** `analytic_perfectoid` includes every perfectoid space; `analytic_empty` includes
the empty space; `analytic_padic_base` includes `Spa(ℚ_p, ℤ_p)` although it is not perfectoid.

### P2.18 absolute products in Perf

**Absolute products in Perf.** (*products-in-perf*) Let `X, Y` be perfectoid spaces of
characteristic `p`. The product `X × Y` exists in `Perf` ([Berkeley] Lecture 8, §8.3, Lem. 8.3.5
and Rem. 8.3.6, printed p. 61; [ECD] §11, proof of Prop. 11.4, p. 55, where products in `Perf` are
what the diamond theory needs). It is also the fibre product `X ×_{Spa(𝔽_p, 𝔽_p)} Y` in
AdicSpaces' category of adic spaces, over the discrete adic space `Spa(𝔽_p, 𝔽_p)`, which is not a
perfectoid (nor Tate, nor analytic) space; for nonempty `X`, the structure maps `X → Spa(𝔽_p, 𝔽_p)` are not adic, so
neither *fibre-products-of-perfectoid-spaces* nor AdicSpacesPartII R0.7
(*fibre-products-existence*) applies, and `Perf` has no
final object ([Berkeley] Rem. 8.3.6), so the absolute product is not a fibre product in `Perf`.
Affinoid charts: for `X = Spa(A, A⁺)`, `Y = Spa(B, B⁺)` affinoid perfectoid of characteristic `p`
with pseudouniformisers `ϖ ∈ A`, `ϖ' ∈ B`, and integers `m, n ≥ 1`, put `E = A ⊗_{𝔽_p} B`,
`u_m = (ϖ^m ⊗ 1)(1 ⊗ ϖ')⁻¹` and `v_n = (1 ⊗ ϖ'^n)(ϖ ⊗ 1)⁻¹` in `E`,
`D_{m,n} = (A° ⊗ B°)[u_m, v_n] ⊆ E`; topologise `E` so that `D_{m,n}` is open with the
`(ϖ ⊗ 1)`-adic topology (equal to the `(1 ⊗ ϖ')`-adic one, since `ϖ^m = u_m ϖ'` and `ϖ'^n = v_n ϖ`
in `D_{m,n}`); let `C_{m,n}` be the completion and `C⁺_{m,n}` the completion of the integral
closure in `E` of the image of `(A⁺ ⊗ B⁺)[u_m, v_n]`, an open integrally closed subring. Then
`(C_{m,n}, C⁺_{m,n})` is a complete perfectoid Tate pair (`C_{m,n}` is a perfect complete Tate
ring); a morphism of pairs `(C_{m,n}, C⁺_{m,n}) → (R, R⁺)` to a complete Huber pair with `R` Tate
is the same as morphisms `φ : (A, A⁺) → (R, R⁺)`, `ψ : (B, B⁺) → (R, R⁺)` with
`φ(ϖ)^m ψ(ϖ')⁻¹ ∈ R⁺` and `ψ(ϖ')^n φ(ϖ)⁻¹ ∈ R⁺`; for `m ≤ m'`, `n ≤ n'`, `Spa(C_{m,n})` is the
rational subset `{|u_m| ≤ 1, |v_n| ≤ 1}` of `Spa(C_{m',n'})`; and
`X × Y = ⋃_{m,n} Spa(C_{m,n}, C⁺_{m,n})`, glued along these rational open immersions (AdicSpaces
layer 5.3). For general `X, Y` the affinoid products glue. The plus ring must contain `u_m` and
`v_n`: the integral closure of the image of `A⁺ ⊗ B⁺` alone, as written in Berkeley's proof, is not
open ([Berkeley] Lecture 8, proof of Lem. 8.3.5, printed pp. 61–62, the chart construction, whose
plus ring is corrected here). The absolute product of two perfectoid fields need not be a point:
`Spa 𝔽_p((t^{1/p^∞})) × Spa 𝔽_p((s^{1/p^∞}))` is the perfectoid punctured open unit disc
`{0 < |s| < 1}` over `𝔽_p((t^{1/p^∞}))` ([ECD] proof of Prop. 11.4), the union of the perfectoid
annuli `{|t|^m ≤ |s| ≤ |t|^{1/n}}`, which is not quasi-compact and has infinitely many points. Here the open-disc notation
means this union of annuli (equivalently the strict radius condition at rank-one
generisations), not the literal inequality `|s| < 1` in every higher-rank value group. For
`X, Y ∈ Perf` over a common perfectoid `Z`, the canonical morphism `X ×_Z Y → X × Y` is not an
isomorphism in general: for `K = 𝔽_p((t^{1/p^∞}))`, `Spa K ×_{Spa K} Spa K = Spa K` is a point,
while `Spa K × Spa K` is not. In `PerfectoidSpace.Perf`, `hasBinaryProducts_perf` supplies
`CategoryTheory.Limits.HasBinaryProducts (Perf p)`. Use Mathlib's `CategoryTheory.Limits.prod`
for `X × Y`, with `prod.fst`, `prod.snd` and `prod.lift` for the projections and the map
from `T ∈ Perf` determined by `a : T → X` and `b : T → Y`. The projection equations and
uniqueness are `CategoryTheory.Limits.prod.{lift_fst, lift_snd, hom_ext}`
in `Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts/BinaryProducts.lean`.
Define `prodChart` as the complete perfectoid Tate pair `(C_{m,n}, C⁺_{m,n})` with its maps
from `(A, A⁺)` and `(B, B⁺)`. Its `ProductChart.universal` field identifies morphisms to
any complete Tate pair with pairs `(φ, ψ)` satisfying the two integrality conditions
([Berkeley] Lemma 8.3.5, proof, printed pp. 61–62). Prove
`prodChart_isRationalSubset` (for `m ≤ m'`, `n ≤ n'`, `Spa(C_{m,n}) → Spa(C_{m',n'})` is the
inclusion of `{|u_m| ≤ 1, |v_n| ≤ 1}`: the ring map `C_{m',n'} → C_{m,n}` fixes the maps
from `A` and `B`, is `CAff.IsRationalLocalization` for numerators `{1, u_m, v_n}` and
denominator `1`, and induces an open embedding with exactly this image; this includes
equal exponents and empty charts when an exponent is zero, [Berkeley] Lemma 8.3.5,
proof, printed pp. 61–62), `prodChartMap_cover` (for affinoid perfectoid `X, Y`,
`X × Y = ⋃_{m,n} Spa(C_{m,n}, C⁺_{m,n})`, an increasing union of affinoid perfectoid opens),
the pullback identification (`X × Y` is a pullback of `X → Spa(𝔽_p, 𝔽_p) ← Y` in adic spaces) and
`prodPerfectoidFieldsIso` (`Spa 𝔽_p((t^{1/p^∞})) × Spa 𝔽_p((s^{1/p^∞})) ≅` the perfectoid punctured
open unit disc over `𝔽_p((t^{1/p^∞}))`). *Needs:* *perfectoid-space*;
*characteristic-p-affinoid-perfectoid-criterion*; *fibre-products-of-perfectoid-spaces*;
*open-subspaces-of-perfectoid-spaces*; AdicSpacesPartII R0.2, R0.6 (*discrete-huber-pair-sheafy*,
*adic-iff-analytic-locus*); layer P1; Mathlib
`CategoryTheory.Limits.HasBinaryProducts`, `UniformSpace.Completion`,
`UniformSpace.Completion.extensionHom`, `integralClosure`; Tau Ceti `TauCeti.Huber.Pair`,
`TauCeti.Huber.IsRingOfIntegralElements`.

**Checks.**

- Computation: `Spa 𝔽_p((t^{1/p^∞})) × Spa 𝔽_p((s^{1/p^∞})) ≅ {0 < |s| < 1} ⊆ Spa 𝔽_p((t^{1/p^∞}))⟨s^{1/p^∞}⟩`, the union of the affinoid perfectoid annuli
  `Spa(C_{m,n}) = {|t|^m ≤ |s| ≤ |t|^{1/n}}`; the absolute product of two perfectoid fields is not
  a point and not quasi-compact.
- `absolute_product_empty` (degenerate) — `X × ∅ = ∅` for every `X ∈ Perf`.
- `not_isPerfectoidSpace_spa_Fp_final` (non-example) — `Spa(𝔽_p, 𝔽_p)` with the discrete topology
  is not an object of `Perf` (its point is not analytic), `Perf` has no final object (Berkeley
  Remark 8.3.6), and `X × Y` is not `X ×_Z Y` for any `Z ∈ Perf` in general; a definition of the
  product as a fibre product over a perfectoid "`Spa 𝔽_p`" is ill-typed.
- `prodChart_plus_open` (non-example) — for `A = B = 𝔽_p((t^{1/p^∞}))` with `ϖ = t`, `ϖ' = s`:
  `(t ⊗ 1)^k u_m^j` is not integral over the image of `A° ⊗ B°` for `j ≥ 1` (reduce modulo
  `1 ⊗ s^j`: `A° ⊗ (B°/s^j)` is free over `A°`), so the integral closure of `A⁺ ⊗ B⁺` is not open in
  `E` and `(E, integral closure of A⁺ ⊗ B⁺)` is not a Huber pair; the plus ring must contain
  `u_m, v_n`.
- Compatibility: `X × Y` with its projections is a pullback of
  `X → Spa(𝔽_p, 𝔽_p) ← Y` in AdicSpaces' category of adic spaces, and it agrees with the absolute
  product of Berkeley Lemma 8.3.5.
- Characterisation: for morphisms `X → Spa K_t`, `Y → Spa K_s` with
  `K_t = 𝔽_p((t^{1/p^∞}))`, `K_s = 𝔽_p((s^{1/p^∞}))` (which exist locally: `t^{1/p^n} ↦ ϖ^{1/p^n}`),
  `X × Y ≅ X ×_{Spa K_t} (Spa K_t × Spa K_s) ×_{Spa K_s} Y`, fibre products over perfectoid bases
  (ECD proof of Proposition 11.4). The same comparison holds for any two maps
  `X → K`, `Y → L` in `Perf`, as in the example.

**Characteristic p: perfectoid spaces are adic spaces locally Spa of perfect Tate rings, and
affinoid perfectoid spaces are the affinoid ones.** (*characteristic-p-affinoid-perfectoid-criterion*)
Let `X` be an adic space with `p = 0` in `O_X(X)`. Prove: (i) `X` is a perfectoid space iff `X` is
covered by open subspaces isomorphic to `Spa(R, R⁺)` with `R` a perfect complete Tate ring; (ii) if
`X` is a perfectoid space, then `O_X(U)` is a perfect ring (Mathlib `PerfectRing (O_X(U)) p`) for
every open `U ⊆ X`; (iii) (Mihara) if `X ≅ Spa(A, A⁺)` for a complete Hausdorff sheafy Huber pair
`(A, A⁺)` with `A` Tate, and `X` is a perfectoid space, then `A` is perfect, hence a perfectoid Tate
ring, and `X` is affinoid perfectoid. The analogue of (iii) in characteristic `0` is not asserted:
whether an affinoid adic space that is a perfectoid space is affinoid perfectoid is recorded as
unclear in [Bhatt] §9.3, Rem. 9.3.11, printed p. 111 ([Bhatt] §9.3, Ex. 9.3.12 (Mihara), printed
p. 111, part (iii), proved by gluing `p`-th roots; [ECD] §3, Proposition 3.5, p. 15, perfect complete
Tate rings of characteristic `p` are perfectoid (uniformity is automatic); [AWS] §2.1, Ex. 2.1.2,
p. 49, for uniform pairs, used in (i) and (iii)). For
`A = 𝔽_p((t^{1/p^∞}))⟨T^{1/p^∞}⟩`, `Spa(A, A⁺)` is a perfectoid space and `A` is perfect; for
`A = 𝔽_p((t))⟨T⟩` (not perfect), `Spa(A, A°)` is not a perfectoid space, by (iii). *Needs:*
*perfectoid-space*; *affinoid-perfectoid-space*; *perfectoid-spaces-and-glued-tilting*; layer P1;
Mathlib `PerfectRing`.

`PerfectoidSpace.hasBinaryProducts_perf` supplies the absolute binary products.
A separate statement retains the failure of quasi-compactness of such products.
**Checks.** `absolute_product_empty` gives the empty product;
`absolute_product_symmetry` interchanges the two factors;
`absolute_product_noncompact` asserts a pair with non-quasi-compact product. The explicit
punctured-disc identification and the bounded ratios in `prodChart` are stronger targets
than existence of this pair.

`PerfectoidSpace.ProductChart` carries the two maps of perfectoid pairs, both integral
bounding ratios, and the representing property against every complete Tate pair.
`prodChart` supplies it in characteristic p. Allowing an exponent zero gives the empty
chart; the product cover uses positive exponents. `prodChartMap_projections` fixes the
maps to the product, `prodChartMap_open` gives the open immersion, and
`prodChartMap_cover` gives the covering. A separate statement asserts the absence of a terminal
object independently of the binary-product instances.
**Checks.** `product_chart_zero_exponent` gives the zero ring when the first exponent is
zero; `product_chart_empty_factor` gives zero for an empty factor;
`product_chart_diagonal` gives the diagonal map for equal parameters and exponents one;
`product_chart_diagonal_ratios` computes the bounding ratio as one on that diagonal.

Lean: `PerfectoidSpace.hasBinaryProducts_perf`.

### P2.19 perfectoid spaces over a perfectoid field

**Scholze's perfectoid spaces over a perfectoid field K are the perfectoid spaces over
`Spa(K, K⁺₀)`.** (*perfectoid-spaces-over-a-perfectoid-field-comparison*) Let `K` be a perfectoid
field (layer P1), `K°` its valuation ring and `K⁺₀ ⊆ K°` the smallest ring of integral elements of
`K`, the integral closure of `ℤ·1 + K°°` in `K` ([Wedhorn] §7.3, Rem. 7.15(2), p. 60); `K⁺₀ = K°`
iff the residue field of `K` is algebraic over `𝔽_p`. (i) Sch12's perfectoid `K`-algebras
(Definition 5.1) and perfectoid affinoid `K`-algebras (Definition 6.1) are the perfectoid Tate
rings, respectively complete perfectoid Tate pairs `(R, R⁺)`, with a continuous `K`-algebra
structure; no condition `K° ⊆ R⁺` is imposed ([Sch12] §2, Def. 2.6(ii), p. 8). (ii) Sch12's category
of perfectoid spaces over `K` ([Sch12] §6, Def. 6.15, p. 38: objects of Huber's category of triples
whose structure sheaf is a sheaf of complete topological `K`-algebras, locally affinoid perfectoid;
morphisms continuous `K`-linear; [Sch12] §2, before Def. 2.18, p. 10, adic spaces over `k` carry a
`k`-algebra structure, not a map to `Spa(k, k°)`) is equivalent to the slice category
`Perfd/Spa(K, K⁺₀)`: a morphism `X → Spa(K, K⁺₀)` is a continuous ring map `K → O_X(X)`
([Wedhorn] Prop. 8.25), and every such map sends `K⁺₀` into `O_X⁺(X)`, which contains all
topologically nilpotent elements and is integrally closed. The full subcategory `Perfd/Spa(K, K°)`
([Bhatt] §9.3, Def. 9.3.9, printed p. 111) consists of those `X` with `K° ⊆ O_X⁺(X)`; it is all of
Sch12's category when the residue field of `K` is algebraic over `𝔽_p` (e.g. `K = ℂ_p`, `ℚ_p^cycl`,
`𝔽_p((t^{1/p^∞}))`), and a proper full subcategory otherwise. (iii) Sch12's tilt (Definition 6.16)
of `X` over `K` is the glued tilt `X♭` with its structure map to `Spa(K♭, K♭⁺₀)` (tilting preserves
the smallest ring of integral elements, ECD Lemma 3.11), and Sch12 Proposition 6.17 is
*tilting-slice-equivalence* for the base `Spa(K, K⁺₀)`: the functorial bijection
`Hom(Spa(R♭, R♭⁺), X♭) = Hom(Spa(R, R⁺), X)` over `K♭`, resp. `K`, holds. (iv) Sch12's fibre
products (Proposition 6.18) are those of *fibre-products-of-perfectoid-spaces*. For `K = ℂ_p`,
Sch12's perfectoid spaces over `ℂ_p` are the perfectoid spaces over `Spa(ℂ_p, O_{ℂ_p})`; for `K` the
completed perfection of `𝔽_p(x)((t))` and `R⁺` the preimage in `K°` of the valuation ring
`𝔽_p[x]^{perf}_{(x)}` of the residue field `𝔽_p(x)^{perf}`, `Spa(K, R⁺)` is a perfectoid space over
`K` in Sch12's sense with no morphism to `Spa(K, K°)`. *Needs:* *perfectoid-space*;
*tilting-slice-equivalence*; *fibre-products-of-perfectoid-spaces*; *affinoid-perfectoid-space*;
layer P1 (P1.1, P1.15); Tau Ceti
`TauCeti.Huber.IsRingOfIntegralElements.mem_of_isTopologicallyNilpotent`; Mathlib
`CategoryTheory.Over`.

`PerfectoidSpace.fieldMinimalPlus` is the integral closure of the subring generated by
the topologically nilpotent elements (which already contains the prime subring).
`fieldBasePair` bundles it. `FieldSpace` records a perfectoid space and a continuous scalar
map into its compatible global sections; continuity uses the subspace topology of the
product of chart rings. Morphisms preserve that map through `globalPullback`.
`fieldSpaceEquivalence_space` and `fieldSpaceEquivalence_scalar` fix the comparison with
the minimal-plus slice. `fieldSpace_maximalPlus_criterion` states the extra integrality
condition for the maximal-plus slice.
**Checks.** `field_minimal_topologically_nilpotent` includes topologically nilpotent scalars;
`field_minimal_inverse_parameter` excludes the inverse parameter;
`field_minimal_cyclotomic` and `field_minimal_cp` identify the minimal and maximal plus rings
in the two algebraic-residue examples. `field_category_empty`,
`field_category_scalar_two_terms` and `field_category_recovers_object` test the empty space,
the scalar addition law and recovery under the equivalence. The proper minimal-plus example uses `ResidueLaurentField`, the localization of the
`t`-adic completion of the monoid algebra over `RationalResiduePerfection`.
`residueValuationRing` is the valuation ring at `x=0`, specified by fractions with
denominator nonzero at zero and their compatible Frobenius roots.
`residueLaurentReduction_spec` fixes reduction on constants and its kernel of
topologically nilpotent elements. `mem_residueLaurentPlus` specifies the inverse
image of that residue valuation ring. `residueLaurentPlus_proper` proves this plus
ring is strictly smaller than the power-bounded ring and contains every
 topologically nilpotent element.

**Checks.** The ten definitions `RationalResiduePerfection`, `residueValuationRing`,
`ResidueLaurentPolynomial`, `residueLaurentMonomial`, `ResidueLaurentIntegral`,
`residueLaurentParameter`, `ResidueLaurentField`, `residueLaurentRoot`,
`residueLaurentReduction`, and `residueLaurentPlus` have shared examples:
`_roots` checks the root relation and the integral pseudouniformizers; `_zero`
checks reduction of zero and the nonzero nonunit integral parameter; `_proper`
puts `x` in the plus ring and `x⁻¹` in its power-bounded complement.

**Checks for the affinoid definitions.** The presentation notation `RationalLoc 𝔭`
abbreviates the carrier of `PairOfDefinition.completionLocObj`; `RationalLoc.pair` and `RationalLoc.structureMap`
bundle the supplier's plus ring and structure morphism. `IsTiltPresentation` means that
both numerator sets span the unit ideal and their rational subsets correspond under `spaTilt`.
The comparison `rationalLocalization.tiltEquiv_affinoid` satisfies
`rationalLocalization.tiltEquiv_affinoid_spec`: continuity in both directions,
equality of the transported plus rings, and commutation with the tilted structure map.
These conditions apply in every characteristic, including the zero-ring localization.

- `spaTilt_test_zero_one` — at any point, `|0| ≤ |1|` and not `|1| ≤ |0|`.
- `spaTilt_test_two_powers` — `|ϖ♭²| < |ϖ♭|`; this fixes the direction of `vle` using two powers.
- `spaTilt_test_charP_eq_id` — in characteristic `p`, transport by `tilt.equivOfCharP` gives the original point.
- `spaTilt_test_no_ringHom` — there is no unital ring map from the characteristic-`p` tilt of `ℚ_p^cycl` to `ℚ_p^cycl`.
- `spaTiltHomeomorph_test_empty`, `spaTiltHomeomorph_test_whole`,
  `spaTiltHomeomorph_test_charP` — the homeomorphism carries the empty and whole subsets
  to themselves and agrees with characteristic-`p` transport on points.
- `rationalLoc_test_empty` — denominator zero gives the zero ring, its full plus ring,
  and the zero structure map.
- `rationalLoc_test_whole` — the presentation `({1},1)` recovers `(R,R⁺)` as a topological
  pair, with the structure map inverse to the comparison.
- `rationalLoc_test_supplier` — the pair's plus ring is `completedPlusSubring` and its
  morphism is `toCompletionLoc`, exactly. Together these three Checks apply to
  `RationalLoc`, `RationalLoc.pair`, and `RationalLoc.structureMap`. The carrier equality
  with `completionLocObj` also holds by reflexivity, as does equality of
  `RationalLoc.instTopologicalSpace` with the supplied object's topology.
  `RationalLoc.instTopologicalSpace_empty`, `RationalLoc.instTopologicalSpace_whole`, and
  `RationalLoc.instTopologicalSpace_supplier` use these same three examples to check
  the zero ring, the homeomorphism for the whole space, and the supplied topology.
  The mirrored example labels for the carrier and pair bundles are
  `RationalLoc_empty`, `RationalLoc_whole`, `RationalLoc_supplier`,
  `RationalLoc.pair_empty`, `RationalLoc.pair_whole`, `RationalLoc.pair_supplier`,
  `RationalLoc.structureMap_empty`, `RationalLoc.structureMap_whole`, and
  `RationalLoc.structureMap_supplier`.
- `isTiltPresentation_test_whole`, `isTiltPresentation_test_empty`,
  `isTiltPresentation_test_mismatch` — `({1},1)` matches itself, `({1},0)` matches itself,
  and these two presentations cannot match each other on a nonempty spectrum.
- `tiltEquiv_test_zero_one`, `tiltEquiv_test_empty`, `rationalLocalization_test_charP` —
  the comparison sends `0,1` to `0,1`, sends every element to zero for two empty
  localizations, and agrees with the characteristic-`p` structure-map comparison.
- `cech_test_two_terms` — with `d⁰(a,b)=b−a`, `(2,5)` maps to `3`, `(5,2)` to `−3`,
  and the augmented section `(2,2)` to zero. This fixes the alternating Čech sign.
- `naiveModel_test_two_exponents` — at `p=2`, multiplication of `(T/ϖ)^{1/2}` by
  `ϖ^{1/2}` gives coefficient exponent `0`, whereas multiplication by `ϖ^{1/4}`
  gives exponent `−1/4`. The inequality `1−2^{-n}>1` never holds, so the family
  `1−p^{-n}>p^{-k}` in the naive-model Check requires `k≥1`.
- `almostExact_test_nonseparated` — on `ℚ_p`, `(p)=ℚ_p` and `⋂_n p^nℚ_p=ℚ_p`:
  reduction of `[ℚ_p→0]` is zero, but the complex has nonzero degree-zero cohomology.

The structure presheaf is the one supplied by Tau Ceti on the underlying `PreAdicSpace`
(`presentationLimitPresheafInCommRingCat`, with `presentationLimitPlusPresheaf`); it is also
described by compatible rational sections, and that description is a comparison, not a second
carrier: `Perfectoid.rationalSections U`, the subring of the product of all `O(V)`, rational
`V ⊆ U`, cut out by restriction compatibility, and `rationalIntegralSections U`, requiring every
component to belong to `O⁺(V)`, are auxiliary family types; `rationalPresheaf` and
`integralPresheaf`, the presheaves of such families with the literal projection along an open
inclusion, are identified with the supplied presheaves by the comparisons below and are not used
downstream in their own right. For a perfectoid pair `R` at a prime `p` and any pair
of definition `D`, `rationalSectionsEquivPresentationLimit` identifies these ordinary
sections with `presentationLimitPresheafInCommRingCat D R.pair.plus` on each open.
`rationalSectionsEquivPresentationLimit_base` sends the family induced by any `a : R`
to Tau Ceti’s canonical `toPresentationLimit` section ([Wedhorn] §8.1, pp. 74–75).
`rationalSectionsEquivPresentationLimit_natural` states compatibility with every restriction;
`rationalPresheaf_iso_presentationLimit` packages its underlying additive maps as a natural
isomorphism. If `D.ringOfDefinition ≤ R.pair.plus`,
`rationalSectionsEquivPresentationLimit_plus` identifies the integral subrings and
`integralPresheaf_iso_presentationLimit` gives the induced natural isomorphism with
Tau Ceti's `presentationLimitPlusPresheaf` ([Wedhorn] §8.1, equation (8.1.1) and its extension
to opens, p. 75; Def. 8.13 and Prop. 8.16, pp. 78–79).
`rationalPresheaf_isSheaf` and `integralPresheaf_isSheaf` give `rationalSheaf` and
`integralSheaf` through these comparisons ([ECD] Thm 3.18, p. 18, and the pointwise integral
condition). Ordinary acyclicity comes from AdicSpacesPartIII R3.1 on the supplied structure
sheaf. `integralSheaf_almost_acyclic` and `integral_cech_almost_acyclic` use Mathlib's
`Sheaf.H` and Čech homology maps; `integralScalar_apply` fixes multiplication on every
rational component ([ECD] Thm 3.24, p. 19). Applying to `rationalPair U` gives the
rational-open forms.

**Checks for the presentation-limit comparison.**
`rationalSectionsEquivPresentationLimit_empty` sends every section of the empty open to zero;
`rationalSectionsEquivPresentationLimit_two_constants` computes `(2 + 5) ↦ 7` on any open;
`rationalSectionsEquivPresentationLimit_nonintegral` excludes the image of `ϖ⁻¹` from the
supplied integral subring on an open containing a point, for a compatible `D`.

**Checks for sections, scalar actions and the Čech sign.** `sections_empty` gives zero
ordinary and integral sections; `sections_rational_recovery` evaluates a compatible family
on its largest rational member and recovers its pair; `sections_nonintegral_constant`
excludes `ϖ⁻¹` on a nonempty open. `sections_restriction_two_constants` and
`integral_restriction_two_constants` compute restriction of sums componentwise.
`integral_scalar_zero`, `integral_scalar_one` and `integral_scalar_two_terms` fix the scalar
endomorphisms. `rationalCechDifference` is `res_V(b) − res_U(a)` on an intersection:
`cech_actual_two_terms` computes `(2,5) ↦ 3`, `cech_actual_reverse` computes `(5,2) ↦ −3`,
and `cech_actual_diagonal` computes `(2,2) ↦ 0` in the actual rational-localization ring.

The global criterion detects `InCharP` on global functions;
`characteristic_p_open_functions` states perfectness of functions on every open.
The nested-restriction statement fixes nested restrictions by their inclusions.
**Checks for chart coordinates.** `chart_identity_sections` recovers a scalar on the identity
chart; `chart_empty_ring` forces every chart of the empty space to have zero ring;
`chart_parameter_nonzero` distinguishes a parameter from zero and from its square.
`characteristic_p_affine_empty` gives the empty characteristic-p affinoid,
`characteristic_p_affine_roots` gives bijective Frobenius on its functions, and
`characteristic_p_affine_agreement` recovers the supplied affinoid ring.
`localization_charp_zero` checks the empty localization without imposing exact `CharP` on
the zero ring; `sections_charp_roots` checks bijective Frobenius on sections.
`perfectoid_cp_point` computes the spectrum of `(ℂ_p, ℂ_p°)` as a singleton.
`tiltCharPIso_sharp` fixes the characteristic-p comparison by sharp on functions, and
`characteristic_p_sharp_two_terms` computes it on a sum of two sections.

### Examples

The worked examples of this layer are the checks above and the running computations in the
subsections: on the perfectoid disc `Spa(ℚ_p^cycl⟨T^{1/p^∞}⟩)` the point `T ↦ ϖ` tilts to `T ↦ ϖ♭`,
`{|T| ≤ |ϖ|}` tilts to `{|T| ≤ |ϖ♭|}`, its localisation is `ℚ_p^cycl⟨(T/ϖ)^{1/p^∞}⟩` with tilt
`ℚ_p^cycl♭⟨(T/ϖ♭)^{1/p^∞}⟩`, the naive model `R°⟨T/ϖ⟩` without roots is not almost integral, the
restriction to `{|T| ≤ |ϖ|²}` sends `(T/ϖ)^{1/p^m}` to `ϖ^{1/p^m}(T/ϖ²)^{1/p^m}`, the completed
residue field at the Gauss point is the Gauss-norm completion of `ℚ_p^cycl(T^{1/p^∞})`, and the
integral Čech complex of the Laurent covering is exact; the ring
`ℤ_p^cycl[[T^{1/p^∞}]]⟨(p/T)^{1/p^∞}⟩[1/T]` of ECD Example 3.4(iv), over no field, whose
localisation at `{|T| ≤ |p|}` inverts `p`; the plus ring `K° + K°°⟨T^{1/p^∞}⟩` which differs from
`R°`; `Spa(ℂ_p, O_{ℂ_p})` as a one-point affinoid perfectoid space, `Spa(ℚ_p, ℤ_p)`, the closed unit
disc over `ℚ_p` and the discrete `Spa(𝔽_p, 𝔽_p)` as adic spaces that are not perfectoid, and the
perfectoid disc with doubled origin which is quasi-compact but not quasi-separated; the tilts
`Spa(ℚ_p^cycl, ℤ_p^cycl)♭ ≅ Spa(𝔽_p((t^{1/p^∞})), 𝔽_p[[t^{1/p^∞}]])` and
`Spa(R⟨T^{1/p^∞}⟩)♭ ≅ Spa(R♭⟨T^{1/p^∞}⟩)`, with the untilt of the perfectoid disc over
`𝔽_p((t^{1/p^∞}))` to the one over `ℚ_p^cycl`; the products
`A⟨T^{1/p^∞}⟩ ⊗̂_A A⟨S^{1/p^∞}⟩ ≅ A⟨T^{1/p^∞}, S^{1/p^∞}⟩` and `L ⊗̂_K L ≅ ∏_G L` for a Galois
extension of perfectoid fields, whose `Spa` has `|G|` points over a one-point fibre product of
spaces, and whose plus ring `(L°)^ℓ` strictly contains the image of `L° ⊗_{K°} L°`;
`ℂ_p ⊗̂_{ℚ_p} ℂ_p`, not uniform, against the perfectoid `ℂ_p ⊗̂ᵘ_{ℚ_p} ℂ_p` and
`ℚ_p^cycl ⊗̂ᵘ_{ℚ_p} ℚ_p^cycl`; and the absolute product
`Spa 𝔽_p((t^{1/p^∞})) × Spa 𝔽_p((s^{1/p^∞}))`, the perfectoid punctured open unit disc, with the
chart whose plus ring must contain `u_m, v_n`.

The Lean `example` of `Suggested.lean` for this layer checks that an empty pre-adic space is
perfectoid exactly when it is sheafy (its sections over `∅` vanish), the chart condition being
vacuous.

### Dependencies

Layers P0 (the basic setup, Serre classes of almost zero modules, inversion of an element of `m`,
almost sheaves and almost sheaf cohomology: P0.1, P0.5, P0.16, P0.29) and P1 (perfectoid Tate
rings and pairs, pseudouniformisers with compatible roots, tilts of pairs and of morphisms, the
perfected Tate algebra, perfectoid fields and the tilting equivalence of rings over a perfectoid
base: P1.1–P1.6, P1.15, P1.18 (*rings-of-integral-elements-under-tilting* (a)), P1.19, P1.27, P1.30) of this roadmap;
P0.29's Cartan criterion, P2's `almost_cartan` and AdicSpacesPartII R0.19.5–R0.19.8's
generic spectral-limit, Tate-pair spectrum and finite-stage-open statements. Other roadmaps, by subsection: P2.3 and
P2.4 use AdicSpaces 3.1 (rational localisations and restriction maps); P2.5 uses AdicSpaces 3.3
(stalks and completed residue fields); P2.6 uses AdicSpaces 4.2 (the Buzzard–Verberkmoes theorem,
stably uniform implies sheafy) and AdicSpacesPartIII R3.1 (Čech acyclicity of the structure sheaf on
sheafy Tate affinoids, *sheafy-cech-acyclicity*), as do P2.7 and P2.19 through P2.6; P2.8 uses
AdicSpacesPartII R0.9 (*reduced-affinoid-supremum-norm*); P2.9–P2.11 use AdicSpaces 3.4, 3.5, 4.1,
5.1 and 5.3 (the structure presheaf and sheafiness, strongly noetherian pairs, pre-adic spaces, open
subspaces and gluing); P2.10 and P2.18 use AdicSpacesPartII R0.2 (*discrete-huber-pair-sheafy*) and
R0.6 (*adic-morphism*, *adic-iff-analytic-locus*); P2.13 uses AdicSpacesPartII R0.1
(*adic-ring-homomorphism*), R0.3 (*completed-tensor-product*,
*completed-tensor-product-universal-property*) and R0.4 for the uniformisation; P2.14 and P2.17 use
R0.4 (*uniformization*, *uniform-completed-tensor-product*); P2.15–P2.17 use R0.3
(*completed-tensor-rational-localisation*) and R0.7
(*affinoid-fibre-product*, *fibre-products-existence*), and AdicEtaleGeometry A0.1
(*tate-completed-tensor-points*) supplies their point surjectivity; P2.16 also uses R0.6
(*adic-morphism*), and P2.17 also R0.6 (analyticity) and AdicSpaces 2.3; the comparisons of
P2.9–P2.11 and P2.16 with AdicEtaleGeometry's analytic Yoneda carrier use its A1.16 (generalized adic
presentations: `YonedaAdicSpace`, `ofAdicSpace`, `ofSpaY`, `mapsFromAdicSpaceEquiv`). None of these
cites this roadmap. P3.17a.4 (sousperfectoid spaces,
*sousperfectoid-adic-space*) uses P2.10; and P5.13.2's *char-p-perfectoid-base-field*
uses P2.8, not conversely. Tau Ceti `TauCeti.AdicSpace`,
`TauCeti.PreAdicSpace`, `PreAdicSpace.Hom`, `PreAdicSpace.IsOpenImmersion`,
`TauCeti.Huber.IsStablyUniform`, `TauCeti.Huber.IsSheafyRing`,
`TauCeti.ValuationSpectrum.isIso_toPresentationLimit_top`,
`TauCeti.ValuationSpectrum.residueFieldValuation`, `TauCeti.Huber.Pair`,
`TauCeti.Huber.Pair.Hom` and its `spaComap`, `TauCeti.Huber.IsTateRing`,
`TauCeti.Huber.powerBoundedSubring`, `TauCeti.Huber.IsBounded`,
`TauCeti.Huber.IsRingOfIntegralElements`, `TauCeti.Huber.IsStronglyNoetherian`,
`TauCeti.Huber.PairOfDefinition.completionLocObj`, `completedPlusSubring` and
`restrictionRingHomOfSubset`, `TauCeti.Huber.restrictedMvPowerSeriesCompletion`,
`TauCeti.ValuationSpectrum.spa`, `rationalSubset` and its lemmas, `isTopologicalBasis_spaRationalFamily`,
`isCompact_of_mem_spaRationalFamily`, `exists_finite_spaRationalFamily_refinement`,
`existsUnique_continuous_ringHom_of_forall_comap_mem_rationalSubset`, `mem_iff_forall_vle_one`,
`spectralSpace_spa_of_pairOfDefinition`, `spaAnalytic_eq_spa_of_isTateRing`, `IsAnalyticPoint`,
`spaLocalizationHomeomorph`, `isSheaf_iff_isSheafFor_rationalCover`; and Mathlib `Valuation`,
`PerfectRing`, `frobeniusEquiv`, `CharP`, `IsAdicComplete`, `UniformSpace.Completion`,
`integralClosure`, `Algebra.TensorProduct.includeLeft`, `Topology.IsStrictMap`, `Homeomorph`,
`HomologicalComplex.ExactAt`, `CategoryTheory.Sheaf.H`, `TopCat.Presheaf.IsSheaf`,
`CategoryTheory.cechComplexFunctor`, `CategoryTheory.ObjectProperty.IsSerreClass`,
`CategoryTheory.ObjectProperty.FullSubcategory`, `ObjectProperty.ι`,
`ObjectProperty.fullyFaithfulι`, `ObjectProperty.IsClosedUnderIsomorphisms`,
`CategoryTheory.GlueData`, `CategoryTheory.Over`, `CategoryTheory.Equivalence`,
`CategoryTheory.IsPullback`, `CategoryTheory.Limits.HasPullbacks`,
`CategoryTheory.Limits.HasBinaryProducts`,
`CategoryTheory.Over.ConstructProducts.over_binaryProduct_of_pullback` and
`CategoryTheory.Adjunction.rightAdjoint_preservesLimits`. Within this roadmap the layer is
consumed by layer P3 (almost purity, the plus ring of completed tensor products).

## Layer P3: almost purity and the étale site

Layer P3 proves the almost purity theorem for perfectoid Tate rings and builds the étale site of a
perfectoid space. It opens with the algebraic prefix — henselian pairs, Elkik's approximation theorem
and its non-noetherian form, and the invariance of finite étale algebras under henselian completion
and under completed colimits — then proves almost purity in characteristic `p`, constructs the
untilting functor on finite étale algebras, treats finite extensions of perfectoid fields and the
Fontaine–Wintenberger theorem, and assembles the almost purity theorem with its three conclusions:
finite étale algebras over a perfectoid Tate ring are perfectoid, tilting is an equivalence of finite
étale algebras, and the integral closure is almost finite étale. On the geometric side it defines
finite étale and étale morphisms of perfectoid spaces, compares them with the adic notions of
AdicEtaleGeometry A1.2–A1.3 (finite étale Huber pairs, finite étale and étale morphisms) and A1.6
(the sites), and builds the étale site with its tilting equivalence and the almost
acyclicity of `O⁺`. AdicEtaleGeometry A1.0.4–A1.0.5 supplies classical henselian approximation
and finite étale completion. The perfectoid-ring results require no perfectoid base field;
the field results explicitly assume a perfectoid field.

### P3.2 perfectoid-field specialization of henselian completion

The perfectoid-field application of henselian completion is supplied by
AdicEtaleGeometry A1.0.5 (*henselian-finite-etale-approximation*): for a perfectoid
field `K`, a pseudouniformiser `ϖ ∈ K°` and a flat `K°`-algebra `A` henselian
along `(ϖ)`, apply its part (a) with `R = A`, `t = ϖ` and `I = A` to obtain
`FÉt(A[ϖ⁻¹]) ≃ FÉt(Â[ϖ⁻¹])`, where `Â` is the `ϖ`-adic completion
([Sch12] §7, Proposition 7.4). *Needs:* P1.10; AdicEtaleGeometry A1.0.5.

### P3.3 almost purity in characteristic p

Almost purity in characteristic `p` is the case in which perfectness does the work: a finite étale
algebra over a perfect complete Tate ring is again perfectoid, and its integral closure is almost
finite étale.

**Almost purity in characteristic `p`: finite étale algebras over a perfectoid Tate ring of
characteristic `p`.** (*finite-etale-covers-in-characteristic-p*) Let `R` be a perfectoid Tate ring of
characteristic `p` (equivalently a perfect complete Tate ring of characteristic `p`, layer P1; no base
field), `ϖ ∈ R` a pseudouniformiser (it has all `p`-power roots `ϖ^{1/pⁿ}` in `R°`), `R⁺ ⊆ R°` a ring
of integral elements, and `S ∈ FÉt(R)` with its natural topology and plus ring `S⁺` = the integral
closure of `R⁺` in `S`. Then: (i) `S` is perfect, complete and uniform, hence a perfectoid Tate ring of
characteristic `p`, and `(S, S⁺)` is a perfectoid Huber pair; (ii) `S°` is the integral closure of `R°`
in `S` and `S°/S⁺`, viewed as an `R⁺`-module, is killed by `R°°`;
thus `S°a` and `S⁺a` agree after transport between the basic setups
`(R⁺, R°°)` and `(R°, R°°)`; (iii) `S°a` is an almost finite étale `R°a`-algebra
(almost finitely presented projective and unramified: the diagonal idempotent `e ∈ S ⊗_R S` lies in
`(S°a ⊗_{R°a} S°a)_*`); (iv) `S°a` is uniformly almost finite projective: there is `n ≥ 0` such that
for every `ε ∈ R°°` there are `R°`-linear maps `S° → (R°)ⁿ → S°` whose composite is multiplication by
`ε`; (v) if `S` is faithfully flat over `R`, then `Tr_{S/R}(S°°) ⊇ R°°` and `R°a → S°a` is almost
faithfully flat and almost split. Almost mathematics is that of the basic setup `(R°, R°°)`
(layer P1), equivalently of the ideal `(ϖ^{1/p^∞})`, and almost finite étale is the notion of
layer P0. Not claimed: `S°` finite, flat or étale over `R°`, or `S` free over `R`. The statement
specialises to the case of a perfectoid `K`-algebra over a perfectoid field `K` of characteristic `p`,
and to the finite separable extensions of such a field ([Sch12] §5, Prop. 5.23, pp. 28–29, over a field; [GR] §3.5,
Thm 3.5.28, pp. 69–71, applied to `R°`, gives the base-field-free argument). *Needs:*
layer P1 (perfectoid Tate rings, the characteristic-`p` criterion, the basic setup `(R°, R°°)`).

`finiteEtale_charP_perfectoid` proves perfectness and supplies the Tate structure,
completeness and Hausdorffness for the finite-module topology in (i).
Its characteristic hypothesis is `pR = 0`, allowing the zero base and zero algebra;
it requires no `CharP S p` instance for the zero algebra.
For the resulting finite étale pair, `Pair.integralClosure_powerBounded`
states (ii), including the annihilation formula `εs ∈ S⁺` for every
`ε ∈ R°°` and `s ∈ S°`. `Pair.integralClosure_almostFiniteEtale` states (iii),
`Pair.integralClosure_uniformlyAlmostFiniteProjective` states the fixed-rank
factorizations in (iv), and `Pair.integralClosure_faithfullyFlat` states (v),
including exact trace lifts of topologically nilpotent elements and linear
splittings after multiplication by each almost scalar. These pair theorems
also give the characteristic-independent conclusions of P3.12. The trace
and splitting clauses follow from the almost purity assertion and
[GR] Cor. 4.1.11 and Thm 4.1.14, pp. 74–75.

Two lemmas feed the proof. Relative Frobenius of a weakly étale map is an isomorphism, and étale
algebras over perfect rings are perfect (*etale-over-perfect-is-perfect*): for a weakly étale map
`A → B` of `𝔽_p`-algebras (flat with flat multiplication `B ⊗_A B → B`) the square of `A → B` with the
absolute Frobenii (Mathlib `frobenius`) is a pushout, so `B ⊗_{A, Frob} A → B` is an isomorphism and `B`
is perfect (Mathlib `PerfectRing`) when `A` is; the same holds for weakly étale morphisms of almost
algebras over a basic setup of characteristic `p`, so for a finite étale `(R°a/ϖ)`-algebra `B`, relative Frobenius identifies
`B/ϖ^{1/p}` with `B`, with the base rings identified by `x ↦ x^p` ([Bhatt] Ch. 4, §4.3, Lem. 4.3.8).
Uniformity and the power-bounded integral-closure comparison for finite étale algebras
are supplied by AdicEtaleGeometry A1.2a (*finite-etale-over-uniform-is-uniform*).

Lean: `perfectRing_of_etale` (*etale-over-perfect-is-perfect* for étale algebras over a perfect ring
of characteristic `p`).

### P3.4 characteristic p: inverting `ϖ` is an equivalence

**Characteristic `p`: inverting `ϖ` is an equivalence between almost finite étale `R°a`-algebras and
finite étale `R`-algebras.** (*characteristic-p-almost-purity-equivalence*) Let `R` be a perfectoid
Tate ring of characteristic `p` with pseudouniformiser `ϖ`, and `(R°a)_afet` the category of almost
finite étale `R°a`-algebras (layer P0). Then the functor `(R°a)_afet → FÉt(R)`, `A ↦ A_*[ϖ⁻¹]`
(`A_*` the ring of almost elements, layer P0), with its natural topology, is an equivalence of
categories, compatible with tensor products; a quasi-inverse is `S ↦ S°a`. Moreover every
`A ∈ (R°a)_afet` is uniformly almost finite projective, `A_*` is perfect, `ϖ`-torsion free and
`ϖ`-adically complete, and `A_*` is the integral closure of `R°` in `A_*[ϖ⁻¹]`; almost faithful
flatness of `A` corresponds to faithful flatness of `A_*[ϖ⁻¹]` ([GR] §3.5, 3.5.26 and Thm 3.5.28,
p. 69). *Needs:* P3.3.

The construction `AlmostFiniteEtale p R` uses P0's category under the almost
base algebra. `almostInversion p R ϖ` sends `A` to the localization of its
actual algebra of almost elements. `almostInversionMap_scalars` identifies
`R° → A_* → A_*[1/ϖ]` with the structural map through `R`;
`almostInversion_isLocalization` specifies its universal property and
`almostInversionMap_natural` its action on arrows. The equivalence instance is
`almostInversion_isEquivalence`; the same construction applies in all
characteristics by P3.11. Parameter comparison `almostInversionParameterIso`
fixes every almost element and satisfies the cocycle equation.

`almostInversion_charP_perfect` states bijectivity of Frobenius on the ring
of almost elements when `pR = 0`, including the zero ring ([GR] proof of Thm 3.5.28,
Claim 3.5.35, pp. 70–71). For every almost finite étale algebra,
`almostInversion_elements_complete` states injectivity of multiplication
by `ϖ` and `ϖ`-adic completeness of its almost elements;
`almostInversion_elements_integralClosure` states that the localization map
is injective and its image consists exactly of elements integral over `R°`.
`almostInversion_uniformlyAlmostFiniteProjective` gives a single finite bound,
and `almostInversion_faithfullyFlat_iff` identifies almost faithful flatness
with faithful flatness after inversion. These four assertions hold in all
characteristics by P3.11 ([Sch12] Lem. 5.3(iv), p. 22, Prop. 5.23 and
Thm 5.25, pp. 28–29, Thm 7.9, p. 42; [GR] Claims 3.5.31–35, pp. 70–71).

**Checks.** `inversion_rank_zero`, `inversion_rank_one`, `inversion_rank_two`
apply this actual functor to the almostifications of `(R°)⁰`, `R°`, `(R°)²`,
with outputs `0`, `R`, `R²`. `inversion_characteristic_p_wrong_parameter`
asserts that the image of `almostSplit p R 1` under `almostInversion p R ϖ`
is nontrivial for a nonzero characteristic-`p` perfectoid pair and admits no
ring equivalence to `Localization.Away (p : R)`. Thus replacing `ϖ` by `p=0`
changes the functor's actual output.

### P3.5 the untilting functor on finite étale algebras

**The untilting functor on finite étale algebras, `T ↦ T♯`, via almost finite étale integral
structures.** (*finite-etale-untilting-functor*) Let `R` be a perfectoid Tate ring with tilt `R♭`
(layer P1) and choose a pseudouniformiser `ϖ♭ ∈ R♭` with `ϖ := (ϖ♭)♯` satisfying `ϖ^p | p` in `R°`
(layer P1), so that `R♭°/ϖ♭ = R°/ϖ` (layer P1). In `Perfectoid.FiniteEtale`, define `untilt` as the
untilting functor `(−)♯ : FÉt(R♭) → FÉt(R)` (`CommAlgCat.FiniteEtale R♭ ⥤ CommAlgCat.FiniteEtale R`),
the composite

`FÉt(R♭) ≃ (R♭°a)_afet → (R♭°a/ϖ♭)_afet = (R°a/ϖ)_afet ≃ (R°a)_afet → FÉt(R), T ↦ T°a ↦ T°a/ϖ♭ ↦ A ↦ A_*[ϖ⁻¹] =: T♯,`

where the first equivalence is *characteristic-p-almost-purity-equivalence*, the middle equivalence
is the unique flat deformation of almost finite étale algebras along the `ϖ`-adically complete flat
`R°a` (layer P0), and the last functor inverts `ϖ` (natural module topology on `T♯`).
Prove `untilt_isPerfectoid` directly for the constructed `T♯`, from layer P1's primitive-ideal
untilting: `T♯ = W(T⁺)[1/[ϖ♭]]/ξ` is the untilt of the perfectoid ring `T` (of characteristic
`p`, `T` being finite étale over the perfectoid `R♭`, hence perfectoid as a uniform complete Tate
ring that is perfect) along the primitive element `ξ`, and untilts of perfectoid rings are
perfectoid (layer P1); no statement about arbitrary finite étale `R`-algebras is used. The
general theorem `FiniteEtale.naturalPerfectoid` — for every finite étale algebra `S` over a
complete Hausdorff perfectoid Tate ring `R`, the supplied `Huber.FiniteEtale R S` with its natural
module topology is perfectoid ([ECD] Theorem 6.1(i), p. 26) — is NOT a target of this subsection:
it is the essential-surjectivity milestone of *finite-etale-tilting-equivalence* (P3.9) and is
exposed, and registered as an instance, only there and in P3.10, after that milestone; nothing in
P3.5–P3.8 depends on it. Prove `tiltUntiltIso`
(a canonical isomorphism `(T♯)♭ ≅ T` of `R♭`-algebras, natural in `T`, so that `(−)♯` is inverse to
tilting on its image and compatible with the untilting of perfectoid algebras of layer P1). Further,
`T♯` is finite étale over `R`; `(T♯)°a = A` is almost finite étale over `R°a`; the functor is
independent of `ϖ♭` up to canonical isomorphism, preserves tensor products, finite products, ranks
(the rank of `T♯` at `x ∈ Spa R` equals the rank of `T` at `x♭`) and faithful flatness; and it
commutes with base change along every map of perfectoid Tate rings `R → R'` (for example a rational
localisation or the completed residue field at a point): `(T ⊗_{R♭} R'♭)♯ ≅ T♯ ⊗_R R'`. Equivalently
`T♯ = W(T⁺)[1/[ϖ♭]]/ξ`, with `T⁺` the integral closure of `R♭⁺` in `T`, for a generator `ξ` of `ker(θ : W(R♭⁺) → R⁺)` (layer P1), which shows finite
étaleness without almost mathematics ([KL15] Lem. 3.6.20; [AWS] Lem. 2.8.10) ([Sch12] §5, Prop. 5.22
and the diagram following it, p. 28). *Needs:* P3.4; layer P0 (flat deformation of almost finite
étale algebras along a complete flat almost algebra); layer P1 (tilts, untilts, `θ`).

**Checks.**

- Degenerate: `untilt R♭ ≅ R` (`R♭` as a finite étale
  `R♭`-algebra), `untilt 0 = 0`, and `untilt (T₁ × T₂) ≅ untilt T₁ × untilt T₂`.
- Computation: for `K = ℚ_p(p^{1/p^∞})^`,
  `K♭ = 𝔽_p((t^{1/p^∞}))^` with `t♯ = p` and `p` odd: `untilt (K♭[u]/(u² − t)) ≅ K[u]/(u² − p)`.
- Computation: for `T = R♭ × R♭`, the sharps of
  the marked idempotents `(1,0)` and `(0,1)` become the respective idempotents of `R × R`.
  The general comparison `tilt (untilt T) ≅ T` is `FiniteEtale.tiltUntiltIso`;
  `FiniteEtale.untiltPairIso` identifies this untilt with P1's untilt.

The split-algebra Checks are `untilt_test_rank_zero`, `untilt_test_rank_one`
and `untilt_test_rank_two`: the finite étale algebras `R♭⁰`, `R♭` and `R♭²` untilt
respectively to the zero algebra `R⁰`, `R` and `R²`.

`FiniteEtale.tiltFunctor`, the geometric tilting functor on all of `FÉt(R)`, has carrier the
tilt of the supplied finite-algebra module topology, with `tiltFunctor_map_coeff` giving its
coordinatewise action on algebra homomorphisms; it is defined only on the full subcategory of
finite étale `R`-algebras that are perfectoid (the essential image of P3.6) until P3.9's
essential-surjectivity milestone shows that this is all of `FÉt(R)`, and it is never used to
prove that perfectoidness. `untilt_isPerfectoid`, `tiltUntiltIso`,
`untiltTiltIso`, and `tiltUntilt_triangle` specify the geometric inverse and
its triangle identity. `finiteEtaleTiltEquiv_comparison` identifies the
categorical inverse with this geometric functor. For every perfectoid Tate
pair `(R,R⁺)` and finite étale `R♭`-algebra `T`, `untiltWittIso` is an
`R`-algebra equivalence from `untilt T` to P1's localized Witt quotient,
using the extension `J` of `ker θ_R` to `W(T⁺)`. Its scalar action comes from
P1's slice structural map through `PrimitiveTriple.untiltCarrierEquiv`.
Write `m_T : (T♯)♭ ≅ T` for `tiltUntiltIso.app T`.
`untiltWittIso_apply` fixes the equivalence by
`(m_T⁻¹(x))♯ ↦ [x] mod J` for every `x ∈ T⁺`, followed by localization.
`untiltWittIso_scalars` sends the scalar `θ_R(w)` to the class of the image
of `w` in `W(T⁺)`, for every `w ∈ W(R♭⁺)`; this also specifies the base
Fontaine map. `untiltPairIso` is the comparison under the base perfectoid
pair, preserving the integral-closure plus ring, and `untiltPairIso_app`
identifies its carrier map with this very Witt equivalence. These are the
finite étale restriction of P1's marked untilting construction ([KL15]
Lemma 3.6.20, p. 97; for general perfectoid Tate rings, [ECD] Theorem 6.1(ii),
p. 26).

For every continuous map `f : R → R'` of perfectoid Tate rings,
`untiltBaseChangeIso` uses the scalar actions from `f` and `f♭`.
`untiltBaseChangeIso_app` sends
`(m_(R'♭ ⊗ T)⁻¹(a ⊗ t))♯` to `a♯ ⊗ (m_T⁻¹(t))♯` for every
`a ∈ R'♭` and `t ∈ T`. `untiltBaseChangeIso_id` identifies the identity
base map with `CommAlgCat.FiniteEtale.baseChangeSelfIso`.
For continuous `R → R' → R''`, `untiltBaseChangeIso_comp` equates the
two successive comparisons with the comparison for the composite, after
`Algebra.TensorProduct.cancelBaseChange` on both sides. No injectivity or
nonzero-ring hypothesis is required ([ECD] Theorem 6.1(ii), p. 26, and the
functorial construction of [Sch12] Proposition 5.22 and Theorem 5.25,
pp. 28–29).

For finite étale `T,U` over `R♭`, `untiltProductIso_app` characterizes
`untiltProductIso` as the ordered pair of the untilted projections.
`untiltProductIso_natural` commutes with arbitrary maps of both factors;
`untiltProductIso_id` and `untiltProductIso_comp` give its identity and
composition equations. These are the product comparison laws for the
same equivalence ([ECD] Theorem 6.1(ii), p. 26). The functor
`finiteEtalePairFunctor` uses the supplied natural module topology, and
`finiteEtalePairFunctor_map` fixes its action on ring maps.

**Checks.** `untiltWittIso_zero`, `untiltWittIso_one`,
`untiltWittIso_teichmuller` evaluate the zero algebra, a scalar sharp in the
rank-one algebra as the localized Teichmüller class, and the marked idempotent
`(1,0)` of a nonzero split algebra. Its localized Teichmüller class is neither
zero nor one. `untiltBaseChangeIso_identity`,
`untiltBaseChangeIso_composite`, `untiltBaseChangeIso_scalar` test the
identity base map, two successive base maps, and a pure scalar tensor.
`untiltProductIso_ordered_idempotent` sends the sharp of the marked
idempotent `(1,0)` to `(1,0)`, distinguishing the ordered comparison from
a swap on a nonzero split algebra. `untiltProductIso_identity` and
`untiltProductIso_composite` test identity and successive product maps.

The characteristic-`p` Check `fontaine_wintenberger_test_char_p` sends a tilt sequence
to its zeroth coefficient, the `p`th power of its first coefficient ([Berkeley]
Lemma 6.2.2, p. 44). `fontaine_wintenberger_test_algebraically_closed` computes
two geometric points for the untilt of the split quadratic algebra ([ECD]
Theorem 6.1(ii), p. 26).

**Checks.** `finite_pair_zero`, `finite_pair_one`, `finite_pair_two` compute
the zero pair, the base pair, and the product plus ring `R⁺ × R⁺`.
`geometric_tilt_zero`, `geometric_tilt_one`, `geometric_tilt_two`
compute the actual geometric functor on the three split algebras.
`untilt_actual_tilt` fixes both ordered idempotents of the split rank-two algebra;
`untilt_arbitrary_product` sends the sharp of the marked `(0,1)` to `(0,1)` for
any two finite étale factors. These values use [ECD] Theorem 6.1(ii), p. 26.
`untilt_test_quadratic` states the quadratic computation over every perfectoid
field with `p ≠ 2` and every nonzero `t ∈ K♭`, hence in particular the Kummer
tower above, as an isomorphism of `K`-algebras.
`untilt_quadratic_nonsplit` adds that nonsquareness of `t` forces
irreducibility of `X²−t♯`, so the comparison includes actual quadratic fields.
`quadratic_characteristic_two_nonexample` asserts that for a perfectoid field
`K` at `p=2`, no object of `CommAlgCat.FiniteEtale (tilt 2 K)` is algebra-isomorphic
to `AdjoinRoot (X²−t)`. This excludes the quadratic algebra from the domain of
`FiniteEtale.untilt`, including `t=0`.

### P3.6 untilting is fully faithful and inverse to tilting on its image

**The untilting functor `FÉt(R♭) → FÉt(R)` is fully faithful, inverse to tilting on its image, with
image the `S` having almost finite étale integral structure.**
(*tilting-finite-etale-algebras-fully-faithful*) Let `R` be a perfectoid Tate ring with tilt `R♭`
(any characteristic, no perfectoid base field). The untilting functor `(−)♯ : FÉt(R♭) → FÉt(R)` of
*finite-etale-untilting-functor* is fully faithful and, on its essential image, inverse to tilting:
for `S = T♯`, `S` is perfectoid and `S♭ ≅ T`. Its essential image consists exactly of the finite
étale `R`-algebras `S` such that `S` (with its natural topology) is perfectoid and `S°a` is almost
finite étale over `R°a`; for such `S`, `S°a` is uniformly almost finite projective over `R°a`. The
functor preserves degrees (rank functions), tensor products and faithful flatness. For `R = K` a
perfectoid field this is the input of [Sch12] Thm 3.7; for `R` of characteristic `p` (`R♭ = R`) it is
the identity ([Sch12] §5, Thm 5.25, p. 29). *Needs:* P3.5.

### P3.7 finite extensions of perfectoid fields

**Finite extensions of perfectoid fields are perfectoid, and tilting is a degree-preserving
equivalence of finite extensions.** (*finite-extensions-of-perfectoid-fields*) Let `K` be a perfectoid
field (characteristic `0` or `p`) with tilt `K♭`. (i) Every finite extension `L/K`, with its natural
topology (the unique extension of the absolute value, Mathlib `spectralNorm`), is a perfectoid field.
(ii) The untilting functor `(−)♯` of P3.5 is an equivalence `FÉt(K♭) ≃ FÉt(K)` with quasi-inverse
`L ↦ L♭`; it restricts to an equivalence between finite field extensions of `K♭` and of `K`,
preserving degrees, automorphism groups and Galois extensions. (iii) `K` is algebraically closed if
and only if `K♭` is: `isAlgClosed_tilt [IsAlgClosed K]` gives
`IsAlgClosed (tiltField p K)`, since the degree-preserving equivalence in (ii) forces every
finite extension of `K♭` to have degree one ([Sch12] §3, Thm 3.7(ii), p. 17).
Here `p` is prime and `K` is complete, nontrivially normed, ultrametric and perfectoid at `p`;
`isAlgClosed_of_tilt` gives the reverse implication ([Sch12] §3, Prop. 3.8, pp. 17–18).
(iv) For every finite extension `L/K`, `L°a` is almost finite étale over `K°a`
(Tate; [GR] §6.6 for deeply ramified fields), uniformly almost finite projective ([Sch12] §3,
Thm 3.7(i), p. 17). *Needs:* P3.6.

Three lemmas accompany the theorem. If the tilt of a perfectoid field is algebraically closed, so is
the field (*untilt-of-algebraically-closed-field*): for `K` a perfectoid field with tilt `K♭`
(layer P1), `IsAlgClosed K♭` implies `IsAlgClosed K` (Mathlib `IsAlgClosed`); in characteristic `p`,
`K♭ = K` and the statement is trivial ([Sch12] §3, Prop. 3.8). The completed cyclotomic extension of
a finite extension of `ℚ_p` is perfectoid (*cyclotomic-completion-of-finite-extension*): for `K` a
finite extension of `ℚ_p` inside a fixed algebraic closure, with completion `C`, the closure `L` of
`K(μ_{p^∞})` in `C` is a perfectoid field, namely the compositum `L = K·ℚ_p^cycl` with the completed
cyclotomic field `ℚ_p^cycl` (layer P1), a finite extension of `ℚ_p^cycl` of degree at most
`[K : ℚ_p]` in which `K(μ_{p^∞})` is dense, and its tilt `L♭` is a finite extension of
`(ℚ_p^cycl)♭ = 𝔽_p((t^{1/p^∞}))^` of degree `[L : ℚ_p^cycl]`, with
`Gal(L/ℚ_p^cycl) ≅ Gal(L♭/(ℚ_p^cycl)♭)` by (ii) ([Sch12] §3, Thm 3.7(i)). The completed tensor
product of plus rings
need not be integrally closed (*completed-tensor-plus-ring-not-integrally-closed*): for `K` the
completion of `ℚ̆_p(μ_{p^∞})`, with algebraically closed residue field and
`|K×| = |p|^{ℤ[1/p]/(p−1)}`,  `ℓ` a prime with `ℓ ∤ p(p − 1)` and `L = K(p^{1/ℓ})` — a cyclic Galois extension of
degree `ℓ`, perfectoid by (i), with trivial residue extension — one has `L ⊗̂_K L ≅ L^ℓ` with plus ring
`(L°)^ℓ`, while `L° ⊗_{K°} L° → (L°)^ℓ` is injective and not surjective; so the map from the `ϖ`-adic
completion of `B⁺ ⊗_{A⁺} C⁺` to `D⁺` in layer P2 is an almost isomorphism that is not an isomorphism,
and the completed tensor product of plus rings is not integrally closed in `D`, hence not a ring of
integral elements. Indeed every tensor has the same residue in all `ℓ` factors,
and this condition persists on completion; the integral idempotent `(1,0,…,0)`
fails it. This calculation uses the Galois product decomposition and P3.12
for the almost comparison.

`completed_tensor_plus_not_integrally_closed` gives the same obstruction for any
nontrivial finite Galois extension of perfectoid fields whose automorphisms act
trivially on the residue field. It specifies the completed-tensor map on pure
tensors and exhibits the integral idempotent of the identity factor outside its
image. `tame_cubic_perfectoid_extension` supplies the residue-characteristic-two,
degree-three instance inside `ℂ₂`. The finite-field theorem and Galois product
decomposition above supply the hypotheses; equal residues exclude that idempotent.

Lean: `IsPerfectoidField.of_finiteDimensional` (finite extensions of perfectoid fields are
perfectoid) and `isPerfectoidField_of_isAlgClosed` (a complete algebraically closed nonarchimedean
field with `‖p‖ < 1` is a perfectoid field).

### P3.8 the Fontaine–Wintenberger theorem

The Fontaine–Wintenberger theorem is the field case of the tilting equivalence, passed to the
limit over all finite extensions.

**Fontaine–Wintenberger: the absolute Galois groups of a perfectoid field and of its tilt are
isomorphic.** (*fontaine-wintenberger*) Let `K` be a perfectoid field with tilt `K♭`. Fix an algebraic
closure `K̄` of `K` with completion `C`, so that `C♭` is an algebraically closed perfectoid field
containing `K♭` (P3.7(iii), `isAlgClosed_tilt`), and let `K♭^sep` be the separable
closure of `K♭` in `C♭`. Then `L ↦ L♭` (for finite Galois `L/K` inside `K̄`, with `L♭ ⊆ C♭` by
functoriality of tilting) defines an isomorphism of profinite groups
`Gal(K̄/K) ≅ Gal(K♭^sep/K♭)` (Mathlib `Field.absoluteGaloisGroup` up to the choice of closures),
compatible with the Galois correspondences (Mathlib
`InfiniteGalois.IntermediateFieldEquivClosedSubgroup`): finite subextensions of `K̄/K` correspond to
finite subextensions of `K♭^sep/K♭` with the same degrees ([Sch12] §3, Thm 3.7(ii), p. 17, the
degree-preserving equivalence of finite extensions; §1, Thm 1.1, p. 2, is the case
`K = ℚ_p(p^{1/p^∞})^`). The
direction of the comparison is fixed by the functor: `L ↦ L♭` goes from finite extensions of `K`
to finite extensions of `K♭`, and its quasi-inverse is the untilt `M ↦ M♯` of P3.5; on Galois groups
the isomorphism `Gal(K̄/K) → Gal(K♭^sep/K♭)` sends `σ` to the automorphism of `K♭^sep` induced on
the inverse limit `lim_{x ↦ x^p}` of `C` by the continuous extension of `σ` to `C`. *Needs:* P3.7;
AdicEtaleGeometry A1.0.5 (henselian completion); layer P1 (the tilt of a perfectoid field).

**Checks.**

- `fontaine_wintenberger_test_kummer_tower` (computation) — `K = ℚ_p(p^{1/p^∞})^`: `K♭ = 𝔽_p((t^{1/p^∞}))^` with `t = (p, p^{1/p}, p^{1/p²}, …)`, `t♯ = p`; `K♭^sep` is the separable closure
  of `𝔽_p((t^{1/p^∞}))^`, and `Gal(K♭^sep/K♭) = Gal(𝔽_p((t))^sep/𝔽_p((t)))` since
  `𝔽_p((t)) ⊂ K♭` is a purely inseparable, then completed, extension, neither of which changes
  finite étale algebras (*finite-etale-under-completed-perfection*, *henselian-finite-etale-approximation*);
  so `G_{ℚ_p(p^{1/p^∞})} ≅ G_{𝔽_p((t))}` (the uncompleted `ℚ_p(p^{1/p^∞})` has the same Galois group
  as its completion by AdicEtaleGeometry A1.0.5(b)). For `p` odd the quadratic extensions correspond: `K(√p) ↔ K♭(√t)`
  (P3.5, `untilt_test_quadratic`).
- `fontaine_wintenberger_test_p_two_cyclotomic` (degenerate, `p = 2`) — `K = ℚ_2(ζ_{2^∞})^`:
  `K♭ = 𝔽_2((t^{1/2^∞}))^` with `t = ε − 1`, `ε = (1, −1, ζ_4, ζ_8, …)` (for `p = 2` the first
  nontrivial root of unity is `ζ_2 = −1`, and `ε` is a compatible system of `2^n`-th roots of `1`),
  and `G_{ℚ_2(ζ_{2^∞})} ≅ G_{𝔽_2((t))}`; in characteristic `2` no polynomial `u² − a` is separable
  (`t^{1/2}` already lies in `K♭`), so the separable quadratic extensions of `𝔽_2((t))` are the
  Artin–Schreier ones `u² − u = f`, and they match the quadratic extensions of `ℚ_2(ζ_{2^∞})`: the
  unramified one `K(ζ_3) = K(√−3)` corresponds to `u² − u = 1`, i.e. `𝔽_4((t))`, and all others are
  wildly ramified (`√−1 = ζ_4` and `√2 = ζ_8 + ζ_8^{-1}` already lie in `K`). The
  Kummer-type check `untilt_test_quadratic` of P3.5 needs `p` odd for this reason.
- `fontaine_wintenberger_test_char_p` (degenerate) — `K` of characteristic `p`: `K♭ = K`, the
  isomorphism is the identity of `Gal(K^sep/K)`.
- `fontaine_wintenberger_test_algebraically_closed` (degenerate) — `K = C` algebraically closed:
  `C♭` is algebraically closed by `isAlgClosed_tilt` (P3.7(iii)) and both groups are trivial;
  conversely a nontrivial `Gal(K̄/K)` forces `K♭` not separably closed.

In characteristic `p`, finite étale algebras are insensitive to completed perfection
(*finite-etale-under-completed-perfection*): for `S` a complete Tate ring of characteristic `p` with
ring of definition `S₀ ∋ ϖ` and `R = (S^{perf})^` the completion of `S^{perf} = colim(S → S → ⋯)`
along Frobenius, for the Tate topology with ring of definition `colim S₀^{1/pⁿ}` — a perfect complete
Tate ring, hence perfectoid (layer P1, P1.2) — base change is an equivalence
`FÉt(S) ≃ FÉt(R)`; in particular, for a `p`-finite perfectoid affinoid `(R, R⁺)`, the completed
perfection of a reduced affinoid `(S, S⁺)` of topologically finite type over a characteristic-`p`
perfectoid field, `R_fét ≃ S_fét`, and the base field is not needed ([Sch12] §7, Lem. 7.5(ii)).

The continuous group isomorphism is `Perfectoid.fontaineWintenberger` on
Mathlib's absolute Galois groups. Its marking by the action on the completed
algebraic closure is the coordinatewise convention above.
`fontaine_wintenberger_test_p_two_unramified` computes the actual untilt over
`ℚ₂^cycl`: the Artin–Schreier algebra `u²−u−1` over its tilt becomes the
algebra `v²+v+1`, adjoining a primitive third root of unity. This is the
unramified quadratic check, not a characteristic-two Kummer polynomial.

`fontaineWintenbergerFiberIso` identifies the geometric fibre of the actual
untilt with the tilted geometric fibre. `fontaineWintenberger_fiber_action`
fixes the action convention by postcomposition on algebra maps into the
chosen algebraic closures. Thus the continuous group isomorphism is tied to
the finite-algebra equivalence; changing compatible fibres conjugates it.

**Check.** `fontaine_wintenberger_p_two_action` identifies the nonidentity
unramified quadratic action `u ↦ u+1` on `u²−u=1` with `ζ₃ ↦ ζ₃²` on
`ζ₃²+ζ₃+1=0`, through the untilt functor over `ℚ₂^cycl`.

`Perfectoid.fontaineWintenberger_nonempty` states the existence of this topological
group isomorphism on Mathlib's `Tilt` and `Field.absoluteGaloisGroup` carriers.
Its hypotheses are P1.11's criterion: `K` is complete, nontrivially normed and
ultrametric, its norms accumulate at `1`, `‖p‖ < 1`, and Frobenius on `K°/p` is
surjective. Here `K°` is `NormedField.valuation.integer`, and the tilt is the
fraction field of `PreTilt K° p`. The marked map and its fibre-action
compatibility are the targets above.

Lean: `Perfectoid.fontaineWintenberger_nonempty`.

### P3.9 tilting is an equivalence of finite étale algebras

**Tilting is an equivalence between finite étale algebras over a perfectoid Tate ring and over its
tilt.** (*finite-etale-tilting-equivalence*) Let `R` be a perfectoid Tate ring ([ECD] Def. 3.1;
layer P1: any characteristic, no perfectoid base field) with tilt `R♭`. Then the untilting functor
`(−)♯ : FÉt(R♭) → FÉt(R)` of *finite-etale-untilting-functor* is an equivalence of categories, with
quasi-inverse the tilting functor `S ↦ S♭` (`S` being perfectoid). It is compatible with tensor
products, with base change along maps of perfectoid Tate rings (in particular rational localisations
and completed residue fields), and preserves ranks and faithful flatness. For every ring of integral
elements `R⁺ ⊆ R°` with tilt `R♭⁺` (layer P1), it induces an equivalence between finite étale
morphisms of Huber pairs `(R, R⁺) → (S, S⁺)` and `(R♭, R♭⁺) → (S♭, S♭⁺)` (plus rings the integral
closures; the finite étale Huber pair of AdicEtaleGeometry A1.2), i.e. between finite étale
affinoid perfectoid spaces over `Spa(R, R⁺)` and over `Spa(R♭, R♭⁺)` ([ECD] §6, Thm 6.1(ii), p. 26).
The content beyond P3.6 is essential surjectivity, stated as its own milestone
(*finite-etale-untilting-essentially-surjective*): every finite étale `R`-algebra `S` (natural
module topology) is perfectoid with `S°a` almost finite étale over `R°a`, hence lies in the
essential image of P3.6. Full faithfulness and the description of the restricted essential image do
not give this; the proof is local-to-global, with the following inputs, each a named statement:
(i) the field case — for a perfectoid field `K`, `FÉt(K♭) → FÉt(K)` is essentially surjective
([Sch12] Thm 3.7 and §7, proof of Thm 7.9, pp. 42–43; P3.7); (ii) the completed-residue-field
case at a point `x ∈ Spa(R, R⁺)` — `S ⊗_R k(x)^` is in the image, by (i) applied to the perfectoid
field `k(x)^` (P3.8); (iii) rational-neighbourhood spreading — the untilt `T_x♯ ≅ S ⊗_R k(x)^`
found in (ii) spreads to a rational neighbourhood `U_x` of `x` with `T_{U_x}♯ ≅ S ⊗_R O(U_x)`, by
the finite-étale approximation of AdicEtaleGeometry A1 (*finite-etale-approximation*, Henselian
spreading of finite étale algebras along the filtered colimit `colim_{U ∋ x} O(U) → k(x)^`,
[Sch12] Lemma 7.5(i)) and the base-change compatibility of P3.5; (iv) overlap compatibility — on
`U_x ∩ U_y` the two untilts are identified by full faithfulness (P3.6) applied to the tilts of the
transition isomorphisms; (v) affinoid gluing — the `T_{U_x}` glue to a finite étale `R♭`-algebra
`T` by finite étale descent on the finite rational cover (A1's *finite-etale-rational-descent*,
with finite projectivity of the pieces), and `T♯ ≅ S` by the sheaf property. The residue-field
spreading statement below is (iii). Only after this milestone are `FiniteEtale.naturalPerfectoid`
(P3.10) and the unrestricted `FiniteEtale.tiltFunctor` (P3.5) exposed as consequences. *Needs:*
P3.6; P3.7; P3.8; P3.5 (base change); AdicEtaleGeometry A1 (*finite-etale-approximation*,
*finite-etale-rational-descent*); P2.3.

Use `Perfectoid.finiteEtaleTiltEquiv` for the inverse equivalence to `untilt`.
**Checks.** `finiteEtaleTiltEquiv_test_rank_zero`, `finiteEtaleTiltEquiv_test_rank_one`
and `finiteEtaleTiltEquiv_test_rank_two` compute its values on `R⁰`, `R` and `R²`
as `R♭⁰`, `R♭` and `R♭²`. `finiteEtaleTiltEquiv_test_direction` checks that tilting
followed by untilting is naturally isomorphic to the identity.

Finite étale algebras over the completed residue field spread out to a rational neighbourhood
(*residue-field-finite-etale-colimit*): for `(A, A⁺)` a complete Tate Huber pair (Tau Ceti
`Huber.Pair`, `Huber.IsTateRing`), `X = Spa(A, A⁺)` (Tau Ceti `ValuationSpectrum.spa`), `x ∈ X` with
completed residue field `k(x)^`, and `O(U)` the rational localisation on rational subsets `U ∋ x` (Tau
Ceti `ValuationSpectrum.rationalSubset`; the Foundations of adic spaces roadmap (AdicSpaces),
layer 3), base change induces an equivalence `2-colim_{U ∋ x} FÉt(O(U)) ≃ FÉt(k(x)^)` compatible with
tensor products: every finite étale `k(x)^`-algebra is `L ⊗ k(x)^` for some `U ∋ x` and
`L ∈ FÉt(O(U))`, every `k(x)^`-algebra map between such comes from an `O(V)`-algebra map over some
rational `x ∈ V ⊆ U`, and two maps agreeing over `k(x)^` agree over some such `V`; no sheafiness of
`(A, A⁺)` is required ([Berkeley] Lecture 7, §7.4, Lem. 7.4.6 and the argument after Thm 7.4.8).

`Perfectoid.rationalResidueMap` evaluates a rational localization of any complete
Tate pair at a point in its rational subset. `rationalResidueMap_spec` states
continuity and agreement with the original residue map; no sheafiness or
perfectoidness is assumed ([Berkeley] Lemma 7.4.6 and the argument after
Theorem 7.4.8, pp. 53–55). Its Checks are
`rationalResidueMap_base` (its kernel on base elements is the valuation support),
`rationalResidueMap_denominator` (the denominator stays nonzero),
`rationalResidueMap_fraction` (`t/s` maps to `t(x)/s(x)`, with the inverted
denominator in the actual completed rational localization), and
`rationalResidueMap_empty` (a zero rational localization cannot contain the point).
The statements `residue_field_finite_etale_essSurj`, `residue_field_finite_etale_full`
and `residue_field_finite_etale_faithful` give objects, morphisms and eventual
equality, with the refinement and scalar maps explicitly recorded.

### P3.10 finite étale algebras over a perfectoid Tate ring are perfectoid

**Finite étale algebras over a perfectoid Tate ring are perfectoid.**
(*finite-etale-over-perfectoid-is-perfectoid*) Let `R` be a perfectoid Tate ring (any characteristic,
no perfectoid base field) and `S` a finite étale `R`-algebra (Mathlib `Module.Finite R S`,
`Algebra.Etale R S`) with its natural topology as a finite `R`-module. The target `Perfectoid.isPerfectoidTateRing_of_finiteEtale` supplies a Tate structure,
completeness and Hausdorffness for that topology, and proves that `S` is perfectoid; `S ≅ (S♭)♯` with `S♭ ∈ FÉt(R♭)`; for every ring of integral elements `R⁺ ⊆ R°`, with `S⁺` the
integral closure of `R⁺` in `S` (the finite étale pair `Huber.Pair.finiteEtale R⁺ S` of
AdicEtaleGeometry A1.2, *finite-etale-affinoid-algebra*), `(S, S⁺)` is a perfectoid Huber pair whose tilt `(S♭, S♭⁺)` has `S♭⁺`
the integral closure of `R♭⁺` in `S♭`. Consequently `Spa(S, S⁺) → Spa(R, R⁺)` is a finite étale
morphism of affinoid perfectoid spaces (AdicEtaleGeometry A1.3). AdicSpacesPartIV R5 uses this as its
finite étale step downstream ([Berkeley] Thm 7.4.5(1); [HK] Prop. 2.15(a)) ([ECD] §6, Thm 6.1(i), p. 26).
This subsection re-exports P3.9's essential-surjectivity milestone
(*finite-etale-untilting-essentially-surjective*): `isPerfectoidTateRing_of_finiteEtale` and the
instance `FiniteEtale.naturalPerfectoid` are that milestone's conclusion, registered here, and are
not a second proof target; nothing in P3.5–P3.9 depends on them. *Needs:* P3.9
(*finite-etale-untilting-essentially-surjective*); AdicEtaleGeometry A1.2 and A1.3.

Lean: `isPerfectoidTateRing_of_finiteEtale` (with the finite-module topology, `S` is a complete
Hausdorff perfectoid Tate ring); the Lean `example` checks the split case `R × R`.

### P3.11 almost purity, categorical form

**Almost purity, categorical form: `(R°a)_afet ≃ FÉt(R)` by inverting `ϖ`, compatibly with
tilting.** (*almost-finite-etale-equivalence*) Let `R` be a perfectoid Tate ring with tilt `R♭`,
`ϖ = (ϖ♭)♯` a pseudouniformiser with `ϖ^p | p`, and `R⁺ ⊆ R°` any ring of integral elements. Then:
(a) the functor `(R°a)_afet → FÉt(R)`, `A ↦ A_*[ϖ⁻¹]` (natural topology), is an equivalence of
categories with quasi-inverse `S ↦ S°a`, identified with `S⁺a` after the
change of basic setup in (c), compatible with tensor products and base change along
maps of perfectoid Tate rings; (b) the six categories

`FÉt(R) ≃ (R°a)_afet ≃ (R°a/ϖ)_afet = (R♭°a/ϖ♭)_afet ≃ (R♭°a)_afet ≃ FÉt(R♭)`

of Scholze's diagram are equivalent by the functors of *finite-etale-untilting-functor*, and the
composite is the tilting equivalence of *finite-etale-tilting-equivalence*; (c)
`(R⁺a)_afet ≃ (R°a)_afet` by extension along `R⁺ → R°`: the quotient
`R°/R⁺` is almost zero as an `R⁺`-module. The two almost categories use the
basic setups `(R⁺, R°°)` and `(R°, R°°)` respectively (layer P1; [ECD] Def. 3.23). For a perfectoid algebra over a characteristic-zero perfectoid field, inverting `p` gives
`R°a_afet ≃ R_fet`; in characteristic `p`, inverting a pseudouniformiser `t` gives `S°a_afet ≃ S_fet`. Tilting gives
`R°a_afet ≃ S°a_afet` for `S = R♭` ([Sch12] §7, Thm 7.9(ii)–(iii) and its proof, p. 42;
[ECD] §6, Thm 6.1 for the general perfectoid Tate ring). *Needs:* P3.9.

For (b), `finiteEtaleTiltEquiv_sixCategories` uses compatible rooted
pseudouniformisers `ϖ♭` and `ϖ = (ϖ♭)♯` with `ϖᵖ | p` in `R°`.
It states that the two reduction functors, each P0's
`Almost.Algebra.finiteEtaleBaseChange` along the actual quotient map, are
equivalences. The middle equivalence transports the scalar action by sharp
modulo the parameters; ring equivalences on almost elements, natural on
arrows, specify that transport. The composite of inverse inversion,
reduction, inverse sharp transport, inverse reduction and inversion is
naturally isomorphic to `(finiteEtaleTiltEquiv p R).functor` ([Sch12] diagram
after Prop. 5.22, pp. 28–29, and Thm 7.9, p. 42).

For (c), `almostFiniteEtale_equivPlus` states the equivalence for every
perfectoid pair `(R,R⁺)`. It specifies the inverse as restriction from
`R°` to `R⁺` through natural ring equivalences on almost elements that
preserve the inclusion's scalar action. Thus its forward functor is
extension along `R⁺ → R°`; it uses P1's `almostSetupPlus` and
`almostModuleEquivPlus` ([ECD] Example 3.22(ii) and Def. 3.23, p. 19).

### P3.12 the integral closure is almost finite étale

**The integral closure is almost finite étale: `S°` over `R°` for `S` finite étale over a perfectoid
Tate ring `R`.** (*integral-closure-almost-finite-etale*) Let `R` be a perfectoid Tate ring (any
characteristic, no perfectoid base field), `R⁺ ⊆ R°` a ring of integral elements, `ϖ` a
pseudouniformiser, `S ∈ FÉt(R)` and `S⁺` the integral closure of `R⁺` in `S`; almost mathematics is
that of `(R°, R°°)` (layer P1). Then: (i) `S°` is the integral closure of `R°` in `S`,
`S°° ⊆ S⁺ ⊆ S°`, and `S°a` is almost finite étale over `R°a`; equivalently,
`S⁺a` is almost finite étale over `R⁺a` under the change of basic setup in P3.11(c): almost finitely
presented, almost projective, and unramified (the diagonal idempotent of `S ⊗_R S` is an almost
element of `S° ⊗_{R°} S°`); (ii) `S°a` is uniformly almost finite projective: there is `n ≥ 0` such
that for every `ε ∈ R°°` there are `R°`-linear maps `S° → (R°)ⁿ → S°` composing to multiplication by
`ε`; (iii) `Tr_{S/R}(S°) ⊆ R°` and the trace pairing induces an almost isomorphism
`S°a → alHom_{R°a}(S°a, R°a)`; (iv) if `S` is faithfully flat over `R`, `S°a` is almost faithfully
flat over `R°a`, `Tr_{S/R}(S°°) ⊇ R°°` (up to almost zero), and `R°a → S°a` is almost split. Not
claimed, and false in general: `S°` finite, flat or étale over `R°`, or `S` free over `R` ([ECD] §6,
Thm 6.1(iii), p. 26, for (i); [Sch12] Thm 5.25 and Thm 7.9(iii), pp. 29, 42, for (ii) over a
perfectoid field; [GR] §4.1, Theorem 4.1.14 and Corollary 4.1.11, pp. 74–75, for (iii)–(iv)).
*Needs:* P3.11.

The proposed Lean hypotheses are `f : R ⟶ S` between `PairCat p` objects and
`Pair.IsFiniteEtale f`, which fixes the finite-module topology and the
integral-closure plus ring. Clause (i) is
`Pair.integralClosure_powerBounded` together with
`Pair.integralClosure_almostFiniteEtale`; clause (ii) is
`Pair.integralClosure_uniformlyAlmostFiniteProjective`, with a bound `n`
chosen before `ε`. Clause (iii), `Pair.integralClosure_trace`, states both
trace containment and an almost isomorphism from `S°` to its `R°`-linear
dual whose evaluation is `Tr(st)`. Clause (iv),
`Pair.integralClosure_faithfullyFlat`, assumes faithful flatness of the
specified ring map and gives almost faithful flatness, a topologically
nilpotent trace preimage for every `ε ∈ R°°`, and `R°`-linear maps
`rε : S° → R°` with `rε(f(a)) = εa`. Sources for the additional clauses are
[Sch12] Prop. 5.23, pp. 28–29, and Thm 7.9(iii), pp. 42–43;
[GR] (4.1.7), Cor. 4.1.11 and Thm 4.1.14, pp. 74–75.

`Perfectoid.FiniteEtale.trace` is the trace of multiplication for a finite
projective étale algebra. `FiniteEtale.trace_retraction` characterizes it
using any retraction `S → Rⁿ → S`: if `aᵢ` are the coordinate functionals
and `bᵢ` the images of the standard basis, its value at `s` is `∑ᵢ aᵢ(s bᵢ)`.
This agrees with P0's trace in the classical basic setup and requires no
freeness assumption ([GR] Def. 4.1.1, p. 72, and (4.1.7), p. 74).
`FiniteEtale.trace_eq_of_free` identifies it with Mathlib’s `Algebra.trace`
whenever the algebra is finite free.

**Checks.** `FiniteEtale.trace_zero` gives zero on the zero algebra;
`FiniteEtale.trace_one` gives `Tr(s)=s` on the identity algebra;
`FiniteEtale.trace_two` gives `Tr(s₀,s₁)=s₀+s₁` on the split rank-two algebra.
`FiniteEtale.trace_nonconstant_rank` uses the projection `K × K → K` for a
field `K`: the module is not free and the trace sends `x` to `(x,0)`.
`integral_trace_zero_not_faithful` excludes the zero algebra from the faithful
clause over a nonzero base. `integral_trace_characteristic_two` gives
`Tr(1,1)=0` but `Tr(1,0)=1` in characteristic two: faithful flatness does not
require the trace of the unit to be a unit.

Lean: `exists_uniformlyAlmostFiniteProjective_powerBounded` (one `n` such that for every `ε ∈ R°°`,
`ε • id_{S°}` factors through `(R°)ⁿ`).

### P3.13 the almost purity theorem

The almost purity theorem collects the three conclusions proved above into one statement over an
arbitrary perfectoid Tate ring.

**The almost purity theorem for perfectoid Tate rings.** (*almost-purity-theorem*) For a perfectoid
Tate ring `R` ([ECD] Def. 3.1; characteristic `0`, `p`, or neither; no perfectoid base field) with
tilt `R♭`, the almost purity theorem ([ECD] §6, Thm 6.1, p. 26) is implied by the conjunction of P3.10 (every
`S ∈ FÉt(R)` is perfectoid), P3.9 (tilting `S ↦ S♭` is an equivalence `FÉt(R) ≃ FÉt(R♭)` with
quasi-inverse the untilting functor) and P3.12 (`S°` is almost finite étale and uniformly almost
finite projective over `R°`, equivalently `S⁺a` over `R⁺a`; categorically P3.11; P3.12's uniform
almost finite projectivity being [Sch12] Thm 5.25 and Thm 7.9(iii), pp. 29, 42), and is not stated
again here; for `R = K` a perfectoid field it is P3.7, and its consequence that (finite) étale
morphisms of perfectoid spaces are strongly (finite) étale is *etale-is-strongly-etale* (P3.17).
*Needs:* P3.10; P3.9; P3.12.

**Checks.**

- `almost_purity_test_identity` (degenerate) — `S = R`: (i) `R` is perfectoid; (ii) `R♭ ↦ R♭`;
  (iii) `R° = R°`, finite étale over itself, so almost finite étale, with `n = 1`: for each `ε ∈ R°°` use
  the identity followed by multiplication by `ε`. The composite sends `x` to `εx`;
  `almost_purity_test_scalar_composite` asserts uniform almost finite projectivity
  with bound one for `Almost.toAlmost (almostSetup p R) R°` and computes
  this composite in the almost category. `almost_purity_test_identity_negative`
  asserts that two identity arrows there do not compose to `0 • id` when `R ≠ 0`.
  Thus both Checks use the almost module occurring in the theorem.
  The example `almost_purity_test_identity` applies
  `Pair.integralClosure_almostFiniteEtale` to `𝟙 R` itself ([ECD] Theorem 6.1(iii), p. 26).
- `almost_purity_test_product` (degenerate) — `S = R × R` (finite étale of rank `2`, the split
  cover): `S` is perfectoid with `ϖ_S = (ϖ, ϖ)` and `S° = R° × R°`
  (`almost_purity_test_product_uniform`, `almost_purity_test_product_perfectoid`);
  the latter Check also computes `S° = R° × R°` and the pseudouniformizer `(ϖ,ϖ)`.
  Here `S♭ = R♭ × R♭`, and `S°` is
  finite étale over `R°` outright, so the almost statement holds with `n = 2`: use the identity followed by componentwise multiplication by
  any `ε ∈ R°°`. The ordinary finite projectivity also gives a splitting for `ε = 1`.
  The example `almost_purity_test_product` applies `Pair.integralClosure_almostFiniteEtale`
  to the diagonal pair map `(R,R⁺) → (R × R,R⁺ × R⁺)`;
  `almost_purity_test_product_finiteEtale` applies `FiniteEtale.isPerfectoid` to this
  split algebra with its product topology ([ECD] Theorem 6.1(i), (iii), p. 26).
- `almost_purity_test_zero` (degenerate) — `S = 0`, the finite étale algebra of rank `0`: the zero
  ring satisfies the definition of a perfectoid Tate ring (`0` is a topologically nilpotent unit,
  `0^p ∣ p`, Frobenius on `0` is bijective) and `0♭ = 0`; the statement is vacuous.
- `almost_purity_test_not_etale_integrally` (non-example of a stronger claim) — `R = K = ℚ_p^cycl`
  and `S = L = K(p^{1/ℓ})` for a prime `ℓ ∤ p(p − 1)`: `|p|^{1/ℓ} ∉ |K^×| = |p|^{ℤ[1/p]/(p−1)}`, so
  `L/K` is totally ramified of degree `ℓ` with value group index `ℓ`; `L° ≠ K°[p^{1/ℓ}]` (an
  element `a p^{1/ℓ}` with `1 < |a| < |p|^{−1/ℓ}`, which exists since `|K^×|` is dense, lies in
  `L°` but not in `K°[p^{1/ℓ}]`) and `L°` is not a finite free `K°`-module, while `L°a` is almost
  finite étale over `K°a` (Tate); (iii) does not assert étaleness or finiteness of `S°` over `R°`.
  The Check supplies a finite étale pair map with `L = K(a)`, `a^ℓ = p`,
  degree `ℓ`, and almost finite étale integral map, while its ordinary integral
  map is neither finite nor étale. Almost purity is [ECD] Theorem 6.1(iii), p. 26;
  the obstruction is the displayed value-group calculation (compare the
  shrinking-different computation in [Berkeley] Example 7.3.3, p. 51).
  The extension `K(√p)` is not an example: `√p ∈ ℚ_p(ζ_p)` for `p ≡ 1 (mod 4)` and
  `K(√p) = K(ζ_4)` is unramified for `p ≡ 3 (mod 4)`.

### P3.14 finite étale morphisms of perfectoid spaces

**Finite étale morphisms of perfectoid spaces.** (*finite-etale-morphism-of-perfectoid-spaces*) Let
`X, Y` be perfectoid spaces (layer P2: the full subcategory of the adic spaces of AdicSpaces locally
of the form `Spa(R, R⁺)`, `R` perfectoid). A morphism `f : Y → X` is finite étale if it is finite
étale as a morphism of analytic adic spaces (AdicEtaleGeometry A1.3: `X` has a cover by open
affinoids `V = Spa(A, A⁺)` with `f⁻¹(V) ≅ Spa(B, B⁺)` over `V`, `B ∈ FÉt(A)` with its natural
topology and `B⁺` the integral closure of `A⁺`). AdicEtaleGeometry states this
predicate in Lean on `TauCeti.AdicSpace` (`AdicSpace.IsFiniteEtale`, A1.3) through finite étale
maps of Huber pairs (`Huber.Pair.Hom.IsFiniteEtale`, A1.2); its form on the analytic Yoneda carrier,
`YonedaAdicSpace.IsFiniteEtale`, is an A1.9 target specified in its README only. In
`PerfectoidSpace`, define `IsFiniteEtale f` as `YonedaAdicSpace.IsFiniteEtale` applied to the image
of `f` under `YonedaAdicSpace.ofAdicSpace` (A1.16; perfectoid spaces being adic spaces);
`IsFiniteEtale.iff_adic` records this identity. Prove `IsFiniteEtale.iff_forall_affinoidPerfectoid` (the defining condition of [ECD]:
for every open affinoid perfectoid `U = Spa(R, R⁺) ⊆ X` the preimage `f⁻¹(U) = Spa(S, S⁺)` is
affinoid perfectoid with `R → S` finite étale and `S⁺` the integral closure of `R⁺` in `S`), which is
equivalent by *perfectoid-vs-analytic-finite-etale*, as are [Sch12] Def. 7.1(ii) (a cover of `X` by
such affinoids) and [Berkeley] Def. 7.5.1(1). `X_fét` denotes the category of perfectoid spaces
finite étale over `X` (all `X`-morphisms); for `X = Spa(R, R⁺)` affinoid perfectoid, global sections
give `X_fét ≃ FÉt(R)^op`. Conventions: the plus ring is the integral closure, never `S°` when
`R⁺ ≠ R°`; `f` need not be surjective; the degree `x ↦ dim_{k(x)}(O_Y ⊗ k(x))` is locally constant
([ECD] §6, Def. 6.2(i), p. 26). *Needs:* AdicEtaleGeometry A1.2 and A1.3 (finite étale Huber pairs
and finite étale morphisms of analytic adic spaces) and A1.9 (their Yoneda form).

**Checks.**

- `PerfectoidSpace.IsFiniteEtale.test_fold` (degenerate) — for every perfectoid space `X`,
  `X ⊔ X → X` and `∅ → X` are finite étale, and the identity is finite étale of degree `1`.
- `PerfectoidSpace.IsFiniteEtale.test_kummer` (computation) — over
  `X = Spa(ℚ_p^cycl⟨T^{±1/p^∞}⟩, ℤ_p^cycl⟨T^{±1/p^∞}⟩)`, `Spa(R[u]/(u^ℓ − T)) → X` (`ℓ ≠ p` prime)
  is finite étale of degree `ℓ`, and its source is perfectoid.
- `PerfectoidSpace.IsFiniteEtale.test_rational` (non-example) — the inclusion of the rational subset
  `{|T| ≤ |p|}` into the perfectoid closed disc is étale but not finite étale.

The global signatures use P2's `PerfectoidSpace.Perfd p`, the full subcategory
of `PreAdicSpace` defined by sheafiness and actual open perfectoid affinoid charts.
`IsFiniteEtale` quantifies over every such chart and requires a cartesian
square with `Spa(S) → Spa(R)` induced by a finite étale pair map.
The perfectoid-pair condition `Perfectoid.Pair.IsFiniteEtale` includes the finite
module, étale algebra, module topology, and integral-closure plus-ring conditions.
`finiteEtale_spa_iff` identifies the affinoid condition with the global one.
`IsFiniteEtale.iff_adic` compares this with the supplied
`YonedaAdicSpace.IsFiniteEtale`, which requires a finite étale cartesian
chart around each point of the target. There is only one analytic finite étale
predicate on the Yoneda carrier.

**Checks.** `finite_identity`, `finite_empty`, `finite_open_not_closed`
compute the identity, the empty inclusion, and the failure of finiteness for
any nonclosed open inclusion. The analogous analytic Checks are
`analytic_finite_identity`, `analytic_finite_empty`, `analytic_finite_nonclosed`.
At pair level, `pair_finite_identity`, `pair_finite_zero`, and
`pair_finite_strict_plus` test the identity, the zero algebra, and a bijective
ring map whose target plus ring is strictly larger than the image of the
source plus ring. The last map is not finite étale.

### P3.15 étale morphisms of perfectoid spaces

Étale morphisms of perfectoid spaces are the local composites of open immersions and finite étale
maps; the strong variants carry the almost finite étale condition explicitly, and the comparison with
AdicEtaleGeometry A1.3 identifies étale with strongly étale, and finite étale with strongly finite étale.

**Étale morphisms of perfectoid spaces.** (*etale-morphism-of-perfectoid-spaces*) A morphism
`f : Y → X` of perfectoid spaces is étale if it is étale in the local sense of AdicEtaleGeometry
A1.3 (`AdicSpace.IsEtaleLocalDescription`, stated in Lean on `TauCeti.AdicSpace`; its Yoneda form
`YonedaAdicSpace.IsEtaleLocalDescription` is an A1.9 target specified in AdicEtaleGeometry's README
only): every `y ∈ Y` has an open neighbourhood `V ⊆ Y` such
that `f|_V` factors as `V ↪ W → U ↪ X` with `V → W` an open immersion, `W → U` finite étale (P3.14)
and `U ⊆ X` open. `W` is then automatically perfectoid. This is [ECD] Def. 6.2(ii) (whose printed
condition `y ∈ Y` is read as `y ∈ V`), [Sch12] Def. 7.1(iii) and [Berkeley] Def. 7.5.1(2). Equivalent
local form: `U` may be taken a rational subset of an affinoid perfectoid open of `X`, `W = Spa(S, S⁺)`
with `S ∈ FÉt(O(U))`, and `V` a rational subset of `W` — here `R = O(U)` is a perfectoid Tate ring,
`S` is finite étale over it, and almost mathematics on the chart is that of `(R°, R°°)` (layer P1) —
so étale morphisms are locally composites "rational embedding, finite étale, rational embedding" of
affinoid perfectoid spaces; they need not be separated or quasi-compact. In `PerfectoidSpace`, define
`IsEtale f` as `YonedaAdicSpace.IsEtaleLocalDescription` applied to the image of `f` under
`YonedaAdicSpace.ofAdicSpace` (A1.16); `IsEtale.iff_adic` records this identity. Prove `IsEtale.of_isOpenImmersion`
(open immersions are étale) and `IsEtale.id` (the identity of any perfectoid space is étale)
([ECD] §6, Def. 6.2(ii), p. 26; [Sch12] Definition 7.1(iii), p. 39). *Needs:* AdicEtaleGeometry A1.3
(étale morphisms by the local description) and A1.9 (its Yoneda form); P3.14; layer P2.

**Checks.**

- `PerfectoidSpace.IsEtale.test_openImmersion` (degenerate) — open immersions of perfectoid spaces
  are étale; the identity is étale.
- `PerfectoidSpace.IsEtale.test_rational_in_cover` (computation) — for `R` perfectoid, `S ∈ FÉt(R)`
  and a rational subset `V ⊆ Spa(S, S⁺)`, the composite `V → Spa(R, R⁺)` is étale (standard affinoid
  étale map).
- Non-example: the inclusion of the open unit disc (a
  non-quasi-compact open) into the closed perfectoid disc is étale but not finite étale; so is the
  quasi-compact open immersion `{|T| ≤ |p|} ↪ {|T| ≤ 1}` (P3.14, `test_rational`), whose image is
  not closed, while finite maps are closed.
- `PerfectoidSpace.IsEtale.test_frobenius` (degenerate) — for `X` of characteristic `p` the
  absolute Frobenius `φ : X → X` is an isomorphism of perfectoid spaces (every `R` is perfect), hence
  finite étale of degree `1` and étale; nothing in this layer calls it inseparable, which it is only
  on the non-perfect `𝔽_p((t))` of P3.8.
- `PerfectoidSpace.IsEtale.test_separatedness_not_required` (characterisation) — the doubled disc
  `X ⊔_U X → X` for a nonclosed rational `U ⊊ X` is étale and not separated: the diagonal
  pulls back on the cross chart `X ×_X X` to the nonclosed inclusion `U ↪ X`.
  Étale morphisms need be neither separated nor quasi-compact, and the étale site of P3.18
  contains such objects.
- Empty-overlap boundary: for nonempty `X` and `U = ∅`, the fold `X ⊔ X → X` is split finite
  étale and separated. Properness of the overlap alone does not imply nonseparatedness.

The proposed `IsEtale` uses open factorizations in `Perfd p` and their equality
with the given map; A1’s `YonedaAdicSpace.IsEtaleLocalDescription` uses the
corresponding factorizations on the analytic Yoneda carrier. `IsEtale.iff_adic` compares these, and
`perfectoid_of_analyticEtale` supplies perfectoidness of an analytic étale
source over a perfectoid base.

**Checks.** `etale_identity`, `etale_empty`, `etale_open_not_finite` test
identity, empty inclusion, and a nonclosed open inclusion which is étale but
not finite. `analytic_etale_identity`, `analytic_etale_empty`,
`analytic_etale_nonclosed` make the same distinctions on the analytic carrier.

### P3.16 strongly finite étale and strongly étale morphisms

**Strongly finite étale and strongly étale morphisms of perfectoid spaces; tilting invariance and
base change.** (*strongly-etale-morphisms-and-base-change*) Over an arbitrary perfectoid base
([Sch12] Def. 7.2 over a perfectoid field): a morphism `(R, R⁺) → (S, S⁺)` of perfectoid Huber pairs
is strongly finite étale if it is finite étale (`S ∈ FÉt(R)` with its natural topology, `S⁺` the
integral closure of `R⁺`) and `S°a` is almost finite étale over `R°a` (layer P0). A morphism
`f : Y → X` of perfectoid spaces is strongly finite étale if `X` has a cover by open affinoid
perfectoid `V` with `f⁻¹(V)` affinoid perfectoid and `(O_X(V), O⁺_X(V)) → (O_Y(f⁻¹V), O⁺_Y(f⁻¹V))`
strongly finite étale; strongly étale if locally on `Y` it factors as an open immersion, a strongly
finite étale map and an open immersion. Define `Perfectoid.Pair.IsStronglyFiniteEtale` (the affinoid
condition on a map of perfectoid pairs), `PerfectoidSpace.IsStronglyFiniteEtale` and
`PerfectoidSpace.IsStronglyEtale` (the two Prop-valued classes on morphisms). Properties: (a) `f` is
strongly (finite) étale iff its tilt `f♭` is (the almost finite étale condition is transported by
*almost-finite-etale-equivalence* (b)); (b) base change along arbitrary maps `Z → X` of perfectoid
spaces preserves both classes, with finite-étale affinoid formula `D = A ⊗_B C` (already complete, being finite
projective over `C`), `D⁺` the integral closure of `C⁺` in `D`, and `|Y ×_X Z| → |Y| ×_{|X|} |Z|`
surjective ([Sch12] Lem. 7.3(i)–(ii), with the printed `|X ×_Z Y|` read as the fibre product of the
base change); (c) composites of strongly (finite) étale maps are strongly (finite) étale ([Sch12]
Cor. 7.8); (d) in characteristic `p` every (finite) étale map is strongly (finite) étale, and by
almost purity this holds in general, by *etale-is-strongly-etale*, so the strong classes coincide with
*finite-etale-morphism-of-perfectoid-spaces* and *etale-morphism-of-perfectoid-spaces* ([Sch12] §7,
Def. 7.2(i), p. 39). *Needs:* P3.14; P3.15; P3.11; P3.12; layer P0
(almost finite étale algebras).

**Checks.**

- `PerfectoidSpace.IsStronglyFiniteEtale.test_charp` (computation) — for a characteristic-`p`
  perfectoid pair `R`, the diagonal `R → R × R` gives a strongly finite étale cover;
  `(a+b)^p = a^p+b^p` and componentwise Frobenius on `R × R` is bijective.
  The characteristic hypothesis is needed for these Frobenius assertions
  ([Berkeley] Proposition 6.1.6, pp. 42–43; [ECD] Theorem 6.1 and Definition 6.2, p. 26).
- `PerfectoidSpace.IsStronglyEtale.test_open` (degenerate) — open immersions are strongly étale;
  they are strongly finite étale only when also closed.
- Compatibility: `IsStronglyFiniteEtale f ↔ IsStronglyFiniteEtale f♭`.

`Pair.almostMap` is almostification of the actual restriction `R° → S°` of
the continuous pair map, over the source basic setup. `Pair.IsStronglyFiniteEtale`
requires it to be almost finite étale, in addition to the finite étale pair
condition. The global strong finite condition uses a covering by such
cartesian affinoid squares; the strong étale condition uses local open
factorizations. `Pair.finiteEtale_iff_strong`, `finiteEtale_iff_strong` and
`etale_iff_strong` are the three comparison theorems.

**Checks.**

- `pair_strong_identity`, `pair_strong_zero`, `pair_strong_tilt` test the
  identity, zero algebra, and preservation and reflection by actual pair tilting.
- `strong_finite_identity`, `strong_finite_clopen`, `strong_finite_tilt` test
  identity, a clopen inclusion, and geometric tilting.
- `strong_etale_identity`, `strong_etale_empty`, `strong_etale_open_not_finite`
  test identity, the empty open, and a nonclosed open which is strongly étale
  but not strongly finite étale.

### P3.17 comparison with AdicEtaleGeometry layer A1 and with finite étale algebras

**Comparison of perfectoid finite étale and étale morphisms with AdicEtaleGeometry layer A1 and with
finite étale algebras.** (*perfectoid-vs-analytic-finite-etale*) Let `X` be a perfectoid space.
(a) (Perfectoidness) If `g : Y → X` is a finite étale (resp. étale) morphism of analytic adic spaces
in the sense of AdicEtaleGeometry A1.3, then `Y` is a perfectoid space. (b) (Morphism classes) For
morphisms of perfectoid spaces the following are equivalent: finite étale in the sense of
AdicEtaleGeometry A1.3 (= *finite-etale-morphism-of-perfectoid-spaces*); [ECD] Def. 6.2(i) (for
every open affinoid perfectoid `U` the preimage is affinoid perfectoid, finite étale with
integral-closure plus ring); [Sch12] Def. 7.1(ii) (for one open affinoid perfectoid cover); [Berkeley]
Def. 7.5.1(1); strongly finite étale. Likewise étale in the sense of AdicEtaleGeometry A1.3 ⇔
[ECD] 6.2(ii) ⇔ [Sch12] 7.1(iii) ⇔ [Berkeley] 7.5.1(2) ⇔ strongly étale. (c) (Algebraic comparison)
For `X = Spa(R, R⁺)` affinoid perfectoid, the functors `X_fét → FÉt(R)^op` (global sections),
`FÉt(R) → {finite étale morphisms of Huber pairs (R, R⁺) → (S, S⁺)}` (`Huber.Pair.finiteEtale`, plus
ring forced to be the integral closure) and the latter `→` Mathlib `CommAlgCat.FiniteEtale R` (forget
topology and plus ring) are equivalences; every finite étale `R`-algebra is perfectoid for its
natural topology. (d) (Sites) The category of perfectoid spaces étale (resp. finite étale) over `X` is
the small étale (resp. finite étale) site of the underlying adic space in the sense of
AdicEtaleGeometry A1.6, with the same coverings; that layer's structure sheaves `O`, `O⁺` on it,
its Galois category of finite étale objects and its basis of affinoid objects apply. (e) (Base change
from adic spaces) If `Y' → X'` is a finite étale (resp. étale) morphism of locally strongly sheafy
analytic adic spaces (e.g. locally noetherian ones) and `Z → X'` is a morphism from a perfectoid
space, then `Y' ×_{X'} Z` exists and is finite étale (resp. étale) over `Z`, with ring on a finite-étale affinoid chart
`D = A ⊗_B C` and `D⁺` the integral closure of `C⁺` — this is AdicEtaleGeometry A1.5
(*etale-base-change*, pullback along finite étale and étale maps), used with its hypotheses; the
clause proved here is that `Y' ×_{X'} Z` is perfectoid ([Sch12] Lem. 7.3(iii)–(iv) and Prop. 7.10)
([ECD] §6, paragraph after Def. 6.2, p. 26). *Needs:* P3.10; AdicEtaleGeometry A1.1 (strongly
sheafy rings), A1.3 (finite étale and étale morphisms), A1.5 (*etale-base-change*) and A1.6 (the
sites).

`perfectoid_of_analyticFiniteEtale` and `perfectoid_of_analyticEtale` take
A1’s `YonedaAdicSpace.IsFiniteEtale` and `.IsEtaleLocalDescription`, respectively,
and a perfectoid target; their conclusion is `isPerfectoidSpace p Y` for the adic space underlying `Y`.
For (e), `perfectoid_finiteEtale_baseChange` and `perfectoid_etale_baseChange`
take locally strongly sheafy adic spaces `Y, X`, a finite étale or étale map
`f : Y → X`, a perfectoid `Z`, and a map
`Z.obj → YonedaAdicSpace.ofAdicSpace.obj X`. They produce a perfectoid `W`,
a projection `W → Z` in the corresponding perfectoid morphism class, and
a cartesian square in `YonedaAdicSpace` over `ofAdicSpace.map f`.
Thus the universal property includes all Yoneda-adic test spaces
([Sch12] Lemma 7.3(iii)–(iv), p. 40, Proposition 7.10, p. 43; [ECD]
Theorem 6.1 and Definition 6.2, p. 26, give the perfectoid Tate-ring form).

For (d), `globalSections_adicComparison` gives ring equivalences from
`globalSections p X = Γ(X, O_X)` (P2, the structure-sheaf sections of the underlying
`PreAdicSpace`) to the global sections of every adic representative
`A` with `ofAdicSpace.obj A ≅ X.obj`. They carry `globalPlus p X` exactly to
`AdicSpace.Oplus A ⊤` and commute with every pair of corresponding morphisms.
`etaleSheaves_adicComparison` gives an equivalence with the small étale
site of a locally strongly sheafy adic representative, preserving and reflecting
covering sieves, and natural isomorphisms identifying `etaleO` and `etaleOPlus`
with A1’s `structureSheafEtale` and `structureSheafPlusEtale` after forgetting
to additive groups and changing universes. The site comparison also transports
A1’s affinoid basis and Galois-category results.
`smallEtale.affinoidCover` gives every étale object an open covering by
affinoid perfectoid étale objects, and `smallFiniteEtale.galoisCategory`
concludes `GaloisCategory (smallFiniteEtale p X)` when
`ConnectedSpace (YonedaAdicSpace.space.obj X.obj)` holds ([Sch12]
7.11–7.13, p. 43; AdicEtaleGeometry A1.6
`AdicSpace.smallEtaleBasis_openCover` and `AdicSpace.smallFiniteEtale.galoisCategory`).

The following results support the comparison. Perfectoid Tate rings are strongly sheafy, and perfectoid
spaces are locally strongly sheafy (*perfectoid-tate-rings-are-strongly-sheafy*): for `R` perfectoid
Tate and every `n ≥ 0`, the Tate algebra `R⟨T_1, …, T_n⟩` (Tau Ceti
`Huber.restrictedMvPowerSeriesCompletion n R`) is stably uniform, hence sheafy for every ring of
integral elements, so `R` is strongly sheafy in the sense of AdicEtaleGeometry A1.1;
`PerfectoidSpace.stronglySheafy p R` concludes A1’s `Huber.IsStronglySheafy R`
for every complete Hausdorff perfectoid Tate ring `R`.
`PerfectoidSpace.locallyStronglySheafy p X` gives an adic space `A` with
`AdicSpace.IsLocallyStronglySheafy A` and an isomorphism
`YonedaAdicSpace.ofAdicSpace.obj A ≅ X.obj`. Every rational localisation and every finite étale algebra of a
perfectoid Tate ring is strongly sheafy, and the local-to-global description of finite étale maps,
fibre products of étale maps and the étale sites of AdicEtaleGeometry A1.3–A1.6 apply to perfectoid
spaces ([Berkeley] Lecture 6, §6.3, Example 6.3.2(1),
Propositions 6.3.3(3) and 6.3.4, pp. 47–48). Preimages of
affinoid perfectoid opens under finite étale maps are affinoid perfectoid, with strongly finite
étale ring map (*strongly-finite-etale-maps-are-affinoid-over-affinoids*): for `f : Y → X` finite
étale (a morphism of perfectoid spaces, or of analytic adic spaces with `X` perfectoid) and
`V = Spa(R, R⁺) ⊆ X` open affinoid perfectoid, `f⁻¹(V) = Spa(S, S⁺)` with `S = O_Y(f⁻¹V) ∈ FÉt(R)`
perfectoid and `S⁺` the integral closure of `R⁺`, `(R, R⁺) → (S, S⁺)` is strongly finite étale,
global sections give `V_fét ≃ FÉt(R)^op`, and strongly finite étale covers of `V` correspond to finite
étale `R`-algebras; in particular every finite étale morphism of perfectoid spaces satisfies [ECD]
Def. 6.2(i) ([Sch12] §7, Prop. 7.6). Every finite étale (resp. étale) morphism of perfectoid spaces is
strongly finite étale (resp. strongly étale) (*etale-is-strongly-etale*): for `f : Y → X` a morphism
of perfectoid spaces, `f` is finite étale iff strongly finite étale and étale iff strongly étale — in
characteristic `p` by *finite-etale-covers-in-characteristic-p*, in general by
*integral-closure-almost-finite-etale* — so the tilting invariance and base-change properties of the
strong classes hold for all (finite) étale morphisms ([Sch12] §7, after Thm 7.9).

<a id="p3-17a-1"></a>

### P3.17a.1 sousperfectoid rings

**Sousperfectoid Tate rings and perfectoid frames.** (*sousperfectoid-ring*) Fix a prime `p`
(the Lean declarations `PerfectoidFrame` and `IsSousperfectoid` take any `p : ℕ`). Let
`R` be a complete Hausdorff Tate ring (Tau Ceti `Huber.IsTateRing`) in which `p` is topologically
nilpotent — equivalently, a Tate ring with a continuous ring map `ℤ_p → R`, that is, a complete Tate
`ℤ_p`-algebra. A perfectoid frame for `R` is a triple `(R̃, ι, σ)` consisting of a perfectoid Tate
ring `R̃` satisfying P1.1's `Perfectoid.IsPerfectoidTateRing p R̃`,
a continuous ring homomorphism `ι : R → R̃`, and a continuous `R`-linear map
`σ : R̃ → R` (`R̃` an `R`-module through `ι`) with `σ ∘ ι = id_R`.
The ring predicate, as P1.1 defines it, is a condition on a complete Hausdorff Tate ring:
`R̃°` is bounded and some pseudouniformiser `ϖ ∈ R̃°` has `ϖ^p ∣ p` with Frobenius
`R̃°/ϖ → R̃°/ϖ^p` bijective; equivalently it includes completeness, Hausdorffness, uniformity,
an explicit prime, a pseudouniformizer with the required divisibility, and surjective
Frobenius modulo `p` (`TauCetiRoadmap.PerfectoidSpaces.Perfectoid.IsPerfectoidTateRing.iff_frobenius_surjective_modP`;
Berkeley Definition 6.1.1).
`R` is sousperfectoid if it admits a perfectoid frame; a Huber pair `(R, R⁺)` is
sousperfectoid if `R` is. Conventions: `ι` is then injective and a homeomorphism onto the closed
subring `ι(R) = ker(id − ι∘σ)`; `σ` need not be multiplicative and need not map any ring of integral
elements into `R⁺`; no perfectoid base field is fixed, and `R̃` is not required to contain one;
characteristic `p` is allowed, and then `R̃` is a perfect complete Tate ring. In `Huber`, define the
structure `PerfectoidFrame` (a perfectoid frame `(R̃, ι, σ)` for `R`) and the class
`IsSousperfectoid` (`R` is a Tate ring admitting a perfectoid frame); prove that a perfectoid
frame makes `R` sousperfectoid, `IsSousperfectoid.of_isPerfectoidTateRing` (perfectoid Tate rings
are sousperfectoid, with the identity frame `PerfectoidFrame.refl`),
`PerfectoidFrame.powerBoundedSubring_eq_comap` (`a ∈ R°` iff `ι(a) ∈ R̃°`: power-boundedness is detected in the
frame, Tau Ceti `Huber.powerBoundedSubring`), `IsSousperfectoid.isUniform` (sousperfectoid rings are
uniform, [*sousperfectoid-uniform*](#p3-17a-2) (P3.17a.2)), `IsSousperfectoid.isReduced` (sousperfectoid rings are reduced,
uniform rings being reduced), invariance under isomorphisms of
topological rings, base change of a frame along a map `R → S` of
complete Tate rings whose completed tensor product `R̃ ⊗̂_R S` is perfectoid (P3.17a.2, through
AdicSpacesPartIII R3.5's [*split-injection-completed-base-change*](../AdicSpacesPartIII/README.md) (AdicSpacesPartIII R3.5)), `isSousperfectoid_ratLoc` (rational
localisations of sousperfectoid rings are sousperfectoid, [*sousperfectoid-rational-localisation*](#p3-17a-2) (P3.17a.2)),
`IsSousperfectoid.finiteEtale` (finite étale algebras over sousperfectoid rings are sousperfectoid,
[*sousperfectoid-finite-etale*](#p3-17a-2) (P3.17a.2)), `IsSousperfectoid.restrictedMvPowerSeries` (Tate algebras over
sousperfectoid rings are sousperfectoid, [*sousperfectoid-rings*](#p3-17a-2) (P3.17a.2)), `isUniform_ratLoc`
(sousperfectoid pairs are stably uniform, [*sousperfectoid-stably-uniform*](#p3-17a-3) (P3.17a.3)) and
`IsSousperfectoid.isSheafyRing` (sousperfectoid pairs are sheafy and strongly sheafy, [*sousperfectoid-sheafy*](#p3-17a-3) (P3.17a.3)).
The closed-embedding assertion is Mathlib `Function.LeftInverse.isClosedEmbedding` applied to
`σ ∘ ι = id_R` and the continuity of `σ` and of `ι`, using the Hausdorff property of `R̃`
(Berkeley Lecture 6, §6.3, Def. 6.3.1, p. 47; HK §7, Def. 7.1, p. 25 (perfectoid frames); BHW §3.1,
Def. 3.2(1), p. 1724; Berkeley Lecture 6, §6.3, Ex. 6.3.2(1), p. 47 (perfectoid rings are
sousperfectoid with the frame `(R, id, id)`); Berkeley Lecture 6, §6.3, Ex. 6.3.2(2), p. 47
(`ℚ_p⟨T⟩` is sousperfectoid with frame `ℚ_p^cycl⟨T^{1/p^∞}⟩` but not perfectoid)). *Needs:*
P1.1 (`Perfectoid.IsPerfectoidTateRing`) and P1.31 (`ℚ_p^cycl`); Tau Ceti `Huber.IsTateRing`,
`Huber.IsHuberRing`; Mathlib `ContinuousLinearMap`. Lean: `Huber.PerfectoidFrame`,
`Huber.PerfectoidFrame.refl`, `Huber.IsSousperfectoid`,
`Huber.IsSousperfectoid.of_isPerfectoidTateRing`,
`Huber.PerfectoidFrame.powerBoundedSubring_eq_comap`, `Huber.IsSousperfectoid.isUniform`,
`Huber.IsSousperfectoid.isReduced`, `Huber.IsSousperfectoid.complete_and_t2`, and
`Huber.isSousperfectoid_iff_frobenius_split` (the characteristic-`p`, finite-Frobenius
characterisation).

**Checks.**

- Computation: `ℚ_p⟨T⟩` is sousperfectoid:
  `ι : ℚ_p⟨T⟩ → ℚ_p^cycl⟨T^{1/p^∞}⟩` with `σ` = (the retraction `ℚ_p^cycl → ℚ_p`) applied to the
  coefficients followed by deleting the non-integral exponents is a perfectoid frame.
- Non-example: `ℚ_p⟨T⟩` is not a
  perfectoid Tate ring: Frobenius on `ℤ_p⟨T⟩/p = 𝔽_p[T]` is not surjective (`T` is not a `p`-th
  power), so "sousperfectoid" is strictly weaker than "perfectoid".
- Non-example: `A = ℚ_p⟨X, Q⟩/(Q²)` is strongly noetherian
  and sheafy (the AdicSpaces layer 4 example) but not sousperfectoid: a sousperfectoid ring is
  uniform, hence reduced, and `Q ≠ 0` is nilpotent. A definition by "sheafy", or by a splitting of
  abstract modules into a non-perfectoid ring, fails this test.
- Degenerate: `ℚ_p` (the Tate algebra in zero variables) is
  sousperfectoid, with frame `ℚ_p → ℚ_p^cycl` split by a continuous `ℚ_p`-linear retraction
  (*sousperfectoid-p-adic-field*).
- Compatibility: every perfectoid Tate ring in the sense of
  layer P1 (for example `ℚ_p^cycl`, or `𝔽_p((t^{1/p^∞}))^∧`) is sousperfectoid via
  the identity frame.
- Characterisation: for a uniform complete Tate ring `A` of
  characteristic `p` whose Frobenius is finite, `A` is sousperfectoid iff Frobenius `A → A` splits as
  `A`-modules (HK §7, Lem. 7.10); for example `𝔽_p((t))⟨T⟩` is sousperfectoid.

- Non-example: the uncompleted field `ℚ_p(ζ_{p^∞})` with its
  `p`-adic topology is a Tate ring that admits no perfectoid frame at all: a
  frame `(R̃, ι, σ)` makes `ι` a homeomorphism onto the closed subgroup `ker(id − ι∘σ)` of the
  complete Hausdorff group `R̃`, so a sousperfectoid ring is complete and Hausdorff, which
  `ℚ_p(ζ_{p^∞})` is not (`Huber.IsSousperfectoid.complete_and_t2`). The class
  `Huber.IsSousperfectoid` therefore omits completeness and
  the Hausdorff property, and omits the topological nilpotence of `p` (which holds in `R̃` since
  `ϖ^p ∣ p` there, and descends along the embedding `ι`): all three are consequences, not
  hypotheses. The same control shows that [*split-injection-completed-base-change*](../AdicSpacesPartIII/README.md) (AdicSpacesPartIII R3.5) cannot produce a
  frame on a non-complete `S`.
- Convention: the identity `σ ∘ ι = id_R` is on `R`;
  on `R̃` the composite `ι ∘ σ` is only an idempotent: for `R = ℚ_p`, `R̃ = ℚ_p^cycl` and
  `σ` = Tate's normalised trace (a continuous `ℚ_p`-linear retraction, `σ = [K_n : ℚ_p]^{-1} Tr`
  on `K_n = ℚ_p(ζ_{p^n})`), `ι(σ(ζ_{p²})) = σ(ζ_{p²}) = 0 ≠ ζ_{p²}` (the primitive `p²`-th roots of
  unity sum to `μ(p²) = 0`), while `σ(ι(1)) = 1`; `ζ_p` is a poorer witness, since for `p = 2`,
  `ζ_2 = −1 ∈ ℚ_2` and `ι(σ(ζ_2)) = ζ_2`. The assertion that the reversed composite
  can fail assumes `R ≠ 0`: `σ ∘ ι = id`, `ι ∘ σ` is idempotent, and over a nonzero
  `R` it is not the identity for every frame.
- Negative control: for `R = 0` the unital map
  `ι : R → R̃` forces `R̃ = 0`, and both composites are identities. A counterexample to
  the reversed identity cannot be demanded over the zero ring.

<a id="p3-17a-2"></a>

### P3.17a.2 stability of the class

**Base change of a frame.** The continuous splitting used here is AdicSpacesPartIII R3.5's
[*split-injection-completed-base-change*](../AdicSpacesPartIII/README.md) (AdicSpacesPartIII R3.5), which this roadmap cites and does not restate. For a
frame `(R̃, ι, σ)` of a Tate ring `R` and a continuous map `R → S` to a complete Tate ring for
which the completed tensor product `R̃ ⊗̂_R S` (AdicSpacesPartII R0, [*completed-tensor-product*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.3))
is perfectoid, `(R̃ ⊗̂_R S, s ↦ 1 ⊗ s, σ ⊗̂ id)` is a frame of `S`, with embedding `s ↦ 1 ⊗ s`
and retraction `x ⊗ s ↦ σ(x)s` on pure tensors (HK Rem. 7.2, p. 25). Completeness and
Hausdorffness of the target are required. Its unit-coefficient, zero-scalar and two-term Checks
are `σ_S(x ⊗ 1) = σ(x)`, `σ_S(x ⊗ 0) = 0` and `σ_S(x ⊗ s + y ⊗ t) = σ(x)s + σ(y)t`.


**Checks.**

- For `S = R` the pure-tensor formula recovers the original retraction.
- A tensor with zero second factor is sent to zero.
- For `x⊗s + y⊗t` the retraction gives `σ(x)s + σ(y)t`.

*Needs:* P3.17a.1; AdicSpacesPartIII R3.5 (*split-injection-completed-base-change*);
AdicSpacesPartII R0.3 (*completed-tensor-product*).

**Sousperfectoid rings are uniform.** (*sousperfectoid-uniform*) Let `R` be a complete Tate ring
that is sousperfectoid with frame `(R̃, ι, σ)`. Prove that `R° = ι^{-1}(R̃°)` and that `R°` is
bounded, that is, `R` is uniform in the sense of HK Definition 2.3 (AdicSpaces layer 4.2); in
particular `R` is reduced (HK §7, Def. 7.1, p. 25 (sousperfectoid implies uniform via strictness);
HK §2, Def. 2.3, p. 3 (uniformity descends along strict inclusions); HK §2, after Def. 2.3, p. 3
(uniform rings are reduced); Berkeley Lecture 7, introduction, p. 49 (perfectoid Tate rings are
uniform by definition)). As predicted, `ℚ_p⟨T⟩° = ℤ_p⟨T⟩` is bounded, while `ℚ_p⟨X, Q⟩/(Q²)` has
the unbounded power-bounded subring `ℤ_p⟨X⟩ + ℚ_p⟨X⟩Q` and so is not sousperfectoid. *Needs:*
[*sousperfectoid-ring*](#p3-17a-1) (P3.17a.1); P1.1 (`Perfectoid.IsPerfectoidTateRing`); Tau Ceti
`Huber.powerBoundedSubring`, `Huber.IsBounded`, `Huber.IsPowerBounded`.
Lean: `Huber.IsSousperfectoid.isUniform`.


**Checks.**

- For an identity frame the power-bounded-subring equality is the identity.
- For `ℚ_p⟨T⟩` the detected subring is `ℤ_p⟨T⟩` and is bounded.
- The nonzero square-zero element in `ℚ_p⟨X,Q⟩/(Q²)` prevents uniformity and hence a frame.

**Rational localisations of sousperfectoid rings are sousperfectoid.** (*sousperfectoid-rational-localisation*) Let `(R, R⁺)` be a Huber pair with `R` sousperfectoid,
`X = Spa(R, R⁺)` and `U = R(T/s) ⊆ X` a rational subset (`T` finite with `T·R = R`). Prove that the
complete Tate ring `𝒪_X(U) = R⟨T/s⟩` (the completed rational localisation of AdicSpaces, Tau Ceti
`Huber.PairOfDefinition.completionLocalization`; AdicEtaleGeometry's `Huber.ratLoc` in Lean) is sousperfectoid. Explicitly: for a frame
`(R̃, ι, σ)` and a ring of integral elements `R̃⁺ ⊆ R̃°` containing `ι(R⁺)` — which can be taken to
be the integral closure in `R̃` of the subring generated by `ι(R⁺)` and the topologically nilpotent
elements `R̃°°`, an open subring contained in `R̃°` — the preimage `Ũ` of `U` in `X̃ = Spa(R̃, R̃⁺)`
is the rational subset `R(ι(T)/ι(s))`, `R̃ ⊗̂_R 𝒪_X(U) ≅ 𝒪_X̃(Ũ) = R̃⟨ι(T)/ι(s)⟩` is perfectoid,
and `(𝒪_X̃(Ũ), ι ⊗̂ id, σ ⊗̂ id)` is a perfectoid frame for `𝒪_X(U)` (Berkeley Lecture 6, §6.3,
Prop. 6.3.3(1), p. 47; Berkeley Lecture 6, proof of Prop. 6.3.3, p. 48 (base change of the frame
along the rational localisation); Berkeley Lecture 6, §6.2, Thm 6.2.6, p. 45 (rational
localisations of perfectoid Tate rings are perfectoid); HK §3, Prop. 3.8, p. 8 (rational
localisations of perfectoid rings, with no base field, are perfectoid)). For `R = ℚ_p⟨T⟩` and
`U = {|T| ≥ 1}`, `𝒪_X(U) = ℚ_p⟨T^{±1}⟩` is sousperfectoid with frame `ℚ_p^cycl⟨T^{±1/p^∞}⟩`; for
`R` perfectoid the statement reduces to the perfectoid case with the identity frame. *Needs:*
[*sousperfectoid-ring*](#p3-17a-1) (P3.17a.1); [*sousperfectoid-uniform*](#p3-17a-2) (P3.17a.2); [*split-injection-completed-base-change*](../AdicSpacesPartIII/README.md) (AdicSpacesPartIII R3.5);
[*completed-tensor-rational-localisation*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.3) (AdicSpacesPartII layer R0); the perfectoid rational-localization argument of P2.3; Tau Ceti
`Huber.PairOfDefinition.completionLocalization`,
`Huber.PairOfDefinition.existsUnique_continuous_ringHom_completion_locTopology`,
`ValuationSpectrum.spaComap_preimage_rationalSubset`, `ValuationSpectrum.rationalSubset`,
`Huber.IsRingOfIntegralElements`. Lean: `Huber.isSousperfectoid_ratLoc`.


**Checks.**

- The whole-spectrum localization retains the given frame.
- The empty rational subset has zero ring and the zero identity frame.
- For `ℚ_p⟨T⟩` and `|T| ≥ 1` the localization is `ℚ_p⟨T±¹⟩` with the perfected Laurent frame.

**Finite étale algebras over sousperfectoid rings are sousperfectoid.** (*sousperfectoid-finite-etale*)
Let `R` be sousperfectoid and `S` a finite étale `R`-algebra (Mathlib `Algebra.Etale` and
`Module.Finite`: finite, flat, finitely presented, unramified), with its natural topology as a finite
`R`-module and `S⁺` the integral closure of `R⁺` in `S` (AdicEtaleGeometry layer A1,
[*finite-etale-affinoid-algebra*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.2)). Prove that `S` is a complete Tate ring and is sousperfectoid, with
frame `S → R̃ ⊗_R S` (`= R̃ ⊗̂_R S`) split by `σ ⊗_R id_S` (Berkeley Lecture 6, §6.3,
Prop. 6.3.3(2), p. 47; Berkeley Lecture 6, proof of Prop. 6.3.3, p. 48 (through almost purity:
finite étale over perfectoid is perfectoid); Berkeley Lecture 7, §7.4, Thm 7.4.5(1), p. 53 (the
almost-purity input); HK §7, Lem. 7.5, p. 25 (finite étale stability and faithfully flat finite
étale descent); HK §2, Prop. 2.15(a), p. 5 (finite étale algebras over perfectoid rings, with no
base field, are perfectoid)). For `R = ℚ_p` and `S = ℚ_p(ζ_p)`, `S` is sousperfectoid with frame
`ℚ_p^cycl ⊗_{ℚ_p} ℚ_p(ζ_p) ≅ (ℚ_p^cycl)^{p−1}`, a finite product of perfectoid fields; for
`R = ℚ_p⟨T^{±1}⟩` and `S = R[T^{1/n}]` with `n` prime to `p`, `S` is sousperfectoid. *Needs:*
[*sousperfectoid-ring*](#p3-17a-1) (P3.17a.1); [*split-injection-completed-base-change*](../AdicSpacesPartIII/README.md) (AdicSpacesPartIII R3.5); AdicEtaleGeometry layer A1
(*finite-etale-affinoid-algebra*); [*completed-tensor-product*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.3) (AdicSpacesPartII layer R0); the finite étale perfectoid-frame argument of HK Lemma 7.5;
P3.10;
Mathlib `Algebra.Etale`, `Module.Finite`, `Module.Projective`, `Module.Flat`.
Lean: `Huber.IsSousperfectoid.finiteEtale`.


**Checks.**

- For `S = R` the tensor frame is the original frame.
- The split algebra `R×R` receives the product frame.
- Over `ℚ_p`, the finite étale cyclotomic algebra uses a finite product of `ℚ_p^cycl` in its frame.

**Descent of the sousperfectoid property along faithfully flat finite étale maps.** (*sousperfectoid-finite-etale-descent*) Let `R → S` be a finite étale and faithfully flat
homomorphism of complete Tate rings, `S` with its natural topology. Prove that if `S` is
sousperfectoid then so is `R`: for a frame `(S̃, ι_S, σ_S)` of `S` and any `R`-linear retraction
`τ : S → R` of `R → S`, the triple `(S̃, ι_S ∘ (R → S), τ ∘ σ_S)` is a perfectoid frame of `R`
(HK §7, Lem. 7.5, p. 25; HK §7, proof of Lem. 7.5, p. 26 (compose the frame of `B` with a splitting
of `A → B`)). Thus if `R → S` is a finite Galois étale cover and `S` is sousperfectoid then `R` is
sousperfectoid. Faithful flatness is needed: `R = ℚ_p × ℚ_p⟨X, Q⟩/(Q²) → S = ℚ_p`, the projection to
a direct factor, is finite étale but not faithfully flat, `S` is sousperfectoid, and `R` is not,
being non-reduced. *Needs:* [*sousperfectoid-ring*](#p3-17a-1) (P3.17a.1); [*sousperfectoid-finite-etale*](#p3-17a-2) (P3.17a.2); Mathlib
`Module.Projective`, `Module.Flat`, `Algebra.Etale`. Lean: `Huber.IsSousperfectoid.of_finiteEtale`.


**Checks.**

- The identity map descends its frame unchanged.
- For the diagonal `R → R×R`, first projection is a module retraction and descends a product frame.
- The projection `ℚ_p×ℚ_p⟨X,Q⟩/(Q²) → ℚ_p` is finite étale but fails faithful flatness and the descent conclusion.

**Tate algebras over sousperfectoid rings are sousperfectoid.** (*sousperfectoid-rings*) Let `R` be
sousperfectoid and `n ≥ 0`. Prove that `R⟨T₁, …, Tₙ⟩` (Tau Ceti
`Huber.restrictedMvPowerSeriesCompletion`) is sousperfectoid: for a frame `(R̃, ι, σ)` of `R`, the
perfectoid Tate ring `R̃⟨T₁^{1/p^∞}, …, Tₙ^{1/p^∞}⟩` — the completion of `R̃[T_i^{1/p^∞}]` for the
topology in which `R̃°[T_i^{1/p^∞}]` is open with its `ϖ`-adic topology — with
`ι⟨T⟩(Σ a_α T^α) = Σ ι(a_α) T^α` and

`σ'(Σ_{α ∈ (ℤ[1/p]_{≥0})^n} x_α T^α) = Σ_{α ∈ ℕ^n} σ(x_α) T^α`

is a perfectoid frame; the case `n = 0` is trivial (Berkeley Lecture 6, §6.3, Prop. 6.3.3(3), p. 47;
Berkeley Lecture 6, proof of Prop. 6.3.3, p. 48 (the frame `R⟨T⟩ → R̃⟨T^{1/p^∞}⟩`); HK §7, Lem. 7.3,
p. 25 (Tate and Laurent algebras over sousperfectoid rings); HK §7, proof of Lem. 7.3, p. 25 (the
splitting of the frame for `A⟨T⟩`)). Parts (1) and (2) of Berkeley's Proposition 6.3.3 are
[*sousperfectoid-rational-localisation*](#p3-17a-2) (P3.17a.2) and [*sousperfectoid-finite-etale*](#p3-17a-2) (P3.17a.2), and Proposition 6.3.4 is
[*sousperfectoid-stably-uniform*](#p3-17a-3) (P3.17a.3). Berkeley's Example 6.3.2(2), `ℚ_p⟨T⟩` sousperfectoid with
`R = ℚ_p` and `R̃ = ℚ_p^cycl`, is the case `n = 1`; for a perfectoid field `L`, `L⟨T₁, …, Tₙ⟩` is
sousperfectoid, which is the polydisc factor of BHW Corollary 3.4; iterating,
`R⟨T⟩⟨S⟩ = R⟨T, S⟩` is sousperfectoid, and relative polydiscs `𝔹^n_X` over sousperfectoid `X` are
sousperfectoid (the ambient spaces of ECD 6.4(iv) in AdicEtaleGeometry layer A3). *Needs:*
[*sousperfectoid-ring*](#p3-17a-1) (P3.17a.1); [*completed-tensor-restricted-power-series*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.3) (AdicSpacesPartII layer R0); P1.1 and P1.30 (perfected Tate algebras); Tau Ceti `Huber.restrictedMvPowerSeriesCompletion`;
Mathlib `ContinuousLinearMap`. Lean: `Huber.IsSousperfectoid.restrictedMvPowerSeries`.


**Checks.**

- With `n = 0` the coefficient retraction is the original `σ`.
- With `R = ℚ_p` and `n = 1` the frame is `ℚ_p^cycl⟨T^(1/p^∞)⟩`.
- Deleting nonintegral exponents fixes `T` and kills a monomial of exponent `1/p` over a nonzero frame ring.

**Laurent algebras and annuli over sousperfectoid rings are sousperfectoid.** (*sousperfectoid-annulus*) Let `R` be sousperfectoid with pseudo-uniformizer `ϖ` and let `a ≤ b` be
integers, so that the annulus is non-empty when `R ≠ 0`; the radii are integral powers of `|ϖ|`.
Prove that `R⟨T^{±1}⟩` and the annulus ring `R⟨T/ϖ^a, ϖ^b/T⟩ := 𝒪(W)` for the rational subset
`W = {|ϖ^b| ≤ |T| ≤ |ϖ^a|}` of `Spa(R⟨S⟩, R⁺⟨S⟩)` with `T = ϖ^a S` are sousperfectoid (HK §7,
Lem. 7.3, p. 25). In `Spa R⟨S⟩` one has `|S| ≤ 1`, so `|T| ≤ |ϖ^a|` is automatic and `W` is the
rational subset `R({ϖ^{b−a}}/S) = {|ϖ^{b−a}| ≤ |S|}` (its weight `ϖ^{b−a}` is a unit, so it
generates the unit ideal and `|S| ≠ 0` is implied). Witness of the convention: for `R = ℚ_p`,
`ϖ = p`, `a = 0`, `b = 1`, `W = {|p| ≤ |T| ≤ 1}` is the closed annulus with `𝒪(W) = ℚ_p⟨T, p/T⟩`;
for `a = 1`, `b = 0` the inequalities `|p^0| ≤ |T| ≤ |p|` have no solution and `W = ∅`, whose ring
`𝒪(∅) = 0` is sousperfectoid for a trivial reason — the hypothesis `a ≤ b` is what makes the
identification of `W` with `R({ϖ^{b−a}}/S)` correct (a formal statement reading the exponent as
`(b − a).toNat` would, for `a > b`, describe the Laurent annulus `R({1}/S)` instead of `∅`).
Over a perfectoid field `L`, the closed annulus `{|ϖ| ≤ |T| ≤ 1}` is a
sousperfectoid, non-perfectoid affinoid. *Needs:* [*sousperfectoid-rings*](#p3-17a-2) (P3.17a.2);
[*sousperfectoid-rational-localisation*](#p3-17a-2) (P3.17a.2); Tau Ceti `ValuationSpectrum.rationalSubset`.

**Checks.** The ambient disc condition
`|S| ≤ 1` is explicit for every valuation of the chosen plus ring. At `(a,b)=(0,1)` rational
membership is then exactly `|p| ≤ |S| ≤ 1`; reversing the exponents would replace `p` by `1`.


**Checks.**

- For `R = ℚ_p` and `(a,b) = (0,1)`, the section ring is `ℚ_p⟨T,p/T⟩`.
- At `a = b`, the Laurent coordinate has absolute value exactly `|ϖ|^a`.
- For `(a,b) = (1,0)` the annulus is empty; replacing `b−a` by its truncated natural value would give a nonempty Laurent annulus.

**Bounded retractions for countable extensions of discretely valued fields.**
(*countable-extension-retraction*) Let `K` be a complete nonarchimedean field with discrete
nontrivial valuation, and `L/K` a complete isometric nonarchimedean field extension whose
underlying `K`-vector space is the closure of the span of a countable family. Construct a
continuous `K`-linear map `σ : L → K` with `σ(a) = a` for `a ∈ K` and `|σ(z)| ≤ |z|`
(HK §7, Rem. 7.8, p. 26, for the countable-extension splitting; Schneider11 §1, Lem. 1.7(ii),
pp. 6–7, for spherical completeness of `K`). Prove the bounded extension step: start with
the identity on `K`, adjoin the countable generators one at a time, and extend a functional
`f` satisfying `|f(v)| ≤ |v|` by choosing its value `b` on the new vector `z` in the nested
closed balls `{b ∈ K : |b − f(v)| ≤ |z − v|}`, `v` in the previous span. The ultrametric
inequality makes these balls nested; spherical completeness gives their intersection.
The resulting compatible functionals have bound `1` on the dense union; completeness of `K`
extends them to `L`, giving the cyclotomic retraction in the field application. *Needs:* Mathlib `ContinuousLinearMap`, `Submodule.span`, `Dense`,
`CompleteSpace`; P1.31 for the cyclotomic completion used in the application.
Lean: `Huber.exists_countable_extension_retraction`.

**Checks.**

- For `L = K`, the retraction is the identity and has operator norm `1`.
- For `K/ℚ_p` finite and `L = K^cycl`, a union of finite cyclotomic levels supplies a
  countable spanning family and the retraction fixes `1`.
- An orthonormal basis for the given norm is not needed: `ℚ_p(√p)` contains a vector of
  norm `p^{-1/2}`, whereas coefficients in an orthonormal `ℚ_p`-basis would give only
  norms in `p^ℤ`. Weighted orthogonal coordinates are compatible with the construction.

**Finite extensions of `ℚ_p` are sousperfectoid.** (*sousperfectoid-p-adic-field*) Let `K` be a
finite extension of `ℚ_p` and `K^cycl` the
completion of `K(μ_{p^∞})`. Prove that `K → K^cycl` admits a continuous `K`-linear retraction, so
that `(K^cycl, inclusion, retraction)` is a perfectoid frame and `K` is sousperfectoid; more
generally `K⟨T₁, …, Tₙ⟩` and every rational localisation and finite étale algebra of it are
sousperfectoid (HK §7, Rem. 7.8, p. 26 (nonarchimedean fields with a perfectoid countable profinite
étale extension are sousperfectoid); Berkeley Lecture 6, §6.3, Ex. 6.3.2(2), p. 47). The
retraction is supplied by [*countable-extension-retraction*](#p3-17a-2) (P3.17a.2): the finite cyclotomic levels have
a countable union dense in `K^cycl`, and the valuation on `K` is discrete. The perfectoidness
of `K^cycl` follows from P1.31 and P3.10, since it is a finite extension of
`ℚ_p^cycl`. Thus
`K` is sousperfectoid but not perfectoid. Write `p = uπ^e` for a uniformizer `π` of `K`.
If `e < p`, no topologically nilpotent `ϖ` satisfies `ϖ^p ∣ p`. If `e ≥ p`, that divisibility
can hold (take `ϖ = π`), but Frobenius on `𝒪_K/p` is not surjective: the class of `π` has
valuation `1`, whereas a nonzero `p`-th power has valuation divisible by `p`, modulo `π^e`.
For `K = ℚ_p`, Frobenius on `𝔽_p` is surjective and the divisibility is the obstruction;
for the totally ramified extension with `π^p = p`, the divisibility holds and Frobenius fails.
Both examples fail the perfectoid input predicate: `ℚ_p` fails the divisibility condition,
and the extension with `π^p = p` fails Frobenius surjectivity modulo `p`. Every affinoid algebra of the form `K⟨T⟩⟨f/g⟩`, and every
smooth affinoid over `K` that is étale over a polydisc through a composite of rational localisations
and finite étale maps, is sousperfectoid. *Needs:* [*sousperfectoid-ring*](#p3-17a-1) (P3.17a.1); [*sousperfectoid-rings*](#p3-17a-2) (P3.17a.2);
[*sousperfectoid-rational-localisation*](#p3-17a-2) (P3.17a.2); [*sousperfectoid-finite-etale*](#p3-17a-2) (P3.17a.2); [*countable-extension-retraction*](#p3-17a-2) (P3.17a.2); the Frobenius criterion of P1.1
(`Perfectoid.IsPerfectoidTateRing.iff_frobenius_surjective_modP`) and `ℚ_p^cycl` (P1.31), with finite étale perfectoid extensions (P3.10).
Lean: `Huber.IsSousperfectoid.of_finite_padic`.

**Checks.**

- `ℚ_p` is sousperfectoid but fails perfectoid divisibility (`Perfectoid.not_isPerfectoidTateRing_padic`).
- The finite extension with `π^p = p` satisfies divisibility and fails Frobenius
  surjectivity modulo `p`.
- `K⟨T⟩` is sousperfectoid; Frobenius on `K°⟨T⟩/p` misses its
  coordinate `T`, so sousperfectoidness does not imply perfectoidness.

<a id="p3-17a-3"></a>

### P3.17a.3 stable uniformity and sheafiness

**Sousperfectoid pairs are stably uniform.** (*sousperfectoid-stably-uniform*) Let `(R, R⁺)` be a
Tate Huber pair with `R` complete Tate and sousperfectoid and `R⁺` any ring of integral elements.
That `(R, R⁺)` is stably uniform — for every rational subset `U ⊆ Spa(R, R⁺)`, `𝒪(U)` is uniform
(HK Definition 3.13; AdicSpaces layer 4.2) — is the splitting criterion *stably-uniform-sheafy*
(AdicSpacesPartIII layer R3; KL15 Rem. 2.8.12) applied to the frame map `ι : R → R̃`, `R̃` perfectoid and hence
stably uniform by the perfectoid rational-localization argument, with the retraction `σ` (Berkeley Lecture 6, §6.3,
Prop. 6.3.4, p. 48). BHW's Proposition 3.3 is this statement; on the closed unit disc over `ℚ_p`
it says that every rational localisation `ℚ_p⟨T⟩⟨f/g⟩` is reduced and uniform. *Needs:*
[*sousperfectoid-rational-localisation*](#p3-17a-2) (P3.17a.2); [*sousperfectoid-uniform*](#p3-17a-2) (P3.17a.2); *stably-uniform-sheafy*
(AdicSpacesPartIII layer R3); the perfectoid rational-localisation argument of P2.3.
Lean: `Huber.isUniform_ratLoc`.


**Checks.**

- The whole-space rational localization recovers the uniform coefficient ring.
- The rational annulus `ℚ_p⟨T,p/T⟩` is uniform by its localized frame.
- The empty rational subset has zero section ring and vacuous boundedness.

**Sousperfectoid pairs are sheafy, strongly sheafy and `𝒪`-acyclic.** (*sousperfectoid-sheafy*) Let
`(R, R⁺)` be a Tate Huber pair with `R` sousperfectoid, as in [*sousperfectoid-stably-uniform*](#p3-17a-3) (P3.17a.3).
Prove that `(R, R⁺)` is sheafy, so that `Spa(R, R⁺)` is an affinoid adic space in the sense of
AdicSpaces layer 5, that `H^i(Spa(R, R⁺), 𝒪) = 0` for `i > 0`, and that `R` is strongly sheafy:
every `R⟨T₁, …, Tₙ⟩` is sheafy — the predicate `Huber.IsStronglySheafy` of AdicEtaleGeometry
AdicEtaleGeometry layer A1 (*strongly-sheafy-huber-pair*) (BHW §3.1, Prop. 3.3, p. 1724 (citing Berkeley 6.3.4); HK §7,
Cor. 7.4, p. 25 (strong sheafiness of sousperfectoid rings); HK §3, Prop. 3.14, p. 9
(Buzzard–Verberkmoes–Mihara)). By this route `Spa(ℚ_p⟨T⟩, ℤ_p⟨T⟩)` is sheafy without using
noetherianity. For nonzero perfectoid `S`, `Spa(S⟨T⟩, S⁺⟨T⟩)` is sheafy although
`S⟨T⟩` is not perfectoid. It can be noetherian: take `S` to be a perfectoid field. *Needs:* [*sousperfectoid-stably-uniform*](#p3-17a-3) (P3.17a.3); [*sousperfectoid-rings*](#p3-17a-2) (P3.17a.2);
*stably-uniform-sheafy*, [*sheafy-cech-acyclicity*](../AdicSpacesPartIII/README.md) (AdicSpacesPartIII R3.1) (AdicSpacesPartIII layer R3); AdicEtaleGeometry layer A1
(*strongly-sheafy-huber-pair*).

The sheafiness statement supplies the affinoid adic space for any
chosen ring of integral elements, and the strong sheafiness is
AdicEtaleGeometry A1's predicate; both are `Huber.IsSousperfectoid.isSheafyRing` in Lean. The acyclicity statement
asserts vanishing of the ordinary additive structure-sheaf cohomology in positive
degree, using Mathlib's `Sheaf.H` and its explicit sheafification and Ext assumptions; it is
specified in this README only.

<a id="p3-17a-4"></a>

### P3.17a.4 sousperfectoid spaces


**Checks.**

- For the closed `ℚ_p` disc, sheafiness follows without a noetherianity premise.
- For nonzero perfectoid `S` the one-variable disc is sheafy although its coordinate is not a p-th power modulo `ϖ`.
- The empty affinoid has zero structure sheaf and vanishing positive-degree cohomology.

**Sousperfectoid adic spaces.** (*sousperfectoid-adic-space*) An adic space `X` (AdicSpaces layer 5:
an object of Huber's category `𝒱` covered by affinoid adic spaces) is sousperfectoid if it admits an
open cover by affinoid adic spaces `Spa(R, R⁺)` with `R` sousperfectoid (*sousperfectoid-ring*);
such `Spa(R, R⁺)` are adic spaces by [*sousperfectoid-sheafy*](#p3-17a-3) (P3.17a.3). Equivalently, the open affinoid
subspaces `Spa(R, R⁺) ⊆ X` with `R` sousperfectoid form a basis of the topology of `X`. No base is
fixed; `X` lives over `Spa ℤ_p` because `p` is topologically nilpotent on every chart. A
sousperfectoid space is analytic, its charts being Tate. A morphism of sousperfectoid spaces is a
morphism of adic spaces. In `AdicSpace`, define the predicate `IsSousperfectoid` (`X` admits an open
cover by `Spa(R, R⁺)` with `R` sousperfectoid); prove `IsSousperfectoid.spa` (`Spa(R, R⁺)` is a
sousperfectoid adic space for `R` sousperfectoid), that
perfectoid spaces are sousperfectoid, that open subspaces of
sousperfectoid spaces are sousperfectoid, that the sousperfectoid
affinoid opens form a basis of the topology of `X`, `IsSousperfectoid.isAnalytic` (every point of a
sousperfectoid space is analytic), that `𝒪_X(U)` is reduced for
every open `U`, `IsSousperfectoid.of_etale` (étale over sousperfectoid is sousperfectoid,
[*sousperfectoid-etale-stability*](#p3-17a-4) (P3.17a.4)), that relative polydiscs exist
(`𝔹^n_X = X ×_{Spa ℤ_p} Spa ℤ_p⟨T₁, …, Tₙ⟩` exists and is sousperfectoid, locally `Spa R⟨T⟩`; on a
chart `Spa(R, R⁺)` the finite-type construction of [*fibre-product-along-finite-type*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.7)
with `M = {1}` gives `W_m = Spa R⟨T⟩` for every `m`, since `M_{L,m} = {1} ∪ Lᵐ` includes the
weight `1`; hence its union is `Spa R⟨T⟩`. This applies also when `p = 0` or when `p` is a
nonzero nonunit in `R`), that sousperfectoid affinoids are stably adic
(for a sousperfectoid affinoid `Spa(R, R⁺)`, the composites of
rational localisations and finite étale maps over `(R, R⁺)` form a stable basis of its étale site in
the sense of Kedlaya–Liu 8.2.19, so it is stably adic, and `X_ét` is covered by the étale sites of
these affinoid opens, [*sousperfectoid-stably-adic*](#p3-17a-4) (P3.17a.4)) and `IsSousperfectoid.exists_fibreProduct_etale` (base
change of an étale map to a locally noetherian space along a map from a sousperfectoid space exists
and is sousperfectoid, [*sousperfectoid-etale-base-change*](#p3-17a-4) (P3.17a.4)) (BHW §3.1, Def. 3.2(3), p. 1724; HK §1,
Introduction, p. 3; Berkeley Appendix to Lecture 19, §19.5, p. 179 (sousperfectoid adic spaces are
locally `Spa` of sousperfectoid rings); BHW §3.1, p. 1724 (why the products must exist as adic
spaces)). Spaces locally admitting charts satisfying the perfectoid input predicate are
sousperfectoid, and so are smooth rigid spaces over a finite extension `K` of `ℚ_p`. *Needs:* [*sousperfectoid-ring*](#p3-17a-1) (P3.17a.1); [*sousperfectoid-sheafy*](#p3-17a-3) (P3.17a.3);
[*sousperfectoid-rational-localisation*](#p3-17a-2) (P3.17a.2); [*affinoid-fibre-product*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.7),
[*fibre-product-along-finite-type*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.7), [*completed-tensor-restricted-power-series*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.3) (AdicSpacesPartII layer R0);
P2.9–P2.10 (affinoid perfectoid and perfectoid spaces); Tau Ceti
`ValuationSpectrum.spa`. Lean: `AdicSpace.IsSousperfectoid`, `AdicSpace.IsSousperfectoid.spa`,
`AdicSpace.IsSousperfectoid.isAnalytic`.

**Checks.**

- Computation: the closed unit disc
  `Spa(ℚ_p⟨T⟩, ℤ_p⟨T⟩)` is a sousperfectoid adic space (Berkeley Lecture 6, §6.3, Ex. 6.3.2(2)).
- Compatibility: every perfectoid space
  (layer P2) is a sousperfectoid adic space, with charts the affinoid perfectoid
  opens.
- Non-example: `Spa(ℚ_p⟨X, Q⟩/(Q²))` is an adic
  space (strongly noetherian) but not sousperfectoid: sections of `𝒪` over any open of a
  sousperfectoid space are reduced (products and equalizers of reduced rings), while `Q ≠ 0` is
  nilpotent in the global sections.
- Degenerate: the empty adic space and
  `Spa(ℚ_p, ℤ_p)` are sousperfectoid.


The predicate `AdicSpace.IsSousperfectoid p X` is exactly the existence, at each
point, of a Part II `AdicSpace.Chart` whose coordinate ring is sousperfectoid
and whose open immersion contains that point in its image.
The empty space satisfies this condition (Lean `example`);
a nonzero nilpotent section on any open excludes it;
the closed disc over `ℚ_p` satisfies it.
Together with the closed-disc and perfectoid Checks above, these distinguish the
local condition from perfectoidness and from merely being analytic.

**Étale spaces over a sousperfectoid space are sousperfectoid adic spaces.** (*sousperfectoid-etale-stability*) Let `X` be a sousperfectoid adic space and `f : Y → X` étale in
the local-description sense of AdicEtaleGeometry layer A1 (Kedlaya–Liu Definition 8.2.16: every
`y ∈ Y` has an open neighbourhood on which `f` factors as an open immersion, a finite étale morphism
and an open immersion, where finite étale means affinoid-locally on the target
`Spa(S, S⁺) → Spa(R, R⁺)` with `R → S` finite étale and `S⁺` the integral closure of `R⁺`); for
locally noetherian targets this is Huber's étale ([*differentials-unramified-smooth-etale*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.16), AdicSpacesPartII layer R0;
the comparison with the local description is AdicEtaleGeometry layer A1,
[*etale-morphisms-local-description-and-comparison*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.4)). Prove that `Y` is an adic space (not
merely a preadic space) and is sousperfectoid. In particular every object of the étale site `X_ét`
is a sousperfectoid adic space, and for a finite étale `R`-algebra `S` over a sousperfectoid `R` and
a composite of rational localisations and finite étale maps `R → S'`, the ring `S'` itself is
sousperfectoid. For a general étale map, sousperfectoidness is a property of the space `Y`, not of a
global ring: `Y` is covered by sousperfectoid affinoids, while the sousperfectoid property of rings
is not known to be of local nature (Berkeley Appendix to Lecture 19, §19.5, p. 179 (étale objects
over a sousperfectoid space are sousperfectoid adic spaces); KL15 §8.2, Def. 8.2.16, p. 162 (étale
morphisms of (pre)adic spaces by local description); KL15 §8.2, Rem. 8.2.18, p. 162 (why a class
stable under finite étale maps is needed); HK §5, Def. 5.1, p. 14 (étale in the naive sense:
locally a composition of rational localisations and finite étale maps); BHW §3.1, proof of
Cor. 3.4, p. 1725 (the étale step, which BHW attribute to 6.3.3(ii), the finite étale case)). Thus
any `Y ∈ X_ét` is itself sousperfectoid and in particular an adic space, and BHW's step "algebras
étale over a sousperfectoid algebra are again sousperfectoid" holds for composites of rational
localisations and finite étale maps and, for general étale maps, in the space-level form above.
*Needs:* [*sousperfectoid-adic-space*](#p3-17a-4) (P3.17a.4); [*sousperfectoid-finite-etale*](#p3-17a-2) (P3.17a.2);
[*sousperfectoid-rational-localisation*](#p3-17a-2) (P3.17a.2); [*sousperfectoid-sheafy*](#p3-17a-3) (P3.17a.3); AdicEtaleGeometry layer A1
(*finite-etale-affinoid-algebra*). Lean: `AdicSpace.IsSousperfectoid.of_etale`.


**Checks.**

- An identity étale map retains the same affinoid charts.
- A rational open in the `ℚ_p` disc is sousperfectoid by rational stability.
- The split two-sheeted finite étale cover uses two copies of each sousperfectoid chart.

**Sousperfectoid spaces are stably adic.** (*sousperfectoid-stably-adic*) Let `X = Spa(R, R⁺)` with
`R` sousperfectoid, with `X_ét` in the local-description sense of AdicEtaleGeometry layer A1, and
let `B_X` be the class of objects `Y → X` of `X_ét` of the form `Y = Spa(S, S⁺)` with
`(R, R⁺) → (S, S⁺)` a finite composite of rational localisations and finite étale maps, the plus
rings being the integral closures. Prove that every such `S` is sousperfectoid (so `Spa(S, S⁺)` is
an affinoid adic space). Define on AdicEtaleGeometry A1's small étale category of `X` the class generated by
the identity: start with the identity object up to isomorphism and close under rational
embeddings, finite étale maps and isomorphisms over `X`. This is `B_X`.
Prove that its members are basic objects in the sense of AdicEtaleGeometry A1's [*basis-comparison*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.6) for an
affinoid base. AdicEtaleGeometry A1's class also starts from arbitrary affinoid open charts; those charts are
not additional starting objects of `B_X`. Prove,
for an explicit base `X = Spa(A₀, A₀⁺)` with `A₀` sousperfectoid (`Huber.IsSousperfectoid p A₀`) and
`U ∈ B_X`: for every sheafy complete affinoid presentation
`U ≅ Spa(A, A⁺)`, the ring `A` is sousperfectoid. Rational localisation and finite
étale stability give this assertion (HK Def. 7.1 and Lemma 7.5(a), p. 25); a cover by
sousperfectoid charts alone does not give it, since locality of the ring property is unknown
(HK Def. 7.1, p. 25).
The stable-basis statement says that `B_X` consists of affinoids, covers
every étale object by open immersions, and admits fibre products over `X` within `B_X`;
it also names the local strong-sheafiness supplied by [*sousperfectoid-sheafy*](#p3-17a-3) (P3.17a.3).
The constructors include every rational subdomain and finite étale extension of a member.
Thus `B_X` is a stable basis in the sense of Kedlaya–Liu Definition 8.2.19, p. 163,
using the identity-generated basis of HK's proof of Proposition 5.5, p. 15.
The covering-reduction criterion is supplied by AdicEtaleGeometry A1.6
(*etale-covering-reduction*). Consequently, by
[*stable-basis-etale-vector-bundles*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.6a) (Kedlaya–Liu Theorem 8.2.22(c),(d)): for every finite projective
`R`-module `M`, `M̃` is an acyclic sheaf on `B_X`, in particular `H^i(X_ét, 𝒪) = 0` for `i > 0`; and
finite projective `R`-modules are equivalent to finite locally free `𝒪_{X_ét}`-modules. For a
general sousperfectoid adic space these statements hold on every sousperfectoid affinoid open, and
such opens cover `X`. The class of all affinoid objects `Spa(S, S⁺)` of `X_ét` with `S`
sousperfectoid is not used as the basis: its closure under fibre products would need
`S₁ ⊗̂_S S₂` to be sousperfectoid for a general étale `S → S₁`, which is not known. Kedlaya–Liu
define `X_ét` and stable bases for preadic spaces over an analytic field; here, as in Berkeley
§19.5, they are used over `ℤ_p`, the proofs of Kedlaya–Liu 8.2.20–8.2.22 not using the base field
(KL15 §8.2, Def. 8.2.19, p. 163 (stable bases and stably adic spaces); KL15 §8.2, Rem. 8.2.18,
p. 162; HK §1, Introduction, p. 3 (finite étale covers of sousperfectoid spaces); KL15 §8.2, after
Def. 8.2.19, p. 163 (a stable basis must be closed under fibre products, which is why the basis is
the composites of rational localisations and finite étale maps); HK §5, proof of Prop. 5.5, p. 15
(the basis `B_X`); HK §5, Prop. 5.8, p. 16 (étale `𝒪`-acyclicity for strongly sheafy, in
particular sousperfectoid, pairs); KL15 §8.2, Thm 8.2.22(c),(d), p. 165). For `X` affinoid
perfectoid this recovers the stably adic property of perfectoid spaces; as in Berkeley §19.5, vector
bundles on `X_ét` for sousperfectoid `X` are finite projective modules on affinoids, the input to
Tannakian `G`-torsors. *Needs:* [*sousperfectoid-etale-stability*](#p3-17a-4) (P3.17a.4); [*sousperfectoid-finite-etale*](#p3-17a-2) (P3.17a.2);
[*sousperfectoid-rational-localisation*](#p3-17a-2) (P3.17a.2); [*sousperfectoid-sheafy*](#p3-17a-3) (P3.17a.3);
[*stable-basis-etale-vector-bundles*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.6a); [*completed-tensor-rational-localisation*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.3) (AdicSpacesPartII layer R0);
[*sheafy-tate-acyclicity-and-kiehl-gluing*](../AdicSpacesPartIII/README.md) (AdicSpacesPartIII R3.5), *finite-projective-etale-descent* (AdicSpacesPartIII layer R3);
AdicEtaleGeometry A1.6 (*basis-comparison*, *etale-covering-reduction*).

**Checks.**

- The identity étale object belongs by the initial
  constructor, including when the base is empty.
- The split two-point cover of `Spa(ℚ₂, ℤ₂)` belongs
  by a finite étale step; it need not be an open immersion.
- The empty finite étale object over `Spa(ℚ₂, ℤ₂)`
  belongs; its coordinate ring is the zero ring.
- The non-quasicompact open unit disc inside the
  closed unit disc is étale but does not belong, since these basic objects are affinoid.
- An affinoid presentation with a nonzero
  nilpotent coordinate cannot belong to `B_X` over a sousperfectoid affinoid base.

**Base change of étale maps to locally noetherian spaces along sousperfectoid spaces.** (*sousperfectoid-etale-base-change*) Let `Z` be a sousperfectoid adic space, possibly non-noetherian. Its charts are stably sheafy; the formula below uses the stably sheafy case of [*fibre-products-existence*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.7), rather than requiring the locally noetherian case; let `W` be a locally noetherian analytic adic space (for example a rigid space),
so that étale maps to `W` have the local open-immersion/finite-étale description (Huber Lemma 2.2.8
through AdicEtaleGeometry layer A1); let `g : Z → W` be a morphism and `f : Y → W` étale (AdicSpacesPartII layer R0).
Prove that the fibre product `Z ×_W Y` exists in the category of adic spaces, that the projection
`Z ×_W Y → Z` is étale in the local-description sense of AdicEtaleGeometry layer A1, and that
`Z ×_W Y` is sousperfectoid. Affinoid formula: if `Y' ⊆ Y` is open and `f|_{Y'}` factors as an open
immersion `Y' → Y''` followed by a finite étale morphism `Y'' = Spa(B, B⁺) → W' = Spa(A, A⁺)` onto
an affinoid open `W' ⊆ W` (`A → B` finite étale, `B⁺` the integral closure of `A⁺`; such
factorisations exist locally by Huber Lemma 2.2.8), and `Z' = Spa(C, C⁺) ⊆ g^{-1}(W')` is an
affinoid open with `C` sousperfectoid, then `C ⊗_A B` is sousperfectoid,
`Z' ×_{W'} Y'' = Spa(C ⊗_A B, integral closure of C⁺ in C ⊗_A B)`, and `Z' ×_W Y'` is its open
subspace lying over `Y'` (Berkeley Lecture 7, §7.5, after Def. 7.5.1, p. 55 (Huber-étale maps of
analytic adic spaces are locally an open immersion followed by a finite étale map); KL15 §8.2,
Def. 8.2.16, p. 162; BHW §3.1, proof of Cor. 3.4, p. 1725 (the étale step)). For `W = 𝔹^n_L`,
`Z = X ×_L 𝔹^n` with `X` affinoid perfectoid and `Y → 𝔹^n` étale, `Z ×_W Y = X ×_L Y` is the BHW
product; for `Y = W'` an open subspace of `W`, `Z ×_W Y = g^{-1}(W')`. *Needs:*
[*sousperfectoid-etale-stability*](#p3-17a-4) (P3.17a.4); [*sousperfectoid-finite-etale*](#p3-17a-2) (P3.17a.2); [*sousperfectoid-sheafy*](#p3-17a-3) (P3.17a.3);
AdicEtaleGeometry layer A1 (*finite-etale-affinoid-algebra*);
[*completed-tensor-product-universal-property*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.3), [*affinoid-fibre-product*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.7),
[*locally-noetherian-adic-space*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.6), [*differentials-unramified-smooth-etale*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.16) (AdicSpacesPartII layer R0); Mathlib
`Algebra.Etale`. Lean: `AdicSpace.IsSousperfectoid.exists_fibreProduct_etale`.
<a id="p3-17b"></a>

### P3.17b perfectoid spaces times polydiscs


**Checks.**

- Base change of an identity map is the original sousperfectoid space.
- For a rational open the pullback is its rational preimage with the localized frame.
- An empty source gives the empty pullback, which is sousperfectoid.

**Perfectoid space times a polydisc: sousperfectoid, not perfectoid.** (*perfectoid-times-polydisc*)
Let `L` be a perfectoid field, `X = Spa(S, S⁺)` an affinoid perfectoid space over `L` and
`𝔹^n_L = Spa(L⟨T₁, …, Tₙ⟩, 𝒪_L⟨T⟩)`. Prove that the fibre product `X ×_L 𝔹^n_L` exists in adic
spaces and equals `Spa(S⟨T₁, …, Tₙ⟩, S⁺⟨T₁, …, Tₙ⟩)`, which is a sousperfectoid affinoid adic space
(BHW §3.1, proof of Cor. 3.4, p. 1725 (the perfectoid-times-polydisc step); Berkeley Lecture 6,
§6.3, Prop. 6.3.3(3), p. 47), and that for `n ≥ 1` and `S ≠ 0`, `S⟨T₁, …, Tₙ⟩` is not perfectoid
(proved here: `T₁` is not a `p`-th power in `S°⟨T⟩/ϖ` for nonzero `S`, the argument of the
`ℚ_p⟨T⟩` non-example of [*sousperfectoid-ring*](#p3-17a-1) (P3.17a.1); Berkeley Lecture 6, §6.3, Ex. 6.3.2(2), p. 47, for
`ℚ_p⟨T⟩`).
Thus `ℚ_p^cycl⟨T⟩` is sousperfectoid and not perfectoid, its perfectoid cover
`ℚ_p^cycl⟨T^{1/p^∞}⟩` being a frame; and a smooth weight disc times a nonempty perfectoid space is not
perfectoid. *Needs:* [*sousperfectoid-rings*](#p3-17a-2) (P3.17a.2); [*sousperfectoid-sheafy*](#p3-17a-3) (P3.17a.3);
[*completed-tensor-restricted-power-series*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.3), [*affinoid-fibre-product*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.7) (AdicSpacesPartII layer R0); P1.1, P1.10 and P2.9; Tau Ceti
`ValuationSpectrum.mem_iff_forall_vle_one`, `Huber.restrictedMvPowerSeriesCompletion`.
Lean: `Huber.not_isPerfectoidTateRing_restrictedMvPowerSeries`.


**Checks.**

- Degenerate: the zero ring is a perfectoid Tate ring in the sense of
  the P1 structure (`0` is a topologically nilpotent unit, `0^p ∣ p = 0`, Frobenius on `0` is
  surjective), hence sousperfectoid with the identity frame; this is why
  [*perfectoid-times-polydisc*](#p3-17b) (P3.17b) needs `S ≠ 0`: `0⟨T⟩ = 0` is perfectoid (`Perfectoid.isPerfectoidTateRing_of_subsingleton`).
- In zero variables the product is the original perfectoid affinoid.
- For a nonempty perfectoid affinoid and one variable, the coordinate is not a p-th power modulo the pseudouniformizer.
- *isLocallyEtaleOverPolydisc_test_perfectoidBase* (characterisation) — for a nonempty affinoid
  perfectoid `Y = Spa(R, R⁺)`, `B^n_Y = Spa(R⟨T⟩, R⁺⟨T⟩) → Y` satisfies the definition although
  `B^n_Y` is not perfectoid for `n ≥ 1` (P3.17b, perfectoid times polydisc).

### P3.18 the étale site and the finite étale site

The étale site of a perfectoid space is the étale site of its underlying adic space; tilting
identifies it with the site of the tilt, and `O⁺` is almost acyclic on affinoids.

**The étale site `X_ét` and the finite étale site `X_fét` of a perfectoid space.**
(*etale-site-of-perfectoid-space*) Let `X` be a perfectoid space. Its étale site `X_ét` is the small
étale site of the underlying locally strongly sheafy analytic adic space (AdicEtaleGeometry A1.6;
*perfectoid-tate-rings-are-strongly-sheafy*): objects the adic spaces étale over `X` — all perfectoid
(*perfectoid-vs-analytic-finite-etale* (a)), i.e. the perfectoid spaces étale over `X` of [ECD] —
with all `X`-morphisms (étale by *etale-morphisms-composition-base-change-open* (cancellation)), and the topology
generated by jointly surjective families (on all points, including higher-rank ones). The finite
étale site `X_fét` is the full subcategory of finite étale objects with the induced jointly surjective
topology (AdicEtaleGeometry A1.6). `X_ét` has finite limits (fibre products of perfectoid spaces,
layer P2) and a basis of affinoid perfectoid objects closed under rational subsets and finite
étale maps. Fibre products admit covers by these objects; over an affinoid perfectoid
base, the fibre product of two affinoid perfectoid objects is affinoid perfectoid; a morphism `X' → X` of perfectoid spaces induces a morphism of sites
`X'_ét → X_ét` by base change, and a morphism to a locally strongly sheafy analytic adic space `Y`
induces `X_ét → Y_ét`; `U ↦ O_U(U)` and `U ↦ O⁺_U(U)` are sheaves of commutative
rings on `X_ét`, as is the sheaf quotient `O⁺/ϖ`. Relative to an explicit almost
basic setup, `O^{+a}` is a sheaf of almost algebras; on each affinoid base
`Spa(R,R⁺)` the geometric setup is `(R⁺, R°°)`; for `X = Spa(R, R⁺)` affinoid perfectoid, `X_fét ≃ FÉt(R)^op` (the contravariance
is global sections: the split cover `X ⊔ X = Spa(R × R, R⁺ × R⁺) → X` corresponds to the algebra
`R × R`, and the map `X ⊔ X → X` to the diagonal `R → R × R`).
The two-component Check `finite_etale_test_contravariance_two_components` applies
`finiteEtaleAffinoidEquiv` to the actual first inclusion
`Spa(R) → Spa(R × R)`, with the finite-algebra pair supplied by
`finiteEtalePairFunctor`. Through `finiteEtaleAffinoidSectionsIso` and the affinoid
section identifications, its image sends `(x,y)` to `x`; its composite with the
fold map is the identity on `Spa(R)` ([ECD] Def. 6.2 and the following affinoid
comparison, p. 26). For `X` connected,
`X_fét` is a Galois category with fundamental group `π₁(X)` (AdicEtaleGeometry A1.6 and A1.8), equal to
`Gal(K^sep/K)` for `X = Spa(K, K⁺)`, `K` a perfectoid field. In `PerfectoidSpace`, define
`smallEtale X` as the category of perfectoid spaces étale over `X` (equivalent to `YonedaAdicSpace.smallEtale X.obj`,
where here and below `X.obj` also denotes the image of the underlying adic space under
`YonedaAdicSpace.ofAdicSpace`) and `smallEtaleTopology` as the Grothendieck topology of jointly surjective
families; `smallEtale.isPerfectoid` takes an object of the supplied
`YonedaAdicSpace.smallEtale X.obj` and proves that its adic source is perfectoid.
Its input has no perfectoid-source hypothesis ([ECD] Theorem 6.1 and Definition 6.2,
p. 26; [Sch12] §7, Definition 7.11, p. 43). *Needs:* *perfectoid-tate-rings-are-strongly-sheafy*; P3.17;
*strongly-finite-etale-maps-are-affinoid-over-affinoids*.

**Checks.**

- Computation: for `K` a perfectoid field,
  `Spa(K, O_K)_fét ≃` finite continuous `Gal(K^sep/K)`-sets (via the Galois category of
  AdicEtaleGeometry A1.8; by *fontaine-wintenberger* these are also the finite continuous
  `Gal(K♭^sep/K♭)`-sets, which is the tilting equivalence `Spa(K, O_K)_fét ≃ Spa(K♭, O_{K♭})_fét` of
  P3.20 on this example).
- `PerfectoidSpace.smallEtale.test_empty` (degenerate) — for `X = ∅` the site has a single object
  (`∅`) covered by the empty family; for `X = Spa(C, C⁺)` with `C` algebraically closed, every finite
  étale object is a finite disjoint union of copies of `X`, so `X_fét ≃ FintypeCat`.
- Computation: for `X = Spa(C, O_C)` with `C`
  algebraically closed, `|X|` is one point and the étale objects are the (arbitrary, not only
  finite) disjoint unions of copies of `X`, with all maps between them: `X_ét ≃ Set` with the
  jointly surjective coverings, so `Sh(X_ét) ≃ Set` and every sheaf has vanishing higher
  cohomology; for `X = Spa(C, C⁺)` with `C⁺ ⊊ O_C`, `|X|` consists of the rank-one point and its
  specialisations, the valuations of the residue field of `O_C` bounded by `1` on `C⁺/C°°`.
  When that residue field is algebraic over `𝔽_p`, integral closedness forces
  `C⁺ = O_C`, so the strict inclusion case does not occur. In general these
  specialisations need not form a chain, and `X_ét` has the open
  subspaces of disjoint unions of `X` as further objects, while `X_fét ≃ FintypeCat` still.
- Compatibility: `PerfectoidSpace.smallEtale p X ≌ YonedaAdicSpace.smallEtale X.obj`,
  with covering sieves preserved and reflected for
  `YonedaAdicSpace.smallEtaleTopology X.obj (isAnalytic_adic p X)`.

Étale and finite étale morphisms of perfectoid spaces satisfy composition, base change,
cancellation, openness and tilting (*etale-morphisms-composition-base-change-open*): since P3.14
and P3.15 agree with the classes of AdicEtaleGeometry A1.3 by the comparisons `IsFiniteEtale.iff_adic` and
`IsEtale.iff_adic`, the stability properties —
composites of (finite) étale morphisms are (finite) étale; for `f : Y → X` (finite) étale and any
`g : X' → X` the fibre product `Y ×_X X'` (layer P2, agreeing with the fibre product of adic spaces)
is (finite) étale over `X'` with finite-étale affinoid formula `D = A ⊗_B C`, `D⁺` the integral closure of `C⁺`,
and `|Y ×_X X'| → |Y| ×_{|X|} |X'|` surjective; cancellation (if `g` and `g ∘ f` are (finite) étale,
so is `f`); openness of étale morphisms, quasi-compact open image when the source is quasi-compact,
and the diagonal of an étale (resp. finite étale) morphism an open (resp. open and closed) immersion
— are AdicEtaleGeometry A1.5 (*etale-composition*, *etale-base-change*, *etale-diagonal*),
used here with the hypotheses stated there (perfectoid spaces are locally strongly sheafy by
*perfectoid-tate-rings-are-strongly-sheafy*); the one clause proved here is tilting: `f` is
(finite) étale iff its tilt `f♭ : Y♭ → X♭` is, which is P3.16(a) together with *etale-is-strongly-etale*
([Berkeley] Lecture 7, §7.5, Proposition 7.5.2(4), p. 55, for tilting;
Proposition 7.5.2(1)–(3) for the stability properties).

`smallEtale` and `smallFiniteEtale` are full subcategories of the actual slice
`Over X`; their morphisms are all commuting triangles. `smallEtaleTopology`
and `smallFiniteEtaleTopology` declare a sieve covering exactly when its maps
are jointly surjective on valuation points. `finiteEtaleAffinoidEquiv` gives
the opposite finite-algebra comparison. `etaleBaseChange` is the geometric
pullback functor. `etaleTiltFunctor` restricts P2's slice tilt, and
`etaleTilt_covers` preserves and reflects covering sieves.
The geometric obligation is `etaleTiltFunctor_isEquivalence` for every perfectoid
`X` ([ECD] Theorem 6.3, p. 26). `etaleTilt_preservesFiniteLimits` applies Mathlib’s
`CategoryTheory.Adjunction.isEquivalencePreservesLimits` to that instance.

**Checks.**

- `site_identity_object`, `site_empty_object`, `site_nonfinite_object` test
  membership of identity, empty inclusion, and nonclosed open inclusion in the
  étale slice. `finite_site_identity_object`, `finite_site_empty_object`,
  `finite_site_excludes_nonclosed_open` distinguish the finite subcategory.
- `topology_empty_cover`, `topology_nonempty_no_empty_cover`,
  `topology_identity_cover`: the empty sieve covers exactly an empty object,
  and the identity sieve covers every object. The finite-site versions are
  `finite_topology_empty_cover`, `finite_topology_nonempty_no_empty_cover`,
  `finite_topology_identity_cover`.
- `site_geometric_point`: the equivalence `etaleGeometricPointEquiv` sends the
  split double cover of `Spa(C,O_C)` to a set of cardinality two. The equivalence
  covers arbitrary permitted disjoint unions and uses the full valuation ring.
- `site_point_galois`: `finiteEtaleGaloisEquiv` sends the split double cover
  of `Spa(K,O_K)` to a two-element set with trivial absolute Galois action
  ([ECD] Definition 6.2, p. 26).
- `tilt_site_identity` and `tilt_finite_identity`: the identity object over
  `Spa(R,R⁺)` tilts to `Spa(R♭,R♭⁺)` with the structure map prescribed by
  `tiltSpaIso`, in the étale and finite étale slices ([ECD] Theorem 6.3, p. 26).

`smallEtaleAdicEquiv` and `smallFiniteEtaleAdicEquiv` compare the perfectoid
sites with the supplied Yoneda-adic sites. Their object equations keep the
underlying spaces and maps, and their covering equations identify the two
joint-surjectivity topologies. `stronglySheafy` supplies the perfectoid input
to the analytic stability theorems of P3.17. `IsEtale.cancel`,
`IsFiniteEtale.cancel`, `IsEtale.isOpenMap` and `IsEtale.diagonal` state
cancellation, openness and the open diagonal. P2.16's `pullback_points`
gives surjectivity on compatible pairs of valuation points for all morphisms of
perfectoid spaces, including étale morphisms.

**Checks.** `site_empty_unique` and `finite_site_empty_unique` identify each
empty-base category with the one-object category; `finite_site_geometric_point`
identifies finite étale objects over `Spa(C,O_C)` with finite sets.

The supplied sites are `YonedaAdicSpace.smallEtale`,
`YonedaAdicSpace.smallFiniteEtale`, `YonedaAdicSpace.smallEtaleTopology`,
and `YonedaAdicSpace.smallFiniteEtaleTopology`; their topologies take the
analyticity proof `isAnalytic_adic p X`.
`smallEtaleAdicEquiv` and `smallFiniteEtaleAdicEquiv` forget the perfectoid
structure, with object, map and covering equations. `finiteEtaleAffinoidSectionsIso`
identifies the algebraic comparison with actual global sections and pullback.

**Checks.** `analytic_site_identity`, `analytic_site_empty`, and
`analytic_site_nonfinite` compute the identity and empty objects and retain
nonfinite open immersions. `analytic_finite_site_identity`,
`analytic_finite_site_empty`, and `analytic_finite_site_nonclosed` make the
corresponding finite-site distinction. `analytic_topology_empty`,
`analytic_topology_nonempty`, and `analytic_topology_identity` test the empty
cover, its failure on a nonempty object, and the identity cover;
`analytic_finite_topology_empty`, `analytic_finite_topology_nonempty`, and
`analytic_finite_topology_identity` test the finite-site topology.

**Checks.** `smallEtaleAdicEquiv_identity`, `smallEtaleAdicEquiv_empty`,
and `smallEtaleAdicEquiv_nonfinite` assert that forgetting to A1 preserves
an identity object, the empty source, and failure of finiteness of an étale object.
`smallFiniteEtaleAdicEquiv_identity`, `smallFiniteEtaleAdicEquiv_empty`, and
`smallFiniteEtaleAdicEquiv_distinct_arrows` preserve identity and empty sources
and distinguish two different arrows in the finite site.

The fold Check `PerfectoidSpace.IsFiniteEtale.test_fold` uses the actual
natural pair of `R²`. The rational-disc Check
`PerfectoidSpace.IsFiniteEtale.test_rational` uses P1's `perfectedTateAlgebra`,
its coordinate, and P2's valuation-space homeomorphism: the subdisc
`|T| ≤ |ϖ|` is étale but not finite. The affine Frobenius Check
`PerfectoidSpace.IsEtale.test_frobenius_affinoid` specifies the ring map
`x ↦ x^p`, its geometric isomorphism, and both étale classes.

The Laurent-torus Check `PerfectoidSpace.IsFiniteEtale.test_kummer` fixes
constants and sends every monomial of exponent `a ∈ ℤ[1/p]` to exponent
`ℓa`, for positive `ℓ` prime to `p`; it is finite étale on pairs and spectra.
`PerfectoidSpace.IsEtale.test_rational_in_cover` tests restriction of any
finite étale map to an open source. The doubled-open Check
`PerfectoidSpace.IsEtale.test_separatedness_not_required` constructs the
pushout of two copies of `X` along a nonclosed open `U`: the fold map is
étale and its diagonal has nonclosed image. This applies to the doubled-disc
example. The two `…BaseChange_map_fst` equations fix base change on morphisms
by the pullback projections.

### P3.19 étale descent

**Étale descent for finite étale morphisms and finite projective modules on perfectoid spaces.**
(*etale-descent-perfectoid*) Let `X` be a perfectoid space with étale site `X_ét`
of *etale-site-of-perfectoid-space*. (a) (Locality) A morphism `f : Y → X` of perfectoid spaces is
finite étale if, for some covering `{X_i → X}` in `X_ét`, every base change `Y ×_X X_i → X_i` is
finite étale; étaleness of `f` is local on `Y` and on `X` for open coverings — for rational
coverings this is AdicEtaleGeometry A1.2 (*finite-etale-rational-descent*, KL 2.6.9), and the
étale-covering case is proved here. (b) (Stack) The prestack `U ↦ U_fét` (finite étale perfectoid
spaces over `U`, with isomorphisms) on `X_ét` is a stack for the étale topology (Mathlib
`CategoryTheory.Pseudofunctor.IsStack`): finite étale spaces with descent data along an étale
covering descend uniquely (the sheaf property of `U ↦ O(U)` on `X_ét` that it rests on is
AdicEtaleGeometry A1.6, *etale-structure-sheaf*). (c) (Vector bundles) For `X = Spa(R, R⁺)`
affinoid perfectoid, the equivalence `M ↦ M ⊗_R O_{X_ét}` between finitely generated projective
`R`-modules and finite locally free `O_{X_ét}`-modules, with `M ⊗ O` a sheaf on `X_ét` and
`M ⊗ O(Y) = M ⊗_R O(Y)` on affinoid perfectoid `Y ∈ X_ét` ([KL15] Thm 8.2.22(c), (d)), is
AdicEtaleGeometry A1.6a (*stable-basis-etale-vector-bundles*, for pairs with the stable étale basis supplied by strong sheafiness;
the rational-site form is AdicSpacesPartIII R3.5 (*sheafy-tate-acyclicity-and-kiehl-gluing*)),
used here with `(R, R⁺)` perfectoid and strongly sheafy by P3.17. The ordinary
finite-étale descent is [KL15] Theorems 1.3.4–1.3.5, and the étale vector-bundle
comparison is Theorem 8.2.22(c)–(d), p. 165. Descent along
v-covers belongs to the downstream consumer DiamondsAndVStacks layer D3.
*Needs:* P3.18;
AdicEtaleGeometry A1.2, A1.6 (*finite-etale-rational-descent*, *etale-structure-sheaf*);
AdicEtaleGeometry A1.6a (*stable-basis-etale-vector-bundles*); AdicSpacesPartIII R3.5
(*sheafy-tate-acyclicity-and-kiehl-gluing*).

The target `finiteEtale_descent` states locality for any jointly surjective
family of étale objects, without a finiteness condition on that family.
For a morphism `f : Y → X` in `Perfd p`,
`etale_iff_openCover_source` and `etale_iff_openCover_target` state that
AdicEtaleGeometry's `YonedaAdicSpace.IsEtaleLocalDescription`, applied to the image of `f` under
`YonedaAdicSpace.ofAdicSpace`, is
equivalent to the same predicate on every source restriction or target
pullback, respectively. Each takes an arbitrary indexed family of opens
covering every valuation point, without quasi-compactness or finiteness
hypotheses ([KL15] Definition 8.2.16 and 8.2.17(c), p. 162).
`TauCetiRoadmap.AdicSpacesPartIII.AdicSpace.vectorBundleEquivalence`
supplies rational gluing on Part II `AdicSpace.Chart`, with a Tate coefficient
ring `c.affinoid.ring` ([KL15] Theorem 2.7.7, p. 58), and
AdicEtaleGeometry A1.6a's *stable-basis-etale-vector-bundles* supplies
(c) on the stable basis of affinoid perfectoid objects ([KL15] Definition
8.3.1, p. 166).

**Checks.** `etale_openCover_whole_source` and
`etale_openCover_whole_target` specialize locality to the singleton cover
by the whole source or target. `etale_openCover_empty_source` uses the
empty covering family of an empty source and concludes that its morphism
to any perfectoid target is étale.

`finiteEtalePrestack` has the actual finite étale slice categories as fibres
and `finiteEtaleBaseChange` as transition functor; `finiteEtale_isStack`
asserts effective descent, including descent of morphisms. Both small
categories have finite limits. `prestack_empty_base` and
`prestack_geometric_point` compute its fibres over the empty space and an
algebraically closed field point.

For any perfectoid `X` and étale object `U → X`,
`finiteEtalePrestack_mapId_hom_app` identifies the component on every finite
étale `T → U` with `Over.pullbackId`. For composable slice arrows
`W → V → U`, `finiteEtalePrestack_mapComp_hom_app` identifies the component
with `Over.pullbackComp`, directed from pullback along the composite to
successive pullbacks. These are equalities of morphisms in the actual slice,
so both projections are fixed. The action on a 2-morphism is `eqToHom` of the
equality extracted by `LocallyDiscrete.eq_of_hom`. This is the pullback
coherence used by `finiteEtale_isStack` ([KL15] Definitions 1.3.1–1.3.2,
pp. 15–16; Theorems 1.3.4–1.3.5, p. 16).

**Checks.** `finiteEtalePrestack_identity_projection` gives the first
projection for pullback along the identity. `finiteEtalePrestack_two_steps`
sends the direct pullback to the iterated pullback while preserving the
projection to `T`. `finiteEtalePrestack_nontrivial_automorphism` shows that
postcomposing the unit component with any nonidentity automorphism of `T`
changes that component.

`etaleO` and `etaleOPlus` evaluate P2's actual global sections and integral
sections on each object as additive groups; their restriction maps are pullbacks
of functions. `etaleO_isSheaf` and `etaleOPlus_isSheaf` assert descent for these
presheaves. `etaleORing` and `etaleOPlusRing` retain multiplication on those same
carriers, with the same ring-homomorphism pullbacks. `etaleORing_isSheaf` and
`etaleOPlusRing_isSheaf` assert ring-valued descent for every perfectoid `X`,
and `etaleRing_forget` identifies their additive presheaves with `etaleO` and
`etaleOPlus` ([Sch12] Proposition 7.13, p. 43; [ECD] Theorem 6.3, p. 26).
The affinoid basis supplies sheafification in the universe containing these
sections and the corresponding Ext groups.

For any integral base section `a`, `etaleOPlusQuotient_ring` equips the
additive sheaf cokernel `etaleOPlusQuotient p X a` with a commutative ring
sheaf lift. The statement identifies the ring quotient arrow with the additive
cokernel arrow on every section. Thus the multiplication descends from `O⁺`,
and the quotient is taken in sheaves, without asserting that every section is
a pointwise quotient. This applies in particular to a locally chosen
pseudouniformiser ([Sch12] Proposition 7.13, p. 43, and the sheaf quotient
construction).

`etaleOPlusAlgebra` uses pullback along `U → X` to make `O⁺(U)` an algebra
over `globalPlus p X`. `etaleOPlusAlgebra_forget` identifies its underlying
ring presheaf and asserts descent. For any basic setup `S` on this base ring,
`etaleOPlusAlmost` applies P0's `Almost.Algebra.toAlmost S` to that algebra
presheaf; `etaleOPlusAlmost_isSheaf` asserts descent in `Almost.Algebra S`.
This retains the algebra unit and multiplication in almost modules.
`etaleOPlusAlmost_affinoidSetup` supplies a setup on
`globalPlus p (ofPair p R)` whose ideal consists exactly of the sections whose
images in `R` are topologically nilpotent. It is P1's `almostSetupPlus p R R.pair`
transported across the integral global section isomorphism. This specifies
`O^{+a}` locally on affinoid bases, with no fixed base field ([GR] §3.3,
3.3.1–3.3.2, p. 47; [ECD] Theorem 6.3, p. 26).

For a perfectoid `X`, a locally strongly sheafy adic `Y`, and
`f : X.obj → ofAdicSpace.obj Y`, `etaleAdicBaseChange` is the functor
`AdicSpace.smallEtale Y → smallEtale p X`. Its object is `U ×_Y X`.
`etaleAdicBaseChange_isPullback` supplies the projection to `U`, its adic
pullback square, and naturality in every arrow of the supplied adic site.
`etaleAdicBaseChange_continuous` and
`etaleAdicBaseChange_preservesFiniteLimits` give the morphism of sites
`X_ét → Y_ét`. The locally strongly sheafy hypotheses use A1's étale base
change and `perfectoid_etale_baseChange`; [Sch12] Lemma 7.3(iii)–(iv), p. 40,
Proposition 7.10 and the paragraph following Definition 7.11, p. 43, give the
locally noetherian case, while [ECD] Theorem 6.1 and Definition 6.2, p. 26,
supply the perfectoid Tate-ring input to this generality.

**Checks.**

- `etaleORing_affinoid`, `etaleORing_empty`, and `etaleORing_first_component`:
  the identity object over `Spa(R,R⁺)` has ring `R`; the empty object's ring
  is zero; restriction from `Spa(R × R)` to the first component sends `(a,b)` to `a`.
- `etaleOPlusRing_affinoid`, `etaleOPlusRing_empty`, and
  `etaleOPlusRing_second_component`: the corresponding integral ring is `R⁺`,
  the empty object has zero ring, and restriction to the second component
  sends `(a,b)` to `b` for `a,b ∈ R⁺`.
- `etaleOPlusAlgebra_scalars`, `etaleOPlusAlgebra_empty`, and
  `etaleOPlusAlgebra_nonzero`: the scalar map is the actual integral pullback;
  the empty object's algebra is zero; on the identity object over a nonzero
  affinoid ring, `1 ≠ 0`.
- `etaleOPlusAlmost_classical`, `etaleOPlusAlmost_zero_ideal`, and
  `etaleOPlusAlmost_classical_nonzero`: the classical setup recovers the
  integral algebra through P0's equivalence, the zero almost ideal kills
  every underlying module, and the classical almostification of a nonzero
  affinoid's identity object remains nonzero.
- `etaleAdicBaseChange_identity_object`, `etaleAdicBaseChange_empty`, and
  `etaleAdicBaseChange_split_two`: pulling back an isomorphism gives an
  isomorphism over `X`; pulling back the empty object gives the empty object;
  pulling back the split two-component cover of `Spa(ℚ₂,ℤ₂)` gives `X ⊔ X`.

<a id="p3-20"></a>

### P3.20 tilting identifies the étale sites

**Tilting identifies the étale sites: `X_ét ≃ X♭_ét`, compatibly with affinoids, fibre products,
coverings and `O⁺/ϖ`.** (*etale-site-tilting-equivalence*) Let `X` be a perfectoid space with tilt
`X♭` (layer P2). The tilting functor `Y ↦ Y♭` of the slice equivalence (layer P2: perfectoid spaces
over `X` `≃` perfectoid spaces over `X♭`) restricts to equivalences of categories `X_ét ≃ X♭_ét` and
`X_fét ≃ X♭_fét` of *etale-site-of-perfectoid-space*. These equivalences preserve and reflect affinoid
perfectoid objects (with `Spa(S, S⁺)♭ = Spa(S♭, S♭⁺)`), fibre products, and covering families
(`|Y| ≅ |Y♭|` compatibly with the maps to `|X| ≅ |X♭|`); hence they are equivalences of sites and
induce equivalences of topoi `Sh(X_ét) ≃ Sh(X♭_ét)` (Mathlib `CategoryTheory.Equivalence.sheafCongr`).
They are functorial in `X`: compatible with the morphisms of sites induced by maps `X' → X` and their
tilts. Locally choose a pseudouniformiser `ϖ♭` with `ϖ = (ϖ♭)♯` and
`ϖ^p | p` in the integral ring (as in P3.5); they identify the sheaves `O⁺_X/ϖ` and
`O⁺_{X♭}/ϖ♭`, and the almost sheaves `O⁺a/ϖ` and `O⁺a_{X♭}/ϖ♭`, on the identified sites. For affinoid
`X = Spa(R, R⁺)` the restriction to `X_fét` is the tilting equivalence `FÉt(R) ≃ FÉt(R♭)` of
*finite-etale-tilting-equivalence* ([ECD] §6, Thm 6.3, p. 26). *Needs:* P3.18.

**Checks.** On `X = Spa(ℂ_p, O_{ℂ_p})`,
`etale_site_tilting_test_wrong_scale` takes `a=p` in `globalPlus p X` and
rejects the chartwise hypothesis `a^p ∣ p` required by
`etaleIntegralQuotientTiltIso`. This tests its stated sufficient hypothesis;
it does not assert failure of every comparison modulo `p`.
`etale_site_tilting_Cp_wrong_scale` evaluates the actual cokernel sheaf
`etaleOPlusQuotient p X (p²)` on the identity étale object: the cokernel
projection sends `p` to a nonzero section. On the corresponding tilted object,
`p` kills every section of every `etaleOPlusQuotient`. Thus these quotient
sheaves cannot be identified at scale `p²` ([Sch12] Thm 5.2 and Lem. 5.3,
p. 22; Thm 7.12 and Prop. 7.13, p. 43).

`finiteEtaleTiltFunctor` is the restriction of P2’s geometric slice tilt to
finite objects; its equivalence instance and `finiteEtaleTilt_covers` give
the finite-site assertion. `etaleOPlusQuotient` is the cokernel of scalar
multiplication in abelian sheaves. `etaleIntegralQuotientTiltIso` compares
these quotient sheaves when `b♯=a`, with `a` a pseudouniformiser and `a^p | p`
on every affinoid chart. `etaleIntegralQuotientTiltIso_sharp` fixes the
comparison on local sections by the sharp map and the cokernel projections.
`etaleTiltSheafEquiv` is Mathlib's `CategoryTheory.Equivalence.sheafCongr`
applied to `etaleTiltFunctor`, using `etaleTilt_covers` to identify the topologies.
For every perfectoid `X` and every sheaf on its tilted étale site,
`etaleTiltSheafEquiv_inverse_obj` identifies inverse image with precomposition
by this geometric site tilt ([ECD] Theorem 6.3, p. 26). This is a sheaf quotient; taking global sections
need not give the ordinary quotient before almostification.

**Checks.** `etaleTiltSheafEquiv_sections` computes sections over an object as
sections over its tilt; `etaleTiltSheafEquiv_restriction` computes restriction
along an arbitrary arrow as restriction along its tilt; `etaleTiltSheafEquiv_empty`
states that sections over an empty object form a singleton.
`integral_quotient_zero`, `integral_quotient_unit`, and
`integral_quotient_empty` give respectively `O⁺`, the zero sheaf, and zero
sections over the empty object. `tilt_site_empty`, `tilt_site_nonfinite`,
`tilt_finite_empty`, and `tilt_finite_nonempty` distinguish empty objects and
preservation of the finite/nonfinite distinction under tilting.

### P3.21 almost acyclicity on the étale site

**Almost acyclicity of `O⁺` and acyclicity of `O` on the étale site of an affinoid perfectoid
space.** (*etale-almost-acyclicity*) Let `X = Spa(R, R⁺)` be an affinoid perfectoid space (`R`
perfectoid Tate, no base field) with étale site `X_ét`, and write `H^i(X_ét, −)` for sheaf cohomology
of abelian sheaves on `X_ét` (Mathlib `CategoryTheory.Sheaf.H`). Then: (a) `H⁰(X_ét, O⁺) = R⁺` and
`H^i(X_ét, O⁺)` is almost zero (killed by `R°°`) for `i > 0`; (b) `H⁰(X_ét, O) = R` and
`H^i(X_ét, O) = 0` for `i > 0`; (c) for every finite covering `{U_j → U}` in `X_ét` of an affinoid
perfectoid object `U` by affinoid perfectoid objects, the augmented Čech complex of `O⁺` is almost
exact and that of `O` is exact; (d) `H⁰(X_ét, O⁺/ϖ)^a = (R⁺/ϖ)^a` and `H^i(X_ét, O⁺/ϖ)` is almost zero
for `i > 0`; (e) for every finitely generated projective `R`-module `M`, `H^i(X_ét, M ⊗_R O) = 0` for
`i > 0`. The same holds on the analytic site (layer P2) and for every affinoid perfectoid object of
`X_ét` ([ECD] §6, Thm 6.3, second paragraph, p. 26 (`O⁺`); [Sch12] Prop. 7.13, p. 43 (`O^{°a}` and
hence `O` on `X_ét`, over a perfectoid field)). *Needs:* P3.18; P3.19 (finite projective coefficients).

`etaleOSheaf` and `etaleOPlusSheaf` are these concrete presheaves with their
sheaf proofs. `etaleIntegralScalar_apply` fixes multiplication by a pulled-back
integral base section. `etaleO_acyclic` and `etaleOPlus_almost_acyclic` use
Mathlib `Sheaf.H` and require `q > 0`; the latter says the induced scalar map
is zero for each topologically nilpotent base element. `etaleTiltSheafEquiv`
is the equivalence of sheaf categories for the displayed covering topologies.
`etaleOPlusQuotient_almost_acyclic` is the corresponding positive-degree
almost vanishing for an integral quotient by a pseudouniformiser.
`etaleOCoefficients` evaluates `M ⊗_R O(U)` with the scalar action pulled
back from the base; `etaleOCoefficients_map` sends `m ⊗ s` to `m ⊗ f*(s)`.
For finite projective `M`, `etaleOCoefficients_isSheaf` and
`etaleOCoefficients_acyclic` give its sheaf property and positive-degree
vanishing. The general vector-bundle equivalence remains the R5 supplier.

**Checks.** `coefficients_zero`, `coefficients_rank_one`, and
`coefficients_rank_two` compute `0`, `O`, and `O ⊕ O`.

**Checks.** `etale_sections_empty` gives zero sections over the empty object;
`etale_sections_affinoid` recovers `R` and `R⁺` on the identity affinoid object;
`etale_sections_two_constants` restricts `(a,b) ∈ R × R` to `a` on the first
component; `etale_integral_two_constants` restricts `(a,b) ∈ R⁺ × R⁺` to `b`
on the second component, through the affinoid section comparison. `etale_H0_not_zero` excludes degree zero from
acyclicity for every nonzero affinoid ring.

`etale_sections_split_two` computes both structure presheaves on the split
cover `Spa(R²)`: their values are `R²` and `(R⁺)²`. Together with the empty
and base-affinoid computations these are three distinct values for each
presheaf. `finite_basechange_split_two` computes the two copies of the new
base obtained by pulling back that split cover.

### P3.22 the tilting equivalence of étale sites with almost acyclicity

**The tilting equivalence of étale sites and the almost acyclicity of `O⁺` on `X_ét`.**
(*etale-site-tilting-and-etale-almost-acyclicity*) [ECD] §6, Thm 6.3, p. 26 — the equivalence of
sites `X_ét ≃ X♭_ét`, `Y ↦ Y♭`, for a perfectoid space `X` over an arbitrary perfectoid base, and
the almost vanishing of `H^i(X_ét, O⁺)` for `i > 0` on an affinoid perfectoid `X` (whence
`H^i(X_ét, O) = 0` by inverting `ϖ`, P3.21(b)) — is the conjunction of P3.20 and P3.21 (with the sheaf property of `O` on
`X_ét` from P3.18) and is not stated again; on `Spa(K, O_K)` for a perfectoid field `K` the
equivalence is the Fontaine–Wintenberger identification of P3.8. *Needs:* P3.20; P3.21.

**Checks.**

- `etale_site_tilting_test_char_p` (degenerate) — for `X` of characteristic `p`, `X♭ = X` and the
  equivalence is the identity of `X_ét`.
- `etale_site_tilting_test_point` (computation) — `X = Spa(K, O_K)` with `K = ℚ_p(p^{1/p^∞})^`:
  `X_fét ≃ X♭_fét` is `FÉt(K) ≃ FÉt(𝔽_p((t^{1/p^∞}))^)`; the objects of `X_ét` are the disjoint
  unions of the `Spa(L, O_L)`, `L/K` finite separable, so `Sh(X_ét)` is the category of discrete
  `G_K`-sets, `O⁺` corresponds to the discrete `G_K`-module `O_{K^sep} = colim_L O_L`, and
  `H^i(X_ét, O⁺) = H^i(G_K, O_{K^sep})`, which is almost zero for `i > 0` by P3.7(iv) and is not
  claimed to vanish exactly; `H^i(X_ét, O) = H^i(G_K, K^sep) = 0` for `i > 0` by the normal basis
  theorem, exactly. The example `etale_site_tilting_test_point` checks the geometric
  comparison for every finite extension of perfectoid fields with maximal plus rings:
  `etaleTiltFunctor` sends `Spa(L,O_L) → Spa(K,O_K)` to the map of tilted pairs,
  through `tiltSpaIso` on both source and base. Its hypothesis is the supplied
  `YonedaAdicSpace.IsFiniteEtale`; the equation fixes the map over the tilted base
  ([Berkeley] Theorem 7.3.1, p. 50; [ECD] Theorem 6.3, p. 26).

### Examples

The worked examples of this layer are the checks above and the standard instances of the theorems:
`ℤ_(3)` is not henselian along `3` (`X² − 7` has a simple root modulo `3` and no root), so `ℚ` with
the `3`-adic topology is not topologically henselian and `FÉt(ℚ) → FÉt(ℚ_3)` is not full, whereas
`FÉt(L) ≃ FÉt(L̂)` for every algebraic extension `L` of a perfectoid field; the Artin–Schreier
extension `S = K[u]/(u^p − u − t^{-1})` of `K = 𝔽_p((t^{1/p^∞}))^`, a perfectoid field with `S°a`
almost finite étale and uniformly almost finite projective (`n = p`) over `K°a` though `S°` is not
finite free over `K°`; the untilt of `K♭(t^{1/2})` being `K(p^{1/2})` for `K = ℚ_p(p^{1/p^∞})^` and
`p` odd; `L = ℚ_p^cycl(p^{1/ℓ})`, `ℓ ∤ p(p − 1)`, perfectoid and totally ramified of degree `ℓ`
with `L°a/K°a` almost finite étale although `L°` is not finite free over `K°` (the quadratic
`ℚ_p^cycl(√p)` is not such an example: `√p ∈ ℚ_p(ζ_p)` for `p ≡ 1 (mod 4)`, and
`ℚ_p^cycl(√p) = ℚ_p^cycl(ζ_4)` is unramified for `p ≡ 3 (mod 4)`); the Fontaine–Wintenberger
isomorphism `G_{ℚ_p(p^{1/p^∞})} ≅ G_{𝔽_p((t))}`, with the `p = 2` cyclotomic case
`G_{ℚ_2(ζ_{2^∞})} ≅ G_{𝔽_2((t))}`; the Frobenius of a characteristic-`p` perfectoid space as an
isomorphism; the almost purity theorem on `S = R`, `S = R × R` and `S = 0`; the Kummer cover
`u^ℓ = T` of the perfectoid torus over `ℚ_p^cycl`, finite étale of degree `ℓ` with perfectoid source
and tilt the Kummer cover of the tilted torus; the open unit disc inside the closed perfectoid disc,
étale but not finite étale; `Spa(K, O_K)_fét ≃` finite continuous `Gal(K^sep/K)`-sets and
`Spa(C, C⁺)_fét ≃ FintypeCat` for `C` algebraically closed; and, for a finite étale Galois cover
`Y → X` of affinoid perfectoids with group `G`, `H^i(G, O⁺(Y))^a = 0` for `i > 0` and
`O⁺(Y)^G = R⁺`.

The two proved Lean examples for this layer show that the split finite étale algebra `R × R`
of a perfectoid Tate ring is perfectoid and that the empty adic space is sousperfectoid.

### Dependencies

Layers P0 (almost mathematics: almost finite étale algebras, almost elements, flat deformation
modulo a tight ideal, almost sheaves), P1 (perfectoid Tate rings, the characteristic-`p` criterion,
pseudouniformisers with `ϖ^p | p`, tilts, untilts and `θ`, perfectoid fields) and P2 (perfectoid
spaces, their tilts, fibre products, the analytic site and the slice equivalence) of this roadmap,
with P3.2's field application using P1.10 and AdicEtaleGeometry A1.0.5.
Other roadmaps, by subsection: AdicEtaleGeometry A1.0.1–A1.0.5 supplies henselian pairs,
algebraic finite étale colimits, smooth lifting and completion comparisons; P3.3 uses A1.2
(finite étale Huber pairs) and A1.2a (uniformity and integral closure); P3.9, P3.10 and P3.14
use A1.2–A1.3 (finite étale and étale morphisms); P3.14–P3.15 use A1.9 (their Yoneda forms);
P3.17 uses A1.1, A1.3, A1.5 and A1.6 (strong sheafiness, stability, sites and structure sheaves);
P3.18 uses A1.5, A1.6 and A1.8 (the Galois category); P3.19 uses A1.2, A1.6 and A1.6a
(rational descent, the étale site and stable-basis vector bundles), together with
AdicSpacesPartIII R3.5 (rational-site vector bundles). P3.17a–P3.17b use AdicSpacesPartII R0
(completed tensors, restricted series and fibre products), AdicSpacesPartIII R3.1 and R3.5
(acyclicity, split base change and finite-projective étale descent), and AdicEtaleGeometry A1
(the étale site and covering reduction). The rational localisations of AdicSpaces 3.1 enter
through P2. Mathlib `HenselianRing`,
`IsAdicComplete.henselianRing`, `CommAlgCat.FiniteEtale`,
`CommAlgCat.FiniteEtale.baseChange`, `Module.Finite`, `Algebra.Etale`, `Algebra.Smooth`,
`PerfectRing`, `spectralNorm`, `IsAlgClosed`, `Field.absoluteGaloisGroup`,
`InfiniteGalois.IntermediateFieldEquivClosedSubgroup`, `CategoryTheory.Pseudofunctor.IsStack`,
`CategoryTheory.Equivalence.sheafCongr`, `CategoryTheory.Sheaf.H`; Tau Ceti `Huber.IsTateRing`,
`Huber.powerBoundedSubring`, `Huber.Pair`, `ValuationSpectrum.spa`, `ValuationSpectrum.rationalSubset`,
`Huber.restrictedMvPowerSeriesCompletion` and the completion of Huber rings.

## Layer P4: injections, immersions and separatedness

The layer builds the points of a perfectoid space with values in affinoid perfectoid fields,
injections together with the qcqs isomorphism criterion, immersions, Zariski closed immersions and
the universal perfectoid space over a Zariski closed subset, the theorem that Zariski closed
immersions are strongly Zariski closed, diagonals, separatedness with its valuative criterion, and
the behaviour of all of these under tilting.

### P4.1 affinoid perfectoid fields and their points

**Affinoid perfectoid fields and `(K, K⁺)`-valued points of perfectoid spaces.**
(*affinoid-perfectoid-field-points*) An affinoid perfectoid field is a pair `(K, K⁺)` of a
perfectoid field `K` (a perfectoid Tate ring that is a nonarchimedean field, [ECD] §3,
Definition 3.6; P1.10) and an open and bounded valuation subring `K⁺ ⊂ K`; equivalently `K⁺` is a
valuation subring with `K°° ⊂ K⁺ ⊂ K° = O_K`, so `(K, K⁺)` is a Huber pair and `Spa(K, K⁺)` is an
affinoid perfectoid space; it is algebraically closed if `K` is. A morphism `(K, K⁺) → (L, L⁺)` is a
morphism of Huber pairs, also called an extension of affinoid perfectoid fields. The points of
`Spa(K, K⁺)` are the valuation subrings `V` with `K⁺ ⊆ V ⊆ K°`; they form a chain, `K⁺` is the
unique closed point, `K°` the unique rank-one point, and the nonempty quasicompact open subsets
are the rational subsets `{|h| ≤ 1} = Spa(K, K'⁺)`, `h ∈ K`, where `K'⁺` is the smallest valuation
subring containing `K⁺[h]`, with `h ∈ K°` for a nonempty subset (a finite union of such sets is again one, the chain being totally
ordered); when `K⁺` has finite rank every valuation subring `K⁺ ⊆ K'⁺ ⊆ K°` is of this form, but
in general `Spa(K, K'⁺)` need not be open, and in particular the rank-one point `K°` need not be
open (see the Checks). For a perfectoid space `X` (P2.10) the set of
`(K, K⁺)`-valued points is `X(K, K⁺) := Hom(Spa(K, K⁺), X)`, covariant in `X` and in extensions of Huber pairs
`(K, K⁺) → (L, L⁺)` (precomposition with `Spa(L, L⁺) → Spa(K, K⁺)`), with the closed-point map `X(K, K⁺) → |X|`; for `X = Spa(R, R⁺)` affinoid perfectoid it
is the set of morphisms of Huber pairs `(R, R⁺) → (K, K⁺)`, and the closed-point map sends `φ` to
`spaComap φ` of the closed point. In `PerfectoidSpace`, define `AffinoidField` (a perfectoid field
with an open and bounded valuation subring), `AffinoidField.toPair` (the underlying Huber pair),
`AffinoidField.ofRankOne` (the pair `(K, K°)`), `AffinoidField.closedPoint` and `fieldPoints`
(`X(K, K⁺)`, with `fieldPoints.map`, `fieldPoints.restrict`, `fieldPoints.closedPointImage`); prove
`AffinoidField.valuationEquiv`, `AffinoidField.isOpen_iff` and `fieldPoints_affinoidEquiv`
(`X(K, K⁺) ≃ Huber.Pair.Hom (R, R⁺) (K, K⁺)` for `X` affinoid) ([ECD] §5, Proposition 5.3 (ii),
p. 21; [Berkeley] Lecture 8, Definition/Proposition 8.3.3 (2)–(3), p. 61). *Needs:* P1.10; P2.10;
Tau Ceti `TauCeti.Huber.Pair`, `TauCeti.Huber.Pair.Hom.spaComap`; Mathlib `ValuationSubring`.

**Checks.**

- `AffinoidField.subsingleton_spa_ofRankOne` (computation) — for every perfectoid field `K`,
  `|Spa(K, K°)|` has exactly one point.
- `AffinoidField.card_spa_eq_two` (computation) — if `K⁺ ⊊ K°` is the preimage of a rank-one
  valuation ring `k⁺ ⊊ k` of the residue field `k = K°/K°°`, then `|Spa(K, K⁺)|` has exactly two
  points and only `K⁺` is closed.
- `AffinoidField.fieldPoints_affinoidEquiv_closedPoint` (compatibility) — for `X = Spa(R, R⁺)`
  affinoid perfectoid and `φ ∈ X(K, K⁺) = Huber.Pair.Hom (R, R⁺) (K, K⁺)`, the closed-point image of
  `φ` is Tau Ceti `Huber.Pair.Hom.spaComap φ` applied to the closed point of `Spa(K, K⁺)`.
- `AffinoidField.not_isOpen_rankOnePoint` (non-example) — let `K⁺` be the preimage of a valuation
  ring `k⁺` of the residue field `k = K°/K°°` with value group `⋃_n ℚⁿ` ordered lexicographically
  with the last coordinate dominant, whose convex subgroups are `0 ⊂ ℚ ⊂ ℚ² ⊂ ⋯` with union the
  whole group. Then `|Spa(K, K⁺)|` is the chain `K⁺ = W_0 ⊂ W_1 ⊂ W_2 ⊂ ⋯ ⊂ K°` with
  `K° = ⋃_n W_n`; each `{V ⊇ W_n} = {|h_n| ≤ 1}` (`h_n` of value `−e_n`, for `n ≥ 1`) is open, but every open
  set containing `K°` contains some `{V ⊇ W_n}`, so `Spa(K, K°) = {K°}` is a quasicompact subset
  that is not open: not every `Spa(K, K'⁺)` is an open subset. The divisible group is essential:
  the residue field of a perfectoid field is perfect, so any valuation on it has
  `p`-divisible value group. The negative control `rank_group_not_integer` takes
  `K : AffinoidField 2` and an actual residue presentation `ρ : K° → k`,
  surjective with kernel the elements of norm less than one. It asserts that no
  valuation `k → WithZero (Multiplicative ℤ)` is surjective, excluding an integer
  residual value group for this carrier.

The structure `AffinoidField` stores a complete nontrivially normed ultrametric field,
its perfectoid-field instance, and a valuation subring satisfying `IsRingOfIntegralElements`.
`AffinoidField.valuationEquiv` gives the valuation-subring classification above;
`AffinoidField.closedPoint_spec` identifies the unique closed point. The two-point test uses
strict inclusion and the absence of intermediate valuation subrings, the equivalent
valuation-ring formulation of the rank-one residual hypothesis.

Additional Checks for the field and its projections:

| Check | Computation or distinction |
| --- | --- |
| `AffinoidField.toPair_rankOne` | The maximal-plus pair has plus ring exactly `K°`. |
| `AffinoidField.toPair_proper` | A strict smaller plus ring gives a different pair on the same field. |
| `AffinoidField.closedPoint_rankOne` | Every point of the maximal-plus spectrum equals its closed point. |
| `AffinoidField.closedPoint_two` | For strict smaller plus, a second point exists and is not closed. |
| `AffinoidField.no_padic` | No affinoid perfectoid field is topologically isomorphic to `ℚ_p`. |
| `fieldPoints_quadratic` | A separable degree-two extension has two distinct geometric field points with the same image. |
| `fieldPoints_map_identity`, `fieldPoints_map_composition` | The identity field point sends each scalar `t` to `t`; postcomposition along `R → S` sends `t` to `a(f(t))` under the affinoid comparison. |
| `fieldPoints_restrict_identity`, `fieldPoints_restrict_composition` | The identity field point extended along `e : K → L` sends `t` to `e(t)`; a further extension `d : L → M` sends it to `d(e(t))`. |
| `fieldPoints_closedPoint_natural` | Postcomposition sends the closed-point image by the underlying continuous map. |

The Check `fieldPoints_restrict_identity` evaluates the identity field point after
extension `e : K → L`: its scalar map sends `t` to `e(t)` ([ECD] §5, p. 21).

`AffinoidField.exists_infiniteRank` supplies this realization for every prime `p`:
a characteristic-`p` affinoid perfectoid field `K`, a surjective residue map
`ρ : K° → k` with kernel `{a | ‖a‖ < 1}`, and a surjective valuation of the perfect
field `k` with value group `⊕ₙ ℚ`, ordered with the largest index dominant.
It identifies `K⁺` with the inverse image of its valuation ring, and supplies
all the proper coarsenings `W_n`, strictly increasing from `K⁺` and exhausting `K°`.
One construction takes the Hahn field `k = 𝔽_p((⊕ₙ ℚ))` and
`K = k((t^{1/p^∞}))^` with its `t`-adic norm. Divisibility of the exponent group
makes `k` perfect; completing the perfect `t`-power tower leaves residue field `k`.
Thus `K` is perfectoid by [ECD] Proposition 3.5, p. 15. The convex subgroups
supported on the first `n` coordinates give the asserted coarsenings; the
valuation-ring description of points is [Berkeley] Example 2.3.8, p. 16.
The example `AffinoidField.not_isOpen_rankOnePoint_realization` applies
`AffinoidField.not_isOpen_rankOnePoint` to this existence theorem and exhibits
an actual non-open rank-one singleton.

### P4.2 quasicompact and quasiseparated morphisms

**Quasicompact and quasiseparated morphisms of perfectoid spaces.**
(*quasicompact-and-quasiseparated-maps*) A perfectoid space `X` is quasicompact (resp.
quasiseparated) if `|X|` is quasicompact (resp. a Mathlib `QuasiSeparatedSpace`). A morphism
`f : Y → X` of perfectoid spaces is quasicompact if `f⁻¹(U)` is quasicompact for every quasicompact
open `U ⊆ |X|`, equivalently for every open affinoid `U ⊆ X`; it is quasiseparated if the diagonal
`Δ_f : Y → Y ×_X Y` (Mathlib `CategoryTheory.Limits.pullback.diagonal` in the category of perfectoid
spaces, whose fibre products are P2.16) is quasicompact; qcqs means both. Equivalently, `f` is
quasiseparated if and only if for every open affinoid `U ⊆ X` and all open affinoids
`V, W ⊆ f⁻¹(U)` the intersection `V ∩ W` is quasicompact; if `X` is affinoid (or quasiseparated),
`f` is quasiseparated if and only if `|Y|` is quasiseparated. The comparison with the topos-theoretic notions for represented sheaves is owned by
AdicSpacesPartII R0.19.10 ([ECD] §8, Proposition 8.3, p. 41). In `PerfectoidSpace`, define `QuasiCompact` and `QuasiSeparated`; prove
`quasiCompact_iff_affinoid` (the test on open affinoids), `quasiSeparated_iff_affinoid_inter`,
`quasiSeparated_iff_quasiSeparatedSpace`, `QuasiCompact.comp`, `QuasiSeparated.comp`,
`QuasiCompact.baseChange` and `QuasiSeparated.baseChange` ([ECD] §5, remark after Proposition 5.11,
p. 25). *Needs:* Mathlib `CompactSpace`, `QuasiSeparatedSpace`,
`CategoryTheory.Limits.pullback.diagonal`; P2.16.

`QuasiCompact` lives on `Perfd`; `quasiCompact_iff_adicCompactOpens` identifies it with
AdicSpacesPartII R0.6's `AdicSpace.QuasiCompact` through `YonedaAdicSpace.ofAdicSpace`. Since a
perfectoid space is itself an adic space on the pre-adic carrier of P2.10, the representatives below
may be taken to be `Y.obj` and `X.obj` themselves, with `eY`, `eX` the identities.
Precisely, for `f : Y → X`, adic representatives `A, B`, isomorphisms
`eY : ofAdicSpace(A) ≅ Y.obj`, `eX : ofAdicSpace(B) ≅ X.obj`, and `g : A → B`
with `ofAdicSpace(g) ≫ eX.hom = eY.hom ≫ f.hom`, it equates `QuasiCompact p f`
with compactness of `g⁻¹(U)` for every compact open `U` of `B`. These are the
compact-open tests in R0 and [ECD] §5, p. 25; the comparison uses
`YonedaAdicSpace.ofAdicSpace_space`, which is natural in the adic space (AdicEtaleGeometry A1.16).

**Checks.**

- `QuasiCompact_affinoid` and `QuasiSeparated_affinoid` (computation) — every morphism `Spa(S, S⁺) → Spa(R, R⁺)` of affinoid
  perfectoid spaces is quasicompact and quasiseparated.
- `not_quasiCompact_openDisc` (non-example) — the open immersion of the open unit disc
  `⋃_n {|T|^n ≤ |ϖ|}` into `Spa(C⟨T^{1/p^∞}⟩, O_C⟨T^{1/p^∞}⟩)` is not quasicompact: the preimage of
  the whole disc is not quasicompact.
- `QuasiCompact_identity` and `QuasiSeparated_identity` (degenerate) — the identity of any perfectoid space is qcqs.

Further Checks use the global `Perfd` carrier:
`QuasiCompact_identity` and `QuasiSeparated_identity` test arbitrary identities;
`QuasiCompact_empty` and `QuasiSeparated_empty` test the empty open subspace;
`QuasiCompact_affinoid` and `QuasiSeparated_affinoid` test all maps between affinoids.
`not_quasiCompact_openDisc` supplies the non-quasicompact counterexample on the P1 root disc.

### P4.3 injections of perfectoid spaces

Injections are the monomorphisms of the category of perfectoid spaces; this subsection and the six
after it define them, build the residue-field points on which they are tested, and characterise
them by points, by residue fields and by base change.

**Injections of perfectoid spaces.** (*injection-of-perfectoid-spaces*) Throughout, `p` is a fixed
prime; perfectoid fields are those of [ECD] Definition 3.6, of either characteristic, with no base
field; `K⁺` is a valuation subring, and open integrally closed subrings that are not valuation
subrings (they occur in [ECD] Lemma 9.9) do not give affinoid perfectoid fields. A morphism
`f : Y → X` of perfectoid spaces is an injection if for every perfectoid space `Z` the map
`f_* : Hom(Z, Y) → Hom(Z, X)` is injective ([ECD] §5, Definition 5.1, p. 21). It suffices to test
affinoid perfectoid `Z`, because two morphisms `Z → Y` that agree on an open affinoid cover of `Z`
agree ([ECD] §5, sentence after Definition 5.1, p. 21). In the category `Perfd` of perfectoid spaces
this says exactly that `f` is a monomorphism (Mathlib `CategoryTheory.Mono`), and, fibre products
existing in `Perfd` (P2.16), it is equivalent to the diagonal `Δ_f : Y → Y ×_X Y` being an
isomorphism (Mathlib `CategoryTheory.Limits.pullback.isIso_diagonal_iff`). In `PerfectoidSpace`,
use `CategoryTheory.Mono` directly for injections. The comparison
`isInjection_iff_affinoid_tests` identifies it with injectivity on affinoid test objects.
Use Mathlib’s `pullback.isIso_diagonal_iff`, monomorphism composition and cancellation, and
split-monomorphism instances for the categorical consequences; sections and diagonals are injections.
The affinoid form `IsInjection_affinoid` is stated for a morphism `φ : (R, R⁺) → (S, S⁺)` of
arbitrary complete Hausdorff Tate Huber pairs (not assumed perfectoid or sheafy) as the
epimorphism condition towards perfectoid Huber pairs `(W, W⁺)`: `a ∘ φ = b ∘ φ` forces `a = b`;
for `R`, `S` perfectoid this is the notion above for `Spa(S, S⁺) → Spa(R, R⁺)`, since
`Hom(Spa(W, W⁺), Spa(S, S⁺)) = Hom((S, S⁺), (W, W⁺))` for affinoid perfectoid pairs (P2.9).
*Needs:* Mathlib `CategoryTheory.Mono`, `CategoryTheory.IsSplitMono`,
`CategoryTheory.Limits.pullback.diagonal`; P2.10; P2.16.

**Checks.**

- `injection_id` (degenerate) — the identity pair map cancels against every perfectoid test
  pair, including when the source ring is zero.
- `injection_rational` (computation) — for `X = Spa(R, R⁺)` affinoid perfectoid and `U ⊆ X`
  rational, the open immersion `U → X` is an injection.
- `isInjection_diagonal` (computation) — for the split double cover `X ⊔ X → X`
  with `X` nonempty, the diagonal is an injection but is not surjective on points.
  In general, for every morphism `f : Y → X` of perfectoid spaces,
  `Δ_f : Y → Y ×_X Y` is an injection, being a split monomorphism.
- `injection_quadratic` and `injection_quadratic_pair` (non-example) — for a perfectoid field `K` and a separable
  quadratic extension `L/K` with continuous structure map (`L` carries the extended valuation and is perfectoid by P3.7), `Spa(L, O_L) → Spa(K, O_K)` is bijective
  on underlying spaces but not an injection: the two `K`-embeddings of `L` into a completed
  algebraic closure `C` give distinct `(C, O_C)`-points with the same image.

Integral perfectoid rings give perfectoid Tate rings
(*integral-perfectoid-criterion-for-annuli*): that `R := R₀[1/π]`, for `R₀` `π`-adically complete
and integral perfectoid with `π` a non-zero-divisor and `π^p | p`, with chosen compatible roots of `π`, is a perfectoid Tate ring with
`R₀ ⊆ R°`, `πR° ⊆ R₀` and `R°/R₀` killed by every `π^{1/pⁿ}` is
*perfectoid-tate-ring-from-integral-perfectoid* (P1.20, [BMS] §3.2, Lemma 3.21, p. 27); it is
applied here to the compatible-root annulus rings `R⟨T^{1/p^∞}, (ϖ^a/T)^{1/p^∞}⟩`, which are
therefore perfectoid. For a nonzero perfectoid field `R = K` and a nondegenerate annulus
(`a` a positive integer), the ordinary annulus `K⟨T, ϖ^a/T⟩` is not perfectoid; the zero-ring case
does not give a non-example.

The carrier `ordinaryAnnulus` is the quotient of the supplier's two-variable restricted
series ring by `TU − ϖᵃ`, with the quotient topology and its additive uniformity.
`ordinaryAnnulus_not_perfectoid` requires `a > 0` and an affinoid perfectoid field.
`rootAnnulus` instead rationally localizes the P1 root disc at `|ϖᵃ| ≤ |T|`.
`rootAnnulus_isPerfectoid` reads the resulting `PairCat.perfectoid` field. The proof
obligation is P2’s theorem `isPerfectoidTateRing_rationalLocalization` on the actual
`RationalLoc` carrier for a presentation whose numerators generate the unit ideal;
`rationalPair_presentation` identifies the bundled annulus with that localization
([ECD] Theorem 3.18, p. 18).
Checks `ordinaryAnnulus_coordinate_product`, `ordinaryAnnulus_nonzero` and
`ordinaryAnnulus_no_coordinate_root` compute the defining product, exclude the zero ring,
and show in characteristic p that `T` has no pth root. Checks `rootAnnulus_zero_radius`,
`rootAnnulus_nonzero` and `rootAnnulus_frobenius` identify the unit annulus at `a = 0`,
exclude the zero ring, and give bijective Frobenius in characteristic p.

### P4.4 filtered intersections of rational subsets

**The perfectoid space attached to a filtered intersection of rational subsets.**
(*intersection-of-rational-subsets*) Let `X = Spa(R, R⁺)` be an affinoid perfectoid space,
`ϖ ∈ R⁺` a pseudouniformizer with `ϖ^p | p` admitting compatible `p`-power roots (P1.3), and `𝒰` a
nonempty family of rational subsets of `X` that is filtered (for `U, V ∈ 𝒰` some `W ∈ 𝒰` lies in
`U ∩ V`); put `T := ⋂_{U∈𝒰} U ⊆ |X|`. Construct `(S_𝒰, S⁺_𝒰)` from the filtered system of section rings
`(O_X(U), O⁺_X(U))`, `U ∈ 𝒰` (P2.3): `S⁺_𝒰` the `ϖ`-adic completion of `colim_{U∈𝒰} O⁺_X(U)` and
`S_𝒰 := S⁺_𝒰[1/ϖ]` with the `ϖ`-adic topology on `S⁺_𝒰`; and let `ι_𝒰 : Z_𝒰 := Spa(S_𝒰, S⁺_𝒰) → X`
be induced by the compatible maps `O_X(U) → S_𝒰`. Prove for this rational-localisation system: (a) `S_𝒰` is a perfectoid
Tate ring, `S⁺_𝒰` is a ring of integral elements, `S⁺_𝒰 ⊆ S_𝒰°` with almost zero cokernel,
`S⁺_𝒰/ϖⁿ = colim_U O⁺_X(U)/ϖⁿ`, and the union of the images of the `O_X(U)` is dense in `S_𝒰`;
(b) `|ι_𝒰|` is a homeomorphism of `|Z_𝒰|` onto `T` with its subspace topology, and `T` is
pro-constructible and stable under generalization; (c) a morphism `g : W → X` from a perfectoid
space factors through `ι_𝒰` if and only if `|g|(|W|) ⊆ T`, and then uniquely, so `ι_𝒰` is an
injection and `Z_𝒰` is the limit of the `U ∈ 𝒰` in `Perfd`; (d) `Z_𝒰` depends, up to unique
isomorphism over `X`, only on `T`. The completions do not depend on the choice of `ϖ`. In
`PerfectoidSpace`, define `proRationalSubspace` (the space `Z_𝒰`) and `proRationalSubspace.ι`
(the morphism `ι_𝒰`). Prove
`proRationalSubspace.integralCompletion_isIntegralPerfectoid`: for a nonempty filtered
`RationalSystem`, an integral pseudouniformizer `π` with `π^p ∣ p` in `R⁺`, the actual
`AdicCompletion (π) (colim O⁺(U))` is integral perfectoid and its image of `π` is a
nonzerodivisor ([ECD] Proposition 6.5, p. 29, and the remark following Lemma 7.6, p. 31).
The construction `proRationalPair` must supply `PairCat.perfectoid` and
`pair.isRingOfIntegralElements` for its localized completion, identified by
`proRationalSubspace.integralCompletion` and `proRationalSubspace.localization`;
`proRationalSubspace.isPerfectoid` exposes the former field. Prove
`proRationalSubspace.plus_mod_pow` (simp:
`S⁺_𝒰/ϖⁿ ≅ colim_U O⁺_X(U)/ϖⁿ`), `proRationalSubspace.range_ι`, `proRationalSubspace.lift` (with
factorization and uniqueness in the `∃!` statement), `proRationalSubspace.isInjection`, `proRationalSubspace.isLimit` and
`proRationalSubspace.congr` ([ECD] §7, remark after Lemma 7.6, p. 31; [ECD] §5, Example 5.2, p. 21;
[Bhatt] §9.4, Proposition 9.4.1 (1)–(3), p. 112 for the Zariski special case). *Needs:* *injection-of-perfectoid-spaces*; P1.3;
P2.3; *almost-integral-perfectoid-criterion* below; Tau Ceti `TauCeti.ValuationSpectrum.rationalSubset`; Mathlib
`AdicCompletion`.

**Checks.**

- `proRationalSubspace_stage_not_dense` (non-example) — take the two-member filtered
  family `{X, U}` with `X` the perfectoid closed disc and `U = {|T| = 1}`. The limit is `U`;
  the map `C⟨T^{1/p^∞}⟩ → C⟨T^{±1/p^∞}⟩` is not dense: every element of the source
  has zero `T⁻¹` coefficient, whereas `T⁻¹` has that coefficient equal to one.
- `proRationalSubspace_singleton` (degenerate) — for `𝒰 = {U}`, `ι_𝒰` is isomorphic over `X` to
  `U → X`.
- `proRationalSubspace_rationalNhds` (compatibility) — for `𝒰` the rational neighbourhoods of
  `x ∈ X`, `Z_𝒰 ≅ Spa(K(x), K(x)⁺)` over `X`, compatibly with *residue-field-point-injection*.
- `proRationalSubspace_annulusClosure` (computation) — for
  `X = Spa(C⟨T^{1/p^∞}⟩, O_C⟨T^{1/p^∞}⟩)` and `𝒰 = {{|ϖ^{1/pⁿ}| ≤ |T|} : n ≥ 0}`:
  `S_𝒰 = C⟨T^{±1/p^∞}⟩`, `S⁺_𝒰` is the `ϖ`-adic completion of
  `O_C[T^{1/p^∞}] + 𝔪_C[T^{−1/p^∞}]`, and `T` is the annulus `{|T| = 1}` together with the rank-two
  point `y` with `|T(y)| = 1⁻`.

Morphisms of perfectoid spaces are generalizing and spectral
(*maps-of-perfectoid-spaces-are-generalizing*): for a morphism `f : Y → X` of perfectoid spaces
(more generally of analytic adic spaces), `|f|` is generalizing (Mathlib `GeneralizingMap`) and is a
spectral map between open affinoids `V ⊆ Y`, `U ⊆ X` with `f(V) ⊆ U` (Mathlib `IsSpectralMap`), so
images of morphisms are stable under generalization ([ECD] §2, sentence before Lemma 2.5, p. 11).

**Almost Frobenius criterion without a base field.** (*almost-integral-perfectoid-criterion*)
Prove the P4 target `perfectoid_of_almost_frobenius` (the almost Frobenius criterion). Let `p` be prime, `A⁺` a
commutative ring and `t : ℕ → A⁺` satisfy `t(n+1)^p = t(n)`. Assume `ϖ := t(0)` is a
non-zero-divisor, `ϖ^p | p`, and `A⁺` is `ϖ`-adically complete and separated. Set
`𝔪 := (t(n) : n ∈ ℕ) = ⋃_n t(n)A⁺`. Assume the Frobenius map
`Φ : A⁺/(t(1)) → A⁺/(ϖ)`, `[a] ↦ [a^p]`, is almost bijective: `𝔪` annihilates its
kernel and, for every `ε ∈ 𝔪` and `y ∈ A⁺/(ϖ)`, `εy` lies in its image
(`Perfectoid.IsAlmostBijective`). The signature also supplies `p ∈ (ϖ)` and
`a ∈ (t(1)) → a^p ∈ (ϖ)` to form `frobeniusQuot`; these follow from the divisibility and
root hypotheses. Then `A := A⁺[1/ϖ]`, with zero-neighbourhood basis `ϖ^nA⁺`, is a complete
Hausdorff perfectoid Tate ring, `A⁺ ⊆ A°`, and `t(n)A° ⊆ A⁺` for every `n`. Thus
`A°/A⁺` is almost zero. This target extends the argument of [Sch12] §5, Lemmas 5.6–5.7,
p. 23, to the root ideal in `A⁺`; those source statements and the P1.28 dictionary have
perfectoid-field scope. Apply this P4 criterion to `S⁺_𝒰` above. *Needs:* P0.2; P1.1;
`frobeniusQuot`; `IsAlmostBijective`; Mathlib `IsAdicComplete` and `Localization.Away`.

The carrier `RationalSystem` records an actual nonempty family of P2 rational subsets,
filtered by intersection. Its `plusDiagram` uses the integral rings of `rationalPair` and
`rationalRes`; `scalar_stage` fixes the base-ring map into its ring colimit. The pair
`proRationalPair` and its map are characterized by `proRationalSubspace.integralCompletion`,
`proRationalSubspace.range_ι` and `proRationalSubspace.lift`. `proRationalStage_base` and
`proRationalStage_comp` fix the stage maps; `proRationalStage_dense` asserts density of their
union. `proRationalSubspace.cone` is a limit cone on `RationalSystem.spaceDiagram`.
`plus_mod_pow` identifies the integral quotient with the quotient of the filtered colimit
by the image of `ϖⁿ`; quotienting commutes with that ring colimit, giving the formula above.
These statements include `n = 0`, when both quotients are the zero ring.

Additional Checks:

| Carrier | Checks and values |
| --- | --- |
| `RationalSystem` | `RationalSystem_singleton`: intersection is `U`; `RationalSystem_empty_member`: one empty member forces empty intersection; `RationalSystem_whole`: the singleton whole space gives the whole space. |
| `proRationalSubspace` and its inclusion | `proRationalSubspace_singleton`: the actual P2 rational localization; `proRationalSubspace_empty_member`: empty space; `proRationalSubspace_whole`: inclusion is an isomorphism. |
| `rationalNeighborhoods` | `rationalNeighborhoods_field_closed`: all of a field spectrum; `rationalNeighborhoods_rankOne`: singleton at a rank-one point; `rationalNeighborhoods_generalizations`: every generalization belongs to the intersection. |

For the disc calculations, `discPair` is the P1 completed root-polynomial algebra with
power-bounded plus ring; `discAnnulus n` is the rational inequality
`|ϖ^{1/pⁿ}| ≤ |T|`, and `annulusSystem` is precisely this family. `unitAnnulus` is `1 ≤ |T|`.
`unitAnnulus_laurent` identifies its topological ring with P1's completed Laurent algebra.
The following Checks pin the carriers independently of their notation:

| Carrier | Checks and values |
| --- | --- |
| `discPair` | `disc_roots_two_terms`: the first two root equations; `disc_origin`: the universal quotient by `T` is the base field and kills every root; `disc_ordinary_quotient_nilpotent`: in the ordinary quotient, `T^{1/p}` is nonzero with pth power zero. |
| `discAnnulus` | `annulus_root_radius`: the next radius shrinks the annulus; `annulus_gauss`: every radius contains the Gauss point; `annulus_excludes_origin`: no radius contains a zero of `T`. |
| `unitAnnulus` | `annulus_gauss`: contains the Gauss point; `annulus_excludes_origin`: excludes the origin; `unitAnnulus_subset_all`: lies in every annulus of the system. |
| `annulusSystem` | `annulusSystem_closed_complement`: its intersection is the complement of the open unit disc; `annulusSystem_proper`: this intersection is proper; `unitAnnulus_subset_all`: the unit annulus belongs to the intersection. |
| `openUnitDisc` | `openUnitDisc_origin`: contains every zero of `T`; `openUnitDisc_excludes_gauss`: excludes the Gauss point; `not_quasiCompact_openDisc`: its inclusion fails quasicompactness. |

`proRationalSubspace.integralCompletion` identifies the plus ring with the actual adic
completion and fixes the image of every base integral scalar. `proRationalSubspace.localization`
specifies inversion of that scalar pseudouniformizer; `proconstructible`, `generalizations`
and `almost_plus` give the topological and almost-integral clauses.

For the annulus-closure computation, `annulusBoundaryPlus` is the closed subring of the
completed Laurent algebra generated by `aTʳ` with `r ≥ 0, |a| ≤ 1`, or `|a| < 1`.
`proRationalSubspace.annulusClosure_plus` identifies the integral ring of the intersection
with this subring. Checks `annulusBoundaryPlus_positive`,
`annulusBoundaryPlus_small_negative` and `annulusBoundaryPlus_negative_not_integral`
show that positive root monomials are integral, negative ones become integral after
multiplication by a topologically nilpotent coefficient, and a negative monomial with
coefficient one is not integral. This distinguishes the boundary plus ring from the
power-bounded subring of the same Laurent algebra ([ECD] §5, Example 5.2, p. 21).

### P4.5 the completed residue field point

**The completed residue field point `i_x : Spa(K(x), K(x)⁺) → X`.** (*residue-field-point-injection*)
For a perfectoid space `X` and a point `x ∈ X` (of any rank), the completed residue field `K(x)`
with its open and bounded valuation subring `K(x)⁺` is an affinoid perfectoid field (P2.5), and
there is a canonical morphism `i_x : Spa(K(x), K(x)⁺) → X` sending the closed point to `x`: on an
affinoid perfectoid neighbourhood `U = Spa(R, R⁺)` of `x` it is given by `(R, R⁺) → (K(x), K(x)⁺)`,
independently of `U`. Moreover ([ECD] §5, Example 5.2, p. 21): (a) `i_x` is isomorphic over `U` to
the pro-rational subspace `ι_𝒰` for `𝒰` the rational neighbourhoods of `x` in `U`
(P4.4), so `K(x)⁺` is the `ϖ`-adic completion of
`colim_{V∋x} O⁺_X(V)` and `K(x)⁺/ϖ = colim_{V∋x} O⁺_X(V)/ϖ`; (b) `i_x` is an injection and `|i_x|`
is a homeomorphism onto the set of generalizations of `x`; (c) every `(K, K⁺)`-valued point `φ` of
`X` whose closed point maps to `x` factors uniquely as `φ = i_x ∘ φ'` with
`φ' : Spa(K, K⁺) → Spa(K(x), K(x)⁺)` given by a morphism of affinoid perfectoid fields; (d) for
`f : Y → X` and `y ∈ Y` with `f(y) = x`, `f ∘ i_y = i_x ∘ Spa(K(x) → K(y))`. In `PerfectoidSpace`,
define `residueFieldPoint` (the morphism `i_x`); prove `residueFieldPoint_closed` (simp: `i_x`
maps the closed point to `x`), `residueFieldPoint_injection`, `residueFieldPoint_range`,
`residueFieldPoint_factor` ((c)), `residueFieldPoint_naturality` ((d)) and
`residueFieldPoint_eq_proRationalSubspace` ((a)). *Needs:* *intersection-of-rational-subsets*;
*affinoid-perfectoid-field-points*; *injection-of-perfectoid-spaces*; P2.5.

**Checks.**

- `residueFieldPoint_closedPoint` (degenerate) — for `X = Spa(K, K⁺)` and `x` the closed point,
  `i_x` is the identity of `X`.
- `residueFieldPoint_gauss` (computation) — for the Gauss point `η` of
  `Spa(C⟨T^{1/p^∞}⟩, O_C⟨T^{1/p^∞}⟩)`, `K(η)` is the completion of `C(T^{1/p^∞})` for the Gauss
  norm, `K(η)⁺ = K(η)°`, and `|i_η|` has image `{η}`.
- `residueFieldPoint_not_isImmersion` (non-example) — `i_η` is an injection whose image `{η}`
  is not open (and not locally closed), so `i_η` is not an immersion.

Completed tensor products over a nonarchimedean field preserve kernels and strictness
(*completed-tensor-over-analytic-field-kernel*): for `K` complete for a nontrivial nonarchimedean
absolute value, Banach `K`-modules `V, W, X` and a bounded `K`-linear `f : V → W`,
`V ⊗_K W → V ⊗̂_K W` is injective, `ker(f) ⊗̂_K X → ker(f ⊗̂ id_X)` is a bijection, and `f ⊗̂ id_X`
is strict when `f` is ([KL15] §2.2, Lemma 2.2.9, p. 31). The stalk of `O⁺/ϖ` at `x` is `K(x)⁺/ϖ`
(*stalk-of-plus-sheaf-modulo-pseudouniformizer*): for `ϖ` a pseudouniformizer near `x`, the map
`O⁺_{X,x} → K(x)⁺` has dense image with kernel `I = ϖI`, so `O⁺_{X,x}/ϖ → K(x)⁺/ϖ` is an
isomorphism ([Torsion] §II.3.1, Lemma II.3.1, p. 16).

Use the sibling declarations `BanachModule.CompletedTensor.injective_tensorProduct_of_field`,
`ker_map_of_field`, `injective_kerMap_of_field` and `isStrictMap_map_of_field` in
AdicSpacesPartII R0.5 (completed tensor products of Banach modules) for the Banach tensor assertions; no further tensor construction is needed.
The kernel assertion has no strictness hypothesis on `f`.

`integralStalk` is the ring colimit of the integral rational-neighbourhood diagram.
`integralStalkToResidue` maps it to the P2 completed residue plus ring;
`integralStalkToResidue_scalar`, `integralStalkToResidue_dense` and
`integralStalkToResidue_kernel` specify its base map, density and `I = ϖI`.
`integralStalk_mod_equiv` and its `_apply` equation give the induced quotient isomorphism.
This uses the rational basis description of the stalk, so it applies in every affinoid
neighbourhood of a point of a general perfectoid space.

Checks `integralStalk_field`, `integralStalk_mod_one` and `integralStalk_two_scalars` compute,
respectively, the stalk at a field's closed point as `K⁺`, the quotient by `1` as zero
in contrast to the nonzero quotient by a pseudouniformizer, and
the two scalar germs `(1,0)` and `(0,1)` on `Spa(K × K,K⁺ × K⁺)` as `1` and `0`
at the closed point of the first component.
`integralStalk_rankTwo_generic` computes the generic stalk of the two-point spectrum
`Spa(K,K⁺)` as `K°`, with every scalar in `K⁺` sent to itself. Here `K⁺ ⊊ K°`
and there are exactly two valuation overrings between them.
`integralStalk_origin` distinguishes a stalk from its residue field: at the origin
of the perfectoid disc the coordinate germ is nonzero. The matching map Checks
`integralStalkToResidue_field` and `integralStalkToResidue_rankTwo_generic` give
bijective evaluation for the two field stalks; `integralStalkToResidue_origin`
sends that nonzero coordinate germ to zero. The field computations use
[ECD] Example 5.2, p. 21; the kernel computation uses [Torsion] Lemma II.3.1,
p. 16, and the root-disc rational sections of P2.
`residueFieldPoint_rankOne_range` computes the singleton image at a rank-one point;
`residueFieldPoint_nonlocallyclosed` excludes an immersion when that singleton is not locally
closed. `gaussPoint_polynomial` specifies the Gauss valuation on finite root polynomials by
the maximum coefficient norm. Its Checks are `gauss_two_coefficients` (the maximum of the
two coefficient norms), `gauss_nonzero_coordinate` (`T` has nonzero value), and
`gauss_not_locallyClosed` (its singleton is not locally closed). Thus the Gauss-point
Checks above refer to the actual root disc and its P2 residue pair.

For a perfectoid pair `R`, a point `x`, and a rational neighbourhood `U` of `x`,
choose a presentation `q` whose numerators generate the unit ideal and whose rational
subset is `U`. Evaluation is `Perfectoid.rationalResidueMap (CAff.of R.pair)`
composed with the comparison supplied by `rationalPair_presentation`.
Its base identity identifies its composite from `R` with
the residue map, and its integrality statement places every integral
section in the completed residue plus ring, and `integralStalkToResidue_stage`
identifies the stalk map on each colimit stage with this same evaluation
([Berkeley] Lemma 7.4.6 and the argument after Theorem 7.4.8, pp. 53–55).
Checks `rationalResidue_zero`, `rationalResidue_scalar` and `rationalResidue_two_terms`
compute a nonzero base function in the valuation support as zero, a base scalar
as its residue value, and `(a+b)/s` as `(a(x)+b(x))/s(x)`.

`gaussRationalField` is the fraction field of the root-polynomial monoid algebra
`C[T^{1/p^∞}]`. `gaussRationalValuation_polynomial` fixes its valuation by the maximum
coefficient norm. `gaussCompletion` is its valued-field completion, and
`residueField_gauss_completion` identifies it topologically with the P2 completed residue
field at the Gauss point, compatibly with constants. Checks `gaussRational_constant`,
`gaussRational_two_terms` and `gaussRational_inverse_monomial` compute a constant, two
distinct exponents, and an inverse root monomial. Checks `gaussCompletion_nonzero`,
`gaussCompletion_constants` and `gaussCompletion_root_monomial` exclude the zero ring,
embed the constant field, and make every root monomial a unit.

### P4.6 the isomorphism criterion for qcqs maps

**Isomorphism criterion for qcqs maps: bijectivity on points and on `(C, C⁺)`-points.**
(*qcqs-isomorphism-criterion*) Let `f : Y → X` be a quasicompact and quasiseparated morphism of
perfectoid spaces (P4.2). The following are equivalent: (i) `f` is
an isomorphism; (ii) `|f| : |Y| → |X|` is bijective and for every rank-one point `y ∈ Y` with image
`x` the map `K(x) → K(y)` is an isomorphism; (iii) for every affinoid perfectoid field `(K, K⁺)` the
map `Y(K, K⁺) → X(K, K⁺)` is a bijection; (iv) for every algebraically closed affinoid perfectoid
field `(C, C⁺)` the map `Y(C, C⁺) → X(C, C⁺)` is a bijection. The qcqs hypothesis makes `|f|` a
spectral map of spectral spaces over an affinoid in (ii)+(iii) ⇒ (i) ([ECD] §5, Lemma 5.4, p. 23).
*Needs:* *quasicompact-and-quasiseparated-maps*; *affinoid-perfectoid-field-points*;
*residue-field-point-injection*; P2.16.

**Checks.**

- `isIso_of_bijective_qcqs_id` (computation) — the identity of `Spa(K,K⁺)` with
  `K⁺ ⊊ K°` and exactly two valuation overrings has two underlying points,
  satisfies the residue-field bijectivity clause (ii), and is bijective on both
  field-valued and algebraically closed field-valued points, (iii)–(iv).
  Its isomorphism assertion is obtained through `isIso_iff_residueFields`.
  `isIso_of_bijective_qcqs_empty` gives the empty-space case, whose sets of
  field-valued points are empty ([ECD] Lemma 5.4, p. 23).
- `not_isIso_openDisc_sum` (non-example, the qcqs hypothesis) — for
  `X = Spa(C⟨T^{1/p^∞}⟩, O_C⟨T^{1/p^∞}⟩)`, `D ⊆ X` the open unit disc and `Z_𝒰 → X` the closed
  immersion onto its complement `T = ⋂_n {|ϖ^{1/pⁿ}| ≤ |T|}` (P4.17), the map `D ⊔ Z_𝒰 → X` is
  bijective on points with isomorphic completed residue fields at every point, so (ii) holds, but it
  is not quasicompact (the preimage of `X` is not quasicompact) and not an isomorphism: `D` is open
  and closed in `D ⊔ Z_𝒰` and not closed in `X` (the point `y` with `|T(y)| = 1⁻` lies in its
  closure).

The proof rests on five results. A qcqs perfectoid space with a unique closed point `x` is
`Spa(K(x), K(x)⁺)` (*qcqs-with-unique-closed-point-is-a-field-point*): `i_x` is an isomorphism, so
`X` is affinoid with `O_X(X) = K(x)` and `O⁺_X(X) = K(x)⁺` ([ECD] §5, Example 5.2, last paragraph,
p. 21). Points of fibre products surject onto compatible pairs (*fibre-product-points-surjective*):
for `W = Y₁ ×_X Y₂` (P2.16), `|W| → |Y₁| ×_{|X|} |Y₂|` is surjective and spectral on spectral opens,
so points `y₁, y₂` over the same `x` lift to `(K(w), K(w)⁺)`-points `pr_i ∘ i_w` with the same
image ([ECD] §5, proof of Proposition 5.3, p. 22). Every affinoid perfectoid field embeds into an
algebraically closed one, and `(K, K⁺)`-points are detected there
(*algebraically-closed-extension-of-affinoid-field*): for `C` the completion of an algebraic
closure of `K` and `C⁺ ⊆ C°` the preimage of a valuation subring of the residue field of `C°`
dominating the image of `K⁺`, `(C, C⁺)` is an algebraically closed affinoid perfectoid field with
`C⁺ ∩ K = K⁺`, and `Y(K, K⁺) → Y(C, C⁺)` is injective for every perfectoid `Y` ([ECD] §5, proof of
Lemma 5.4, p. 23). A map of perfectoid fields that is injective on geometric points is an
isomorphism (*rank-one-residue-field-rigidity*): a continuous homomorphism `K → L` of perfectoid
fields such that any two continuous homomorphisms `L → C` agreeing on `K` are equal, for every
algebraically closed perfectoid field `C`, is an isomorphism ([ECD] §5, proof of Proposition 5.3,
p. 22). A homeomorphism inducing isomorphisms of completed residue pairs is an isomorphism
(*isomorphism-from-homeomorphism-and-residue-pairs*): if `|f|` is a homeomorphism and
`(K(x), K(x)⁺) → (K(y), K(y)⁺)` is an isomorphism for every `y` over `x`, then `f` is an
isomorphism ([ECD] §5, proof of Lemma 5.4, p. 23; *stalk-of-plus-sheaf-modulo-pseudouniformizer*
for the sheaf half).

The declarations are `isIso_iff_fieldPoints`, `isIso_iff_geometricPoints` and
`isIso_iff_residueFields`, each with explicit quasicompactness and quasiseparatedness.
`isIso_residueFieldPoint_of_uniqueClosed`, `AffinoidField.exists_algebraicallyClosed`,
`AffinoidField.rigidity` and `isIso_of_homeomorphism_residuePairs` state the supporting
interfaces. Fibre-product point surjectivity is already `pullback_points`
in P2.16; the spectral assertion on affinoids follows from `QuasiCompact_affinoid`.
`not_isIso_openDisc_sum` uses the genuine coproduct of the open disc and its pro-rational
complement, and states bijectivity, residue-pair isomorphisms and failure of quasicompactness
and of being an isomorphism together.

### P4.7 characterisations of injections

**Characterisations of injections of perfectoid spaces.** (*injection-criteria*) Let `f : Y → X` be
a morphism of perfectoid spaces, with no quasicompactness hypothesis. The following are equivalent:
(i) `f` is an injection (P4.3); (ii) for every affinoid perfectoid
field `(K, K⁺)`, `Y(K, K⁺) → X(K, K⁺)` is injective; (ii') the same for every algebraically closed
affinoid perfectoid field `(C, C⁺)`; (iii) `|f|` is injective and `K(x) → K(y)` is an isomorphism
for every rank-one point `y ∈ Y` with image `x`; (iv) `|f|` is injective and, for every perfectoid
space `Z`, `Hom(Z, Y) → Hom(Z, X) ×_{C⁰(|Z|, |X|)} C⁰(|Z|, |Y|)` is a bijection, i.e. `f` is final
among morphisms `g : Z → X` for which `|g|` factors through a continuous map `|Z| → |Y|`. This is
the continuous-factorisation contract of [ECD] Proposition 5.3(iv): continuity of `|Z| → |Y|` is for
the topology of `|Y|`, which need not be the subspace topology of the image. In
particular injections `Y → X` are determined, up to unique isomorphism over `X`, by the continuous
injection `|Y| → |X|`, not by its image set; `|Y|` need not carry the subspace topology of `|X|`.
Only when `|f|` is in addition a topological embedding does the factorisation condition reduce to
`|g|(|Z|) ⊆ |f|(|Y|)`, so that the injection is determined by its image subset with the subspace
topology ([ECD] §5,
Proposition 5.3 (iii)–(iv) and the paragraph after it, p. 21; [Berkeley] Lecture 8,
Definition/Proposition 8.3.3 (3), p. 61 for (ii')). *Needs:* *injection-of-perfectoid-spaces*;
*affinoid-perfectoid-field-points*; *residue-field-point-injection*; *qcqs-isomorphism-criterion*;
P2.16.

The step (iii) ⇒ (iv) uses the `(K, K⁺)`-points of a map that is injective on points with
isomorphic rank-one residue fields (*field-points-of-residually-injective-maps*): under (iii), for
every affinoid perfectoid field `(K, K⁺)` the map `f_* : Y(K, K⁺) → X(K, K⁺)` is injective with image
the `φ` whose closed point maps into `f(|Y|)` ([ECD] §5, proof of Proposition 5.3, (iii) ⇒ (iv),
p. 22).

The four criteria are `injection_iff_fieldPoints`, `injection_iff_geometricPoints`,
`injection_iff_residueFields` and `injection_iff_continuous_lift`; the affinoid comparisons
are `isInjection_iff_affinoid_tests` and `isInjection_affinoid_iff`.
`fieldPoints_image` specifies the image as precisely the points whose closed point lands in
that of the injection.

**Checks.**

- Same image, not isomorphic: the injection of P4.6 from the open disc together with its
  pro-rational complement has the same image as the identity of the closed disc, but the two are not
  isomorphic over the disc, since its `|f|` is a continuous bijection and not a homeomorphism; so an
  injection is not determined by its image set.
- Embedding case: a rational open immersion is an injection whose `|f|` is an open embedding, so it is
  determined by its image with the subspace topology.

### P4.8 base change of injections

**Base change of injections, and the underlying space of a pulled-back injection.**
(*injections-base-change-composition*) Let `f : Y → X` be an injection of perfectoid spaces and
`X' → X` any morphism, with base change `f' : Y' = X' ×_X Y → X'`. Then `f'` is an injection, and
the natural map `|Y'| → |X'| ×_{|X|} |Y|` is a homeomorphism (the target with the fibre product
topology). Moreover composites of injections are injections, and if `g ∘ f` is an injection then so
is `f` (use the supplier monomorphism instances for these categorical assertions) ([ECD] §5,
Corollary 5.5 (i), p. 24). *Needs:* *injection-of-perfectoid-spaces*; *injection-criteria*;
*maps-of-perfectoid-spaces-are-generalizing*; P2.16.

`injectionPullbackHomeomorph` identifies the actual categorical pullback with the
subspace of pairs `(y,z)` having the same image, and `injectionPullbackHomeomorph_apply`
fixes both projections in the order `(pullback.fst, pullback.snd)`.

### P4.9 universally injective maps

**Injections are exactly the universally injective maps.** (*injection-iff-universally-injective*)
A morphism `f : Y → X` of perfectoid spaces is an injection if and only if it is universally
injective: for every morphism `X' → X` the base change `f' : X' ×_X Y → X'` is injective on
underlying spaces; nothing beyond the existence of fibre products is assumed ([ECD] §5,
Corollary 5.5 (ii), p. 24). *Needs:* *injections-base-change-composition*; *injection-criteria*.

### P4.10 immersions, open immersions and closed immersions

Immersions are injections that are topological embeddings with locally closed image; this
subsection and the next define them and match the open ones with the open immersions of AdicSpaces.

**Immersions, open immersions and closed immersions of perfectoid spaces.** (*perfectoid-immersion*)
A morphism `f : Y → X` of perfectoid spaces is an immersion if `f` is an injection
(P4.3) and `|f| : |Y| → |X|` is a locally closed immersion, i.e. a
topological embedding (Mathlib `Topology.IsEmbedding`; `|Y|` carries its own spectral topology, and
the embedding condition says it is the subspace topology) with locally closed image (Mathlib
`IsLocallyClosed`). It is an open immersion (resp. closed immersion) if moreover `|f|` is an open
(resp. closed) embedding (Mathlib `Topology.IsOpenEmbedding`, `Topology.IsClosedEmbedding`). No
quasicompactness is assumed. The definition is possible because perfectoid spaces carry no nilpotent
structure: an injection is determined by its continuous injection `|Y| → |X|`
(*injection-criteria* (iv)), and when `|f|` is an embedding by its image with the subspace topology,
so an immersion is determined up to unique isomorphism over `X` by the locally closed subset
`f(|Y|)`. Neither condition on `|f|` follows from P4.7, so the definition is not redundant: the
injection of P4.6 from the open disc and its pro-rational complement is a bijection onto the closed
set `|X|` but not an embedding, and `i_η` at the Gauss point (P4.5) is an injection that is not an
immersion.
Closed immersions are strictly more general than Zariski closed immersions (P4.17). In
`PerfectoidSpace`, define `IsImmersion`, `IsOpenImmersion` and `IsClosedImmersion`.
`IsImmersion.isInjection` projects the `Mono f` conjunct of `IsImmersion`. Prove `IsImmersion.ext` (two immersions into `X` with the same image are
uniquely isomorphic over `X`), `IsImmersion.stableUnderGeneralization_range`,
`IsClosedImmersion.quasiCompact` and `IsOpenImmersion.of_isOpen` (for `U ⊆ |X|` open, the open
perfectoid subspace `U → X` is an open immersion) ([ECD] §5, Definition 5.6 and the sentence after
it, p. 24). *Needs:* *injection-of-perfectoid-spaces*; Mathlib `Topology.IsEmbedding`,
`IsLocallyClosed`, `Topology.IsOpenEmbedding`, `Topology.IsClosedEmbedding`; P2.10.

**Checks.**

- `isOpenImmersion_rationalSubset` (computation) — for `X = Spa(R, R⁺)` affinoid perfectoid and
  `U ⊆ X` rational, `U → X` is an open immersion.
- `isClosedImmersion_origin` (computation) — the point `T^{1/pⁿ} = 0` (all `n`), i.e.
  `Spa(C, O_C) → Spa(C⟨T^{1/p^∞}⟩, O_C⟨T^{1/p^∞}⟩)`, is a closed immersion.
- `IsOpenImmersion_empty`, `IsClosedImmersion_empty`, `IsOpenImmersion_identity` and `IsClosedImmersion_identity` (degenerate) — the empty perfectoid space maps to every `X` by an open and
  closed immersion; the identity is both.

Additional Checks for these predicates are
`IsImmersion_identity`, `IsImmersion_empty`, `IsImmersion_open`;
`IsOpenImmersion_identity`, `IsOpenImmersion_empty`, `IsOpenImmersion_open`;
and `IsClosedImmersion_identity`, `IsClosedImmersion_empty`, `IsClosedImmersion_principal`.
They compute identity and empty inclusions, arbitrary open restrictions, and the universal
closed inclusion for a principal ideal. The Gauss-point and annulus-closure Checks provide
the distinctions from a general injection and from a Zariski closed immersion.

### P4.11 open immersions agree with those of AdicSpaces

**Open immersions of perfectoid spaces agree with AdicSpaces' open immersions.**
(*open-immersion-comparison*) The P4 notion lives on `Perfd`:
`openImmersion_iff_adic` identifies it with `YonedaAdicSpace.IsOpenImmersion` on the
underlying sheaf morphism. For any adic representatives `g : A → B`, `eY`, `eX`
compatible with `f : Y → X` as in P4.2, `openImmersion_iff_preAdic` identifies it with
`TauCeti.PreAdicSpace.IsOpenImmersion g.hom` (AdicSpaces §5.1), through
`YonedaAdicSpace.ofAdicSpace`. On the pre-adic carrier of P2.10 this second identification is
definitional: open immersions of `Perfd` are `TauCeti.PreAdicSpace.IsOpenImmersion` between
perfectoid spaces. AdicEtaleGeometry A1.16's *adic-spaces-in-yoneda-adic-spaces*, by which `ofAdicSpace` preserves and
reflects open immersions, relates these two adic formulations. Equivalently, `f` identifies `Y` with the open
subspace `X|_U`, where `U = f(|Y|)`. Every open subset of `X` has its perfectoid
restriction (P2), and rational subsets of affinoid perfectoid spaces are affinoid
perfectoid. Thus open immersions into `X` correspond to its open subsets
([ECD] Proposition 5.3(iv), p. 21, and Definition 5.6, p. 24).
*Needs:* *perfectoid-immersion*; *injection-criteria*; layer P2;
`TauCeti.PreAdicSpace.IsOpenImmersion`; AdicEtaleGeometry's
*adic-spaces-in-yoneda-adic-spaces*.

### P4.12 Zariski closed subsets and Zariski closed immersions

Zariski closed subsets of an affinoid perfectoid space are the zero loci of ideals; this subsection
and the next define the two notions of closed immersion attached to them, which P4.15 shows agree.

**Zariski closed subsets and Zariski closed immersions into affinoid perfectoid spaces.**
(*zariski-closed-immersion*) Let `X = Spa(R, R⁺)` be an affinoid perfectoid space. For an ideal
`I ⊆ R` put `V(I) := {x ∈ |X| : |g(x)| = 0 for all g ∈ I}`; a subset `Z ⊆ |X|` is Zariski closed if
`Z = V(I)` for some ideal `I` ([Torsion] §II.2, Definition II.2.1, p. 13). A morphism `f : Z → X` of
perfectoid spaces is a Zariski closed immersion (defined by `I`) if `f` is a closed immersion
(P4.10) and `f(|Z|) = V(I)` ([ECD] §5, Definition 5.7 (i), p. 24). Conventions:
`V(I)` depends only on the closure of `I` and on its radical; `V(I) = ⋂_{F ⊆ I finite} U_F` with
`U_F := {x : |g(x)| ≤ 1 for g ∈ F}` rational, because `I` is an ideal and `|ϖ(x)|ⁿ → 0` cofinally at
every continuous valuation; `V(I)` is closed, pro-constructible and stable under generalization;
`V(I + J) = V(I) ∩ V(J)` and `V(IJ) = V(I ∩ J) = V(I) ∪ V(J)`; for a closed ideal `J`, `V(J)` is
the range of the closed embedding `spaComap` of the quotient pair (Tau Ceti
`Huber.Pair.Hom.range_spaComap_quotientHom`, `ValuationSpectrum.isClosedEmbedding_spaComap_quotientMk`),
which gives `isClosed_zeroLocus`. Every Zariski closed subset is the
image of exactly one Zariski closed immersion up to unique isomorphism (P4.14). A Zariski closed immersion defined by `I` need not be
`Spa(R/I, ·)`: the perfectoid structure on `V(I)` kills all compatible `p`-power roots. The global
notion over non-affinoid `X` (locally Zariski closed) is layer [P8](../PerfectoidTowers/README.md#layer-p8-finite-quotients-and-closed-perfectoid-loci-in-towers). In `PerfectoidSpace`, define
`zeroLocus` (`V(I)`), `IsZariskiClosed` and `IsZariskiClosedImmersion`; prove
`zeroLocus_eq_iInter_rational` (`V(I) = ⋂_F U_F`), `isClosed_zeroLocus`,
`stableUnderGeneralization_zeroLocus`, `zeroLocus_sup`, `zeroLocus_mul`, `zeroLocus_closure`
(simp: `V(closure I) = V(I)` and `V(√I) = V(I)`).
`IsZariskiClosedImmersion.isClosedImmersion` projects the closed-immersion conjunct
of [ECD] Definition 5.7(i), p. 24. *Needs:* *perfectoid-immersion*; Tau Ceti
`TauCeti.ValuationSpectrum.rationalSubset`, `TauCeti.Huber.Pair.Hom.range_spaComap_quotientHom`,
`TauCeti.ValuationSpectrum.isClosedEmbedding_spaComap_quotientMk`.

**Checks.**

- `zeroLocus_T` (computation) — for `R = C⟨T^{1/p^∞}⟩`, `V((T)) = {origin}`, and the Zariski closed
  immersion is `Spa(C, O_C) → X`; its kernel is the closure of the ideal generated by all
  `T^{1/pⁿ}`, not `(T)`.
- `zeroLocus_T_sub_one` and `zeroLocus_T_sub_one_char_p` (computation) — for `C = ℂ_p`, with the power-bounded plus ring and a chosen
  compatible system of primitive roots `ζ_{pⁿ}`, `V((T − 1)) ⊆ Spa(C⟨T^{±1/p^∞}⟩, C⟨T^{±1/p^∞}⟩°)` is homeomorphic to `ℤ_p`
  (compatible systems of `p`-power roots of unity, `γ ↦ (ζ_{pⁿ}^γ)_n`); for a perfectoid field `C` of characteristic `p`,
  `T − 1 = (T^{1/pⁿ} − 1)^{pⁿ}` and `V((T − 1))` is the single point `T^{1/pⁿ} = 1` for all `n`.
- `zeroLocus_bot_top` (degenerate) — `V(0) = |X|` and `V(R) = ∅`.

Additional Checks: `zeroLocus_bot_top` computes the whole and empty spaces;
`zeroLocus_two_ideals` computes `V(a,b) = V(a) ∩ V(b)` and
`V(ab) = V(a) ∪ V(b)`; `zeroLocus_nilpotent_quotient` computes
`V(aⁿ) = V(a)` for `n > 0`. Checks `isZariskiClosed_whole`, `isZariskiClosed_empty` and
`isZariskiClosed_union` test the subset predicate; `zariskiImmersion_id`,
`zariskiImmersion_empty` and `zariskiImmersion_nonclosed` test identity, empty source and the
obstruction of a nonclosed image. These use valuation support ideals, as in the supplier's
quotient-spectrum range formula.

### P4.13 strongly Zariski closed immersions

**Strongly Zariski closed immersions: surjective perfectoid quotients.**
(*strongly-zariski-closed-immersion*) Let `X = Spa(R, R⁺)` be an affinoid perfectoid space. A
morphism `f : Z → X` is strongly Zariski closed if `Z = Spa(S, S⁺)` is affinoid perfectoid, `f` is
induced by a morphism `(R, R⁺) → (S, S⁺)` with `R → S` surjective, and `S⁺` is the integral closure
of the image of `R⁺` in `S` ([ECD] §5, Definition 5.7 (ii), p. 24); `S` perfectoid is part of the
data. Equivalently (by the open mapping theorem, Tau Ceti `Huber.IsTateRing.isQuotientMap`), with
`I := ker(R → S)` closed, `(S, S⁺)` is isomorphic to the library's quotient pair
`TauCeti.Huber.Pair.quotient (R, R⁺) I`, whose plus ring is the integral closure of the image of
`R⁺` (`Huber.Pair.quotient_plus`), and `R/I` is perfectoid. The plus-ring clause is part of the
definition: it is what makes a strongly Zariski closed map a closed immersion (P4.21).
The algebraic predicate `IsStronglyZariskiClosed_affinoid` is defined for arbitrary Huber pairs
by these two clauses, without completeness or perfectoidness; its geometric interpretation and
the base-change theorem here require perfectoid Tate rings. For base change along `(A,A⁺) → (C,C⁺)`,
the resulting ring map is `C → B ⊗̂_A C`, the supplier’s `completedTensor.inr` (in `Huber.Pair`). In `PerfectoidSpace`, define `IsStronglyZariskiClosed`;
prove `stronglyZariskiClosed_affinoid_iff` (the affine predicate supplies both surjectivity
and the plus-ring equality through its fields), `stronglyZariskiClosed_isClosed_ker`,
`stronglyZariskiClosed_quotient_comparison` and `stronglyZariskiClosed_iff_zariskiClosed`
(the quotient immersion is defined by `ker(R → S)`). *Needs:* Tau Ceti
`TauCeti.Huber.Pair`, `TauCeti.Huber.Pair.quotient`, `TauCeti.Huber.Pair.quotient_plus`,
`TauCeti.Huber.IsTateRing.isQuotientMap`.

**Checks.**

- `completedTensor_inr_two_terms` (convention) — base change sends `c + d ∈ C` to
  the completion of `1 ⊗ c + 1 ⊗ d ∈ B ⊗_A C`; with `c = 2`, `d = 3` its value is `5`.
- `isStronglyZariskiClosed_origin` (computation) — `C⟨T^{1/p^∞}⟩ → C`, `T^{1/pⁿ} ↦ 0`, with plus
  rings `O_C⟨T^{1/p^∞}⟩ → O_C`, is strongly Zariski closed.
- `stronglyClosed_id` (degenerate) — the identity of `X` and the map from the empty space
  `Spa(0, 0)` are strongly Zariski closed.
- `stronglyClosed_quotient` (supplier agreement) — for any Huber pair `P` and any ideal `I`,
  `Pair.quotientHom P I` satisfies the algebraic predicate; `I = 0` and `I = ⊤` give
  the identity quotient and the zero quotient. No perfectoidness of the quotient is inferred.
- `not_isStronglyZariskiClosed_ofPowerBounded` (non-example) — for `R⁺ ⊊ R°`,
  `Spa(R, R°) → Spa(R, R⁺)` has `R → R` surjective and `R⁺ → R°` almost surjective ([ECD] Example
  3.22 (ii)), but `R°` is not the integral closure of `R⁺`, so it is not strongly Zariski closed
  (nor a closed immersion).

The geometric predicate `IsStronglyZariskiClosed` requires an affinoid presentation of the
source and the actual algebraic strong-quotient condition on its presenting map.
Checks `strongImmersion_id`, `strongImmersion_empty`, `strongImmersion_larger_plus` give
identity, empty source, and failure for a strict enlargement of the plus ring on the same
ring. `stronglyZariskiClosed_affinoid_iff` compares it to the affine predicate.

### P4.14 the universal perfectoid space over a Zariski closed subset

The universal perfectoid Zariski closed subspace is the pro-rational subspace of the rational
neighbourhoods `U_F` of `V(I)`; this subsection builds it and the next two show it is a surjective
perfectoid quotient and match it with the closed immersions of AdicSpaces.

**The universal perfectoid space over a Zariski closed subset.**
(*universal-perfectoid-zariski-closed*) Let `X = Spa(R, R⁺)` be an affinoid perfectoid space with a
pseudouniformizer `ϖ ∈ R⁺` with `ϖ^p | p` admitting compatible `p`-power roots, and `I ⊆ R` an ideal (not necessarily
closed or finitely generated). Let `𝒰_I := {U_F : F ⊆ I finite}`,
`U_F = {x : |g(x)| ≤ 1 for g ∈ F}`, and use `zariskiClosedSubspace.presentation_affinoid D F` for the presentation
`({1} ∪ F)/1`. This definition only needs a commutative topological ring and a pair of
definition `D`, without a perfectoid hypothesis. Let `ι_I : Z_I = Spa(S_I, S_I⁺) → X` be the pro-rational
subspace of `𝒰_I` (P4.4): `S_I⁺` is the `ϖ`-adic completion of
`colim_F O⁺_X(U_F)` and `S_I = S_I⁺[1/ϖ]`. Then: (a) `Z_I` is affinoid perfectoid and `R → S_I`
has dense image; (b) `|ι_I|` is a homeomorphism onto `V(I)`; (c) a morphism `g : W → X` from a
perfectoid space factors through `ι_I` if and only if `g^*(h) = 0` in `O_W(W)` for all `h ∈ I`, if
and only if `|g|` lands in `V(I)`, and then uniquely; (d) `ι_I` is a Zariski closed immersion
defined by `I` (P4.12); (e) `I^perf := ker(R → S_I)` is a closed ideal
containing `I`, equal to `{h ∈ R : |h| = 0 on V(I)}`, and `Z_{I^perf} = Z_I`; (f) `Z_I` depends
only on `V(I)`. In `PerfectoidSpace`, define `zariskiClosedSubspace` (`Z_I`) and
`zariskiClosedSubspace.ι` (`ι_I`); prove `zariskiClosedSubspace.range_ι` (homeomorphism onto
`V(I)`), `zariskiClosedSubspace.lift` (with factorization and uniqueness in the `∃!` statement),
`zariskiClosedSubspace.isZariskiClosedImmersion`, `zariskiClosedSubspace.denseRange`,
`zariskiClosedSubspace.ker_eq` ((e)), `zariskiClosedSubspace.congr` ((f)) and
`zariskiClosedSubspace_bot` (simp: `Z_0 = X`, `Z_R = ∅`) ([Torsion] §II.2, Lemma II.2.2, p. 13, and
Remark II.2.3, p. 14; [Bhatt] §9.4, Proposition 9.4.1 (1), p. 112). *Needs:*
*intersection-of-rational-subsets*; *zariski-closed-immersion*; *perfectoid-immersion*.

**Checks.**

- `presentation_empty` (degenerate) — `F = ∅` gives numerators `{1}` and denominator `1`.
- `presentation_zero` (computation) — `F = {0}` gives `{1,0}/1`; the extra inequality is
  `0 ≤ 1`, so its rational subset is the whole spectrum.
- `presentation_two_terms` (convention) — `F = {a,b}` gives `{1,a,b}/1`, imposing both
  `|a| ≤ 1` and `|b| ≤ 1`. In `presentation_two_term_bounds`, `(a,b) = (0,1)`
  gives the whole spectrum, while `(0,ϖ⁻¹)` gives the empty set for a topologically
  nilpotent unit `ϖ`: the second value is greater than one.
- `presentation_membership` (supplier agreement) — membership is exactly membership in
  `spa(R⁺)` together with `x.toValuativeRel.vle a 1` for every `a ∈ F`.
- `zariskiClosedSubspace_T` (computation) — `R = C⟨T^{1/p^∞}⟩`, `I = (T)`: `S_I = C`, and
  `ker(R → S_I)` is the closure of `(T^{1/pⁿ} : n ≥ 0) ⊋ (T)`; `R/(T)` is not reduced.
  The example uses `zariskiClosedMap`, a continuous ring isomorphism `S_I ≃ C`
  fixing constants, and equality with that closed ideal. Evaluation of the root
  polynomials at zero gives the calculation; the universal perfectoid quotient
  is [Torsion] Lemma II.2.2 and Remark II.2.3, pp. 13–14.
- `zariskiClosedSubspace_T_sub_one` (computation) — `C = ℂ_p`, `R = C⟨T^{±1/p^∞}⟩`, `I = (T − 1)`:
  `S_I = C⁰(ℤ_p, ℂ_p)`, with `T^{1/pⁿ} ↦ (γ ↦ ζ_{pⁿ}^γ)` ([AWS] Lecture 2, Example 2.4.10); the
  image is dense (the characters of `ℤ/pⁿ` span the functions on `ℤ/pⁿ`, `p` being invertible),
  `R → S_I` is surjective by P4.15, and `I^perf ⊋ (T − 1)` although `(T − 1)` is closed (P4.15).
- `zariskiClosedSubspace_bot` (degenerate) — `Z_0 = X` and `Z_R = ∅`.

Every Zariski closed immersion is the universal one (*zariski-closed-immersion-is-universal*): for
a Zariski closed immersion `f : Z → X = Spa(R, R⁺)` defined by `I`, there is a unique isomorphism
`Z ≅ Z_I` over `X`; consequently `Z` is affinoid perfectoid, `Z = lim_{F ⊆ I finite} U_F`, and a
Zariski closed immersion is determined by its image `V(I)` ([Torsion] §II.2, after Remark II.2.3,
p. 14).

`zariskiClosedPair_universal` characterizes the ring map by unique factorization among
perfectoid pairs annihilating `I`; `zariskiClosedSubspace.lift` gives the corresponding
factorization for every perfectoid space. `range_ι`, `ker_eq`, `congr` and `universal`
fix the underlying subset, the vanishing ideal and the isomorphisms over `X`.
Checks `zariskiClosedSubspace_bot`, `zariskiClosedSubspace_top` and
`zariskiClosedSubspace_radical` compute identity, empty source and invariance under radical;
`disc_origin` and `disc_ordinary_quotient_nilpotent` distinguish the perfectoid quotient
from the ordinary quotient by `(T)`.

`zariskiClosedSubspace_T_sub_one` specifies the continuous-function quotient over `ℂ_p`
with a chosen compatible sequence of primitive roots, sending the nth root of `T` to
`γ ↦ ζ_{pⁿ}^{γ mod pⁿ}`. It also states the strict enlargement of the kernel beyond
`(T−1)`. `zariskiClosedSubspace_T_sub_one_char_p` computes the contrasting characteristic-p
quotient: every root of `T` maps to `1`.

### P4.15 Zariski closed immersions are strongly Zariski closed

The proof route is [BS22] §7.1–§7.3. P4.15.1 consumes the generic derived-algebra inputs
and supplies the additional compatibilities specified below; P4.15.2–P4.15.5 own the
prismatic construction. They use P1 and P0.19's supplied ordinary cotangent complex, independently of the
Zariski-closed theorem in P4.15.6 and of its consumers in P6, PerfectoidTowers and
DiamondsAndVStacks. The algebraic constructions apply to integral perfectoid rings,
including rings with p-torsion and the zero ring.

#### P4.15.1 derived completion and complete flat descent

**Derived algebra for the semiperfectoid construction, consumed and specialized.**
(*semiperfectoid-derived-completion-descent*) Use DerivedModuleAlgebra's generic
carriers and maps in the following table. For a commutative ring `B` write
`D(B) = DerivedCategory (ModuleCat B)` with cohomological indexing; connective means
`H^i = 0` for `i > 0`. For finite `J ⊂ B`, `Λ_J` means that supplier's completion
reflector, with its specified unit.

| Supplier subsection | Imported API and use |
| --- | --- |
| DerivedModuleAlgebra §§0.1–0.4 and 0.14 | `Algebra.CotangentComplex.PolynomialResolution`, its `comparison` and `toQuasiIso`, `DerivedDeRham.derivedTensor`, `DerivedDeRham.leftDerivedNonadditive`, `DerivedDeRham.derivedExteriorPower` and the ordinary cotangent complex supply the resolution and power models for P4.15.2–P4.15.3. |
| DerivedModuleAlgebra §§1.1–1.6 | `DerivedDeRham.derivedLimit`, `DerivedDeRham.derivedCompleteness`, `DerivedDeRham.derivedCompletion`, `DerivedDeRham.completionUnit`, `DerivedDeRham.completionAdjunction`, `DerivedDeRham.completeNakayama`, `DerivedDeRham.koszulCompletionTower`, `DerivedDeRham.ordinaryQuotientCompletion` and `DerivedDeRham.simplicialRingCompletion` supply completion and its algebra unit. |
| DerivedModuleAlgebra §§1.7–1.9 | `DerivedDeRham.completeFlatness`, its discrete predicate `DerivedDeRham.IsCompletelyFlatModule`, `DerivedDeRham.completeFlatBaseChange`, `DerivedDeRham.boundedTorsionCriterion` and `DerivedDeRham.completeFlatDescent` supply complete flatness and descent for p-complete ordinary rings with bounded p-power torsion. |

Convention: `M` is J-completely flat precisely when `M ⊗_B^L B/J` is a flat
`B/J`-module in degree zero; complete faithful flatness additionally means that this
reduction is faithfully flat. Neither predicate includes completeness. The equivalent
J-power-torsion-module criterion is the supplier's §1.7. A completed tensor is
`Λ_J(M ⊗_B^L N)`. Realizations use direct-sum totalization of simplicial module models,
while Čech descent uses product totalization of cosimplicial complexes; completion is
applied to the resulting derived objects. Retain simplicial algebra models and their
multiplication through `simplicialRingCompletion`, rather than treating normalization as
a strictly commutative differential graded algebra. For discrete modules, ordinary
completion agrees with derived completion when the higher Koszul homology towers are
pro-zero; in particular a principal ideal and bounded power torsion on the module suffice
([BS22] §1.2, pp. 10–12, footnotes 5–7;
[Stacks, tag 0920](https://stacks.math.columbia.edu/tag/0920)).

The additional targets below use these same carriers and units.

- `semiperfectoidFiniteIdealDescent`: for a commutative ring `B`, finite
  `J = (f₁, …, f_r)` and a J-completely faithfully flat map `E• → F•` of simplicial
  commutative B-algebras, prove effective descent for derived J-complete modules along
  its derived J-completed Čech nerve. Thus the complete module category over `E•` is
  equivalent to the limit of those over that nerve, and a complete module is recovered
  by product totalization of its completed base changes. Complete flatness is measured
  after reduction to `π₀(E•)/Jπ₀(E•)`. This is strictly more general than the supplier's
  §1.9: it permits arbitrary finite ideals and simplicial bases without a bounded-torsion
  hypothesis. Supply the derived module models over simplicial bases and their coherent
  descent data, using simplicial resolutions and quasi-isomorphisms on underlying module
  complexes. Build the adapter to compatible systems over the simplicial Koszul quotients
  on `(f₁^n, …, f_r^n)` with the supplier's module-limit models. Prove simplicial
  faithfully flat descent at each Koszul level, then take the inverse limit. On §1.9's domain,
  identify this equivalence and its augmentation with `DerivedDeRham.completeFlatDescent`
  ([BS22] §1.2, footnote 7, p. 12). Composition, completed base change and completed
  filtered colimits preserve complete flatness; complete faithful flatness is preserved
  for diagrams of covers over a fixed base, as detected by faithfully flat reduction.
- `semiperfectoidCompletionSurjectivity`: for a p-completely faithfully flat map of
  ordinary rings `B → C`, and a map `M → N` of discrete derived p-complete B-modules,
  surjectivity on `H⁰` after derived p-completed base change implies surjectivity of
  `M → N`. Use the supplier's derived Nakayama and complete-flat reduction on the
  cokernel. Degree-zero completion is right exact on connective complexes for every
  finite ideal: completing a triangle of connective complexes gives the right exact
  sequence on `H⁰` ([BS22] §1.2 and proof of Theorem 7.4, p. 62;
  [Stacks, tag 0G1U](https://stacks.math.columbia.edu/tag/0G1U) and
  [tag 0AAJ](https://stacks.math.columbia.edu/tag/0AAJ)).
- `semiperfectoidCompletionTransfinite`: for a fixed finite ideal of `B`, derived
  completion commutes with `ω₁`-filtered colimits in `D(B)` and in simplicial B-algebras,
  compatibly with `completionUnit` and `ringCompletionUnit`. The finite-generator Koszul
  tower has countably many levels; `ω₁`-filtered colimits commute with the countable
  products in its derived-limit model. Apply this to `(p,d)` in P4.15.2 and `(p)` in
  P4.15.4 ([BS22] proof of Proposition 7.2, p. 55).

*Needs:* DerivedModuleAlgebra §§0.1–0.4, 0.14 and 1.1–1.9 as listed above;
Mathlib `DerivedCategory`, `CategoryTheory.SimplicialObject` and `TensorProduct`.

The simplicial module and faithfully flat descent adapters are included in
`semiperfectoidFiniteIdealDescent` here, as in [BS22] §1.2, footnote 7.
P4.15.2–P4.15.5 consume this subsection; the transfinite constructions use P1's
integral perfectoid interfaces.

**Checks.**

- For `J = (p)` and `B = ℤ_p`, completion fixes `B` in degree zero.
- The derived p-completion of `ℚ` is zero; `ℚ` is not itself derived p-complete.
- Over `ℤ_p`, the derived tensor of `𝔽_p` with `𝔽_p` has nonzero first Tor,
  so `ℤ_p → 𝔽_p` is not p-completely flat despite its flat ordinary reduction.
- Finite-ideal descent along the identity of the simplicial algebra
  `𝔽_p ⊗^L_ℤ 𝔽_p`, with `J = (p)`, retains its nonzero `H^−1`; descent does not
  replace a simplicial base by its degree-zero ring.
- For `J = (p,d)` and `B = ℤ_p[[d]]`, descent along `B → B × B` recovers every
  complete B-module with its canonical augmentation, using both ideal generators.
- For nonzero derived p-complete `M`, derived p-completed base change to `B[1/p]`
  kills `M`, so it cannot reflect surjectivity of `0 → M`; its zero reduction is not
  faithfully flat when `B/p ≠ 0`.
- Degree-zero completion sends the surjection `ℤ → ℤ/p` to `ℤ_p → ℤ/p`.
- A constant `ω₁`-filtered diagram has the identity comparison with completion.
- In the `ω₁`-filtered union of countable subsets `S` of `ω₁`, the union of the
  p-completions of `⨁_{s∈S} ℤ` is the p-completion of `⨁_{s<ω₁} ℤ`: every completed
  element has countable support.
- Countable filtered colimits alone fail: the union of `ℤ_p^n` under zero-extension
  is not p-complete; the compatible reductions of `(1,p,p²,…)` are missing.

#### P4.15.2 universal prisms and perfection

**Prisms for semiperfectoid rings.** (*semiperfectoid-universal-prism*) Develop `δ`-rings,
free `δ`-algebras, distinguished elements, and their divided-power comparison as in
[BS22] §2, restricted to the constructions below. A prism `(B, J)` has `B` a `δ`-ring,
`J` a Cartier-divisor ideal, derived `(p, J)`-complete `B`, and `p ∈ J + φ(J)B`;
bounded means `B/J` has bounded p-power torsion, and perfect means `φ` is invertible.
Prove rigidity `JB' = J'` for maps of prisms. For a bounded prism `(B,J)` and a
`(p,J)`-completely flat `δ`-`B`-algebra `D`, construct the prismatic envelope of an ideal
`K ⊇ JD` locally generated by `J` and a `(p,J)`-completely regular sequence relative
to `B`; prove its complete flatness over `B`, completed base change in bounded prisms
and compatibility with flat localization in `D`. Prove the equivalence between
perfect prisms and P1.24's integral perfectoid rings.
For any prism construct its initial perfect prism by Frobenius colimit and
`(p, J)`-completion; this construction includes the bounded-torsion comparison that
makes the completed perfection an ordinary ring ([BS22] §3, Definition 3.2,
Lemmas 3.5, 3.8–3.9, Theorem 3.10 and Proposition 3.13, pp. 28–35).

For an integral perfectoid `R`, prove that `(A_inf(R), ker θ)` is initial among prisms
with a map from `R` to their quotient ([BS22] Lemma 4.8, pp. 38–39). For a derived
p-complete quotient `A = R/K`, construct the initial prism `(Δ_init(A), (d))` with
`A → Δ_init(A)/d`: freely `δ`-adjoin `k/d` for lifts of generators of `K`, kill the
`δ`-ideal generated by d-power torsion, and iterate degree-zero derived `(p,d)`-completion
and torsion removal transfinitely until d is regular. Prove independence of the
presentation by initiality. Perfect this prism and take its quotient to construct
`semiperfectoidPerfectoidization` and its universal map ([BS22] Proposition 7.2 and
Corollary 7.3, pp. 55–56). This construction proves initiality without surjectivity.
*Needs:* P1.16–P1.18 and P1.24; P4.15.1; P0.19's cotangent and lifting API for Lemma 4.8.

**Checks.**

- For a perfect `𝔽_p`-algebra `k`, the corresponding perfect prism is `(W(k), (p))`.
- For integral perfectoid `A`, the initial prism is `(A_inf(A), ker θ)`, and its
  perfect-prism quotient recovers `A` with the identity structural map.
- For `A = 0`, the initial prism, its perfection and its quotient are zero;
  regularity is interpreted in the zero ring as well.

#### P4.15.3 the affine Hodge–Tate input

**Affine prismatic comparison and quasisyntomic covers.**
(*semiperfectoid-affine-hodge-tate*) For a bounded oriented prism `(B, (d))` and a
p-completely smooth `B/d`-algebra `C`, construct `Δ_{C/B}` with Frobenius by the
Čech–Alexander complexes of prismatic envelopes. Prove the affine Hodge–Tate comparison
`H^i(Δ_{C/B} ⊗_B^L B/d) ≅ Ω̂^i_{C/(B/d)}`, using the orientation to trivialize the
twists, and completed base change along maps of bounded oriented prisms. Here
`Ω̂^i_{C/(B/d)} := H^0(Λ_(p)(L∧^i_C L_{C/(B/d)}))` is the module of continuous
p-completed differential forms: use the ordinary cotangent complex of P0.19 and
DerivedModuleAlgebra's `derivedExteriorPower` and `derivedCompletion`, as supplied
by P4.15.1. For p-completely smooth `C`, this completed derived exterior power is
concentrated in degree zero; for `C = (B/d)⟨T⟩`, its degree-one module is `C dT`.
This subsection
owns the characteristic-p polynomial divided-power calculation and the universal
oriented-prism reduction used to prove that comparison ([BS22] §4.3,
Construction 4.18 and Lemmas 4.19–4.21; §5, Corollary 5.5; §6, Proposition 6.2 and
Theorem 6.3, pp. 42–53). No global prismatic-cohomology output is required.

Extend this affine construction by polynomial simplicial resolutions and derived
completion, as in [BS22] Construction 7.6, pp. 56–57. Prove resolution independence,
completed base change and preservation of homotopy colimits. The Hodge–Tate reduction
has an exhaustive conjugate filtration with graded pieces the derived p-completions of
`∧^i L_{C/(B/d)}[-i]` in cohomological indexing, with twists trivialized by `d`.
A quasisyntomic `B/d`-algebra here is p-completely flat over `B/d`, derived p-complete
with bounded p-power torsion, and has p-complete relative cotangent Tor amplitude
`[-1,0]`; a cover is also p-completely faithfully flat.
Construct the covers obtained by adjoining all p-power roots of polynomial generators.
For bounded-torsion `C` with p-completely flat `L_{C/(B/d)}[-1]`, prove that the
Hodge–Tate reduction is discrete and p-completely faithfully flat over `C`, and that
the prismatic complex is discrete with the `δ`-structure and prism of Lemma 7.7.
For quasiregular semiperfectoid `A` (bounded p-power torsion and p-completely flat
`L_{A/ℤ_p}[-1]`), prove that `Δ_{A/A_inf(R)}` and its Hodge–Tate reduction are
discrete, identify the resulting prism with `Δ_init(A)`, and prove that
`A → Δ_init(A)/d` is p-completely faithfully flat. Include the regular-quotient envelope
calculation and the idempotent-retract argument needed for this identification
([BS22] Lemmas 7.7–7.8, Example 7.9 and Proposition 7.10, pp. 57–60).
*Needs:* P4.15.1–P4.15.2; P0.19's full cotangent complex with transitivity and filtered
colimits; P1's perfected polynomial algebras. These prismatic and quasisyntomic inputs
are owned here, rather than assigned to the cotangent supplier.

**Checks.**

- For `(B,d) = (ℤ_p,p)` and `C = 𝔽_p`, the prismatic complex is `ℤ_p` in degree zero.
- For `C = 𝔽_p[T]` over that prism, the Hodge–Tate reduction has `H^0 = C`,
  `H^1 = C dT`, and no higher cohomology.
- For the bounded oriented prism `B = ℤ_p[[u]]`, `d = u-p`, `φ(u) = u^p`,
  with `B/d = ℤ_p`, and `C = ℤ_p⟨T⟩`, the Hodge–Tate reduction has `H^0 = C`,
  `H^1 = Ω̂^1_{C/ℤ_p} = C dT`, and no higher cohomology. For `F = exp(p²T) ∈ C`,
  completion kills `dF − p²F dT`, a nonzero class in ordinary algebraic
  `Ω^1_{C/ℤ_p}`; thus the ordinary differential module cannot replace `Ω̂^1`.
- For `A = 𝔽_p[T^{1/p^∞}]/(T)`, the quasiregular comparison applies although `A`
  is nonreduced: the class of `T^{1/p}` is nonzero with p-th power zero.

#### P4.15.4 André's root-adjoining cover

**Completely faithfully flat perfectoid root cover.** (*semiperfectoid-andre-cover*)
For every integral perfectoid `R`, construct a p-completely faithfully flat map
`R → R'` with `R'` integral perfectoid and every monic polynomial of positive degree
over `R'` having a root in `R'`: for `P : R'[X]`, assume `P.Monic` and `0 < P.natDegree`.
In particular every element of `R'` admits compatible p-power roots.
Supply Proposition 7.11's input: for a bounded oriented prism `(B,(d))` and a
quasisyntomic `B/d`-algebra `C`, construct a prism `(D,dD)` with a p-completely
faithfully flat map `C → D/dD`. The prism map is completely flat, and is completely
faithfully flat if `B/d → C` is p-completely faithfully flat; these properties persist
on perfection when `B` is perfect. Build it using the root-adjoining quasisyntomic cover and the discrete
Hodge–Tate comparison of P4.15.3. Apply this to adjoining roots of all positive-degree
monic polynomials, and iterate with completed transfinite colimits to obtain `R'`
([BS22] Proposition 7.11, p. 60; Theorem 7.14 and its proof, p. 61).
*Needs:* P4.15.1–P4.15.3; P1.24. No Zariski-closed or tower theorem is an input.

**Checks.**

- For an algebraically closed characteristic-p field, the identity is a cover with
  the asserted positive-degree monic-root property.
- The zero ring admits the identity cover, and its unique element has zero roots.
- The monic polynomial `1` has degree zero and no root over any nonzero ring. In particular,
  for `R = 𝔽_p`, faithful flatness forces the cover ring to be nonzero; the root property
  excludes `1`.
- Merely adjoining a single root is insufficient: `𝔽_p[X]/(X^p−1)` has nonzero
  nilpotent `X−1`, so is not integral perfectoid. The prism and perfection steps
  supply the perfectoid target of the cover.

#### P4.15.5 rooted quotients and the surjectivity reduction

**Rooted quotient and descent of the perfectoidization map.**
(*semiperfectoid-rooted-quotient-reduction*) Let `R` be integral perfectoid and
`r : ℕ → R` satisfy `r(n+1)^p = r(n)`. Put `J = (r(n) | n ∈ ℕ)` and
`Q = (R/J)^∧_p`. Prove `isIntegralPerfectoid_completedRootQuotient`: `Q` is integral
perfectoid; also prove `surjective_completedRootQuotientMap` for the canonical
`R → R/J → Q`. Prove reducedness of integral perfectoid rings using the perfect-prism
comparison. Hence every map from `R/(r(0))` to an integral perfectoid ring kills `J`.
Completeness of the target then gives the
unique factorization through `Q`. Thus `Q` is its perfectoidization, with surjective
structural map ([BS22] proof of Theorem 7.4, p. 62).

For a p-completely faithfully flat map `R → R'` of integral perfectoid rings and
a derived p-complete quotient `A` of `R`, let `A' = (A ⊗_R^L R')^∧_p`.
Prove the flat base-change comparison
`(A_perfd ⊗_R^L R')^∧_p ≅ A'_perfd`, with the structural maps fixed. This includes
discreteness of both base changes and integral perfectoidness of the completed tensor
on the left, using the perfect-prism comparison and flat completed base change of
P4.15.2–P4.15.3. Descend surjectivity by P4.15.1. Supply the completed filtered-colimit
reduction to finitely generated defining ideals and the successive-principal-quotient
reduction, carrying derived p-completeness at each step. Use P4.15.4 to root the
principal defining element after flat base change and apply the rooted quotient
calculation. This proves `semiperfectoidPerfectoidization_surjective` from its
construction, without using the Tate-space application ([BS22] proof of Theorem 7.4,
p. 62, with the base-change machinery of Construction 7.6(4)–(5), pp. 56–57).
*Needs:* P4.15.1–P4.15.4; P1.24's integral perfectoid theory.

**Checks.**

- For `r(n) = 0`, `Q = R` and the structural map is the identity.
- For `r(n) = 1`, `Q = 0` and the structural map is the surjection to zero.
- For `R = 𝔽_p[T^{1/p^∞}]` and `r(n) = T^{1/p^n}`, `Q = 𝔽_p`. The ordinary
  quotient `R/(T)` is nonreduced and is not this perfectoid quotient.

#### P4.15.6 the Zariski-closed application

**Zariski closed immersions are strongly Zariski closed.** (*zariski-closed-is-strongly-zariski-closed*)
Let `X = Spa(R, R⁺)` be an affinoid perfectoid space and `I ⊆ R` an ideal. The universal perfectoid
space `Z` over the Zariski closed subset `V(I)` (P4.14) is affinoid
perfectoid, `Z = Spa(S, S⁺)`, with `R → S` surjective, `S⁺` the integral closure of the image of
`R⁺`, and `R⁺ → S⁺` almost surjective; in particular `|Z| → V(I)` is a homeomorphism. Hence every
Zariski closed immersion is strongly Zariski closed, and the two notions coincide.
The direction "Zariski closed implies strongly Zariski closed" is stated for
`f : Y ⟶ ofPair p R`, and `stronglyZariskiClosed_iff_zariskiClosed` retains the equivalence; the
ring-level core, the universal perfectoid quotient of `R` killing `I`, is the Lean
`Perfectoid.exists_surjective_perfectoidQuotient`. The input is
the perfectoidization of semiperfectoid rings, applied to the `ϖ`-adic completion
`A := (R⁺/(I ∩ R⁺))^∧_ϖ`, as in [BS22] Remark 7.5: the target
`semiperfectoidPerfectoidization` of P4.15.2 constructs, for a derived `p`-complete ring `A` that is a
quotient of an integral perfectoid ring, the initial integral perfectoid ring `A_perfd` under `A`.
P4.15.5 proves `semiperfectoidPerfectoidization_surjective`; its kernel consists of the elements killed
by every map from `A` to an integral perfectoid ring. Also prove
`semiperfectoidPerfectoidization_tensor`: for every integral perfectoid `A`-algebra `B`, the
multiplication map `A_perfd ⊗_A B → B` is an isomorphism. Applied to the completed quotient
above, this gives `S = A_perfd[1/ϖ]` and `S⁺` the integral closure of the image of `R⁺` in `S`
([BS22] §7, Corollary 7.3, Theorem 7.4 and Remark 7.5, p. 56; proof of Theorem 7.4, p. 62;
[ECD] §5, Theorem 5.8, p. 25). *Needs:*
*universal-perfectoid-zariski-closed*; *strongly-zariski-closed-immersion*;
*zariski-closed-immersion*; P1.24; P4.15.1–P4.15.5.

A surjection of perfectoid Tate rings is almost surjective on integral elements
(*surjective-perfectoid-map-almost-surjective-on-plus-rings*): for a surjective continuous
homomorphism `ψ : R → S` of perfectoid Tate rings and rings of integral elements `R⁺`, `S⁺` with
`ψ(R⁺) ⊆ S⁺`, the cokernels of `R° → S°` and `R⁺ → S⁺` are killed by every `ϖ^{1/pⁿ}`
([AWS] Lecture 2, Lemma 2.8.5, pp. 70–71). Zariski closed and strongly Zariski closed immersions are stable
under base change (*zariski-closed-base-change*): for `Z → X` Zariski closed defined by `I` and
`X' = Spa(R', R'⁺) → X` any morphism of affinoid perfectoid spaces, `Z ×_X X' → X'` is Zariski
closed defined by `IR'`, with `Z_I ×_X X' ≅ Z_{IR'}`, and strongly Zariski closed when `Z → X` is
([Torsion] §II.2, Lemma II.2.9 (i)–(ii), pp. 15–16).

The structure `SemiperfectoidRing` requires both a derived p-complete ring and an actual
surjective map from an integral perfectoid ring. `DerivedPComplete` uses the equivalent
sequence test: `(x_n) ↦ (x_n − p x_{n+1})` is bijective. This is the telescope criterion for
`RHom(ℤ[1/p], A) = 0` ([Stacks, §15.93, tag 091N](https://stacks.math.columbia.edu/tag/091N)).
`semiperfectoidPerfectoidization_universal`, `_surjective`, `_kernel` and `_tensor`
state universality, the quotient property, the universal vanishing kernel, and the tensor
isomorphism, with the latter fixing `1 ⊗ b ↦ b`.

Checks:

| Definition | Checks and results |
| --- | --- |
| `DerivedPComplete` | `derivedComplete_characteristic_p`: every characteristic-p ring passes; `derivedComplete_padic`: `ℤ_p` passes; `derivedComplete_rational_negative`: `ℚ` fails, since `(p^{-n})` gives a nonzero kernel sequence. |
| `SemiperfectoidRing` | `SemiperfectoidRing_characteristic_p`: a perfect characteristic-p ring; `SemiperfectoidRing_quotient`: any quotient of such a ring; `SemiperfectoidRing_no_rational`: no underlying ring is isomorphic to `ℚ`. |
| `semiperfectoidPerfectoidization` | `semiperfectoid_fixed`: an integral perfectoid input is fixed; `semiperfectoid_zero`: the zero input stays zero; `semiperfectoid_kills_roots`: in characteristic p, an element whose p-power is zero maps to zero. |

`surjective_almostSurjective_plus` quantifies every chosen root of a pseudouniformizer;
`stronglyZariskiClosed_iff_almostSurjective` retains the integral-closure condition as well.
`zariskiClosedSubspace.baseChange` identifies the actual pullback with the universal
subspace for the extended ideal, over the new base.

`completedIntegralQuotient` has the actual carrier
`AdicCompletion (ϖ) (R⁺/(I ∩ R⁺))`; its structure records the hypotheses needed by
semiperfectoid perfectoidization. `completedIntegralQuotientMap` is the canonical quotient
followed by completion. `zariskiClosedPair_from_perfectoidization` identifies
`A_perfd[1/ϖ]` with the universal Zariski quotient, fixes the map on `R⁺`, and identifies
its plus ring with the integral closure of the image. Here `ϖ ∈ R⁺` is a
pseudouniformizer as in [BS22] Remark 7.5, p. 56, chosen moreover with `ϖ^p | p` (possible since
`R` is perfectoid).
Checks `completedIntegralQuotient_zero_ideal`, `completedIntegralQuotient_unit_ideal`
and `completedIntegralQuotient_can_have_nilpotents` recover `R⁺` at the zero ideal,
give the zero ring at the unit ideal, and exhibit a nonreduced completed quotient before
perfectoidization.

Lean: `exists_surjective_perfectoidQuotient` (a surjection `R → S` onto a perfectoid Tate ring
killing `I`, initial among continuous maps to perfectoid Tate rings that kill `I`). The unit-ideal
Check fixes the vanishing convention of the universal quotient.

### P4.16 comparison with the closed immersions of AdicSpaces

For a morphism `f : Y ⟶ ofPair p R` of perfectoid spaces, the reverse implication
(*strongly-zariski-closed-is-zariski-closed*) goes from `IsStronglyZariskiClosed p R f`
to `IsZariskiClosedImmersion p R f` ([ECD], following Definition 5.7, p. 24).

**Comparison with AdicSpaces' closed immersions.** (*adicspaces-closed-immersion-comparison*)
(a) For `X = Spa(R, R⁺)` affinoid perfectoid, a morphism `f : Z → X` of perfectoid spaces is
strongly Zariski closed if and only if it is isomorphic over `X` to the library's closed immersion
of affinoid adic spaces `Spa(Huber.Pair.quotient (R, R⁺) J) → Spa(R, R⁺)` (Tau Ceti
`Huber.Pair.quotient` with `TauCeti.Huber.Pair.quotient_plus`, the closed embedding
`Pair.Hom.isClosedEmbedding_spaComap_quotientHom` with range `V(J)`,
`range_spaComap_quotientHom`) for a closed ideal `J ⊆ R` with `R/J` perfectoid; then
`J = ker(R → S)`.
(b) For arbitrary `X`, the quotient-defined adic notion is AdicSpacesPartII R0.8's
`AdicSpace.IsClosedImmersion`. P4's `IsAdicClosedImmersion` lives on `Perfd` and agrees
with that condition through `YonedaAdicSpace.ofAdicSpace`, as expressed by
`isAdicClosedImmersion_iff_adicCharts`: on compatible adic representatives `g : A → B`,
every point of `B` has a complete sheafy affinoid chart whose entire inverse image
is affinoid, with an open quotient coordinate homomorphism and the integral closure
of the image plus ring. Equivalently, the map is locally on `X` strongly Zariski closed;
such a morphism is a closed immersion in the sense of [ECD]
Definition 5.6 whose restrictions over the affinoid perfectoid opens contained in a chart of the
cover are Zariski closed. With its quantifiers: a map `f : Z → X` of perfectoid spaces is a closed
immersion in the cover-local sense of R0.8 (some affinoid cover of `X` whose preimages are quotient
charts) if and only if there exists an open cover of `X` by affinoid perfectoid `U_i` with every
`Z ×_X U_i → U_i` strongly Zariski closed; then the same holds over every affinoid perfectoid open
contained in a chart of the given cover. The converse direction uses the open mapping theorem
(AdicSpaces 0.6): the surjection `O(U_i) → O(Z ×_X U_i)` is open, so the preimage of `U_i` is a
quotient chart; the forward direction uses P4.12 and P4.15, the image over an affinoid perfectoid
`W` inside a chart `Spa(A, A⁺)` with ideal `J` being `V(J·O(W))`. This is the only global
identification made here. It is not PerfectoidTowers P8.14's notion of Zariski-closed embedding,
which asks for strong Zariski closedness over every open affinoid perfectoid of `X`; the two global
notions are not identified in this roadmap. (c) AdicSpaces' closed
immersion defined by a closed ideal `J` of a perfectoid `R` need not have perfectoid source: for
`R = C⟨T^{1/p^∞}⟩` and `J = (T)`, `R/J` is not reduced, whereas the Zariski closed subspace `V(T)` is
`Spa(C, O_C)` ([AWS] Lecture 2, Example 2.4.9). (d) That every Zariski closed immersion into an
affinoid is of the form (a) is [ECD] Theorem 5.8 (P4.15). *Needs:*
*strongly-zariski-closed-immersion*; P4.15; P4.12; AdicSpacesPartII R0.8
`AdicSpace.IsClosedImmersion`; AdicSpaces 0.6 (the open mapping theorem); Tau Ceti `TauCeti.Huber.Pair.quotient`,
`TauCeti.Huber.Pair.quotient_plus`, `TauCeti.Huber.Pair.Hom.isClosedEmbedding_spaComap_quotientHom`,
`TauCeti.Huber.Pair.Hom.range_spaComap_quotientHom`.

Immersions are stable under base change and composition, and closed immersions are quasicompact
(*immersions-base-change-and-composition*): immersions, open immersions and closed immersions are
stable under composition and under base change along arbitrary morphisms, the image of the base
change of `f` being the preimage of `f(|Y|)`; closed immersions are quasicompact ([ECD] §5,
Corollary 5.5 (i), p. 24). A surjective perfectoid quotient is a Zariski closed immersion
(*strongly-zariski-closed-is-zariski-closed*): for `ψ : R → S` a surjective continuous
homomorphism onto a perfectoid Tate ring, `I := ker ψ` and `S⁺` the integral closure of `ψ(R⁺)`,
the ideal `I` is closed, `(S, S⁺) ≅ Huber.Pair.quotient (R, R⁺) I`, and `Spa(S, S⁺) → X` is a
Zariski closed immersion defined by `I`, isomorphic to `Z_I` over `X`; thus every strongly Zariski
closed map is Zariski closed, the direction that needs the plus-ring clause ([ECD] §5, after
Definition 5.7, p. 24; Tau Ceti `TauCeti.Huber.Pair.Hom.isClosedEmbedding_spaComap_quotientHom`).

`stronglyZariskiClosed_quotient_comparison` identifies a strong quotient with
`Huber.Pair.quotient` by its kernel, including the topology, plus ring and the quotient map.
The global geometric equivalence is `stronglyZariskiClosed_iff_zariskiClosed`.
The composition and base-change declarations are `IsImmersion.comp`, `.baseChange`,
`IsOpenImmersion.comp`, `.baseChange`, `IsClosedImmersion.comp`, `.baseChange`;
`IsClosedImmersion.quasiCompact` gives quasicompactness.

`IsAdicClosedImmersion` expresses the adic cover-local condition of (b) on the actual `Perfd`
morphism, with the same quantifier everywhere: there exists an affinoid cover of the target such
that over each chart of that cover the source is an affinoid quotient, with the quotient topology
and the integral closure of the image as plus ring. The equivalence
`isAdicClosedImmersion_iff_locally_strong` identifies this with the existence of an affinoid
perfectoid cover on whose charts the map is strongly Zariski closed (the statement of (b)); there
is no `isAdicClosedImmersion_iff_strong`: for an affinoid target the direct comparison is (a),
whose source is supplied as an affinoid quotient, and no upgrade from a cover-local immersion into
an affinoid target to a quotient presentation over the whole target is asserted or used (see the
note after the chart condition below).
`IsAdicClosedImmersion.isClosedImmersion` compares the cover-local quotient condition with
P4's topological closed immersion. Checks `adicClosed_identity`, `adicClosed_empty`,
`adicClosed_principal` and `adicClosed_annulus_negative` give the identity, empty source,
a principal perfectoid quotient, and the annulus boundary that fails the quotient
condition. `isAdicClosedImmersion_iff_adicCharts` uses the complete sheafy charts
`AdicSpacesPartII.AdicSpace.Chart` and `AdicSpace.Affinoid.map` to express R0's quotient-chart
condition on adic representatives, again as the existence of a cover by quotient charts; all
representative isomorphisms commute with the morphism being tested.

**No local-to-global upgrade is asserted.** No implication from the cover-local predicate to a global
quotient presentation over an arbitrary affinoid target is used in this roadmap. If a global closed
ideal `J ⊆ R` with image `V(J)` is supplied, P4.15 identifies the immersion with the universal
perfectoid quotient, whose kernel is `J^perf` (equal to `J` only when `J` is already perfectoid-saturated,
as the root-disc example `R/(T)` shows). PerfectoidTowers P8.14 retains its separate every-affinoid
condition (Hansen–Johansson, Definition 5.4 and the caution following it). Any further upgrade of a
cover-local immersion into an affinoid target to a global quotient would need an independent
descent theorem for arbitrary closed ideals — the restricted kernels must be identified with the
extended ideals after closure and perfectoid saturation, the locally given ideals must descend (the
obstruction lives in `H^1(X, J)`, not in the acyclicity of `O` or `O⁺`), and `J = J^perf` must be
shown — and such kernels need not be finitely generated (`ker(K⟨T^{1/p^∞}⟩ → K)` is a nonzero
idempotent ideal of a domain), so AdicSpacesPartIII R3.5's finite-projective glueing does not
apply; this is not deduced here from acyclicity, is not a target, and is not used to satisfy any
downstream contract. [ECD] Definitions 5.6–5.7 and Theorem 5.8, pp. 24–25,
distinguish this condition from P4's broader `IsClosedImmersion`, as does the remark
after [Berkeley] Definition 17.4.2, p. 155. The affine construction uses the pinned
quotient-pair suppliers above.

### P4.17 closed immersions versus Zariski closed immersions

**Closed immersions and Zariski closed immersions are different notions.**
(*closed-immersion-vs-zariski-closed*) (a) Every strongly Zariski closed map is Zariski closed, and
every Zariski closed immersion is a closed immersion (*strongly-zariski-closed-is-zariski-closed*,
*zariski-closed-immersion*). (b) The image of an immersion is stable under generalization, so a
closed subset of `|X|` that is not (the closed point of `Spa(K, K⁺)` for `K⁺` of rank two) is not
the image of any immersion: the Zariski statements do not extend to arbitrary closed subsets. (c) A
closed immersion need not be Zariski closed: for `X = Spa(C⟨T^{1/p^∞}⟩, O_C⟨T^{1/p^∞}⟩)` with `C` a
perfectoid field and `ϖ ∈ C` a pseudouniformizer with compatible roots, the subset
`T := ⋂_n {|ϖ^{1/pⁿ}| ≤ |T|}` (the closed annulus `{|T| = 1}` together with the rank-two point `y`
with `|T(y)| = 1⁻`) is the complement of the open unit disc, hence closed, and `ι : Z_𝒰 → X` for
`𝒰 = {{|ϖ^{1/pⁿ}| ≤ |T|}}` (P4.4) is a closed immersion with image
`T`; but `T` is not Zariski closed, since every `g ∈ C⟨T^{1/p^∞}⟩` vanishing on `T` vanishes at the
Gauss point, hence is `0` (the Gauss norm is the spectral norm), while `T ≠ X`. (d) Equivalently, a
closed immersion `Spa(S, S⁺) → Spa(R, R⁺)` of affinoid perfectoid spaces does not force `R → S` to
be surjective ([Berkeley] Lecture 17, remark after Definition 17.4.2, p. 155). *Needs:*
*intersection-of-rational-subsets*; *perfectoid-immersion*; *zariski-closed-immersion*.

**Checks.**

- `not_isZariskiClosed_annulusClosure` (non-example) — the closed immersion `Z_𝒰 → X` of (c):
  `T = {|T| = 1} ∪ {y}` is closed, `S_𝒰 = C⟨T^{±1/p^∞}⟩` (P4.4), `R → S_𝒰` is the
  (non-surjective) inclusion `C⟨T^{1/p^∞}⟩ ⊆ C⟨T^{±1/p^∞}⟩`, and the only `g ∈ R` with
  `|g| = 0` on `T` is `g = 0`, so `V(I) = T` has no solution `I`.
- `not_range_isImmersion_closedPoint` (non-example) — for `K⁺ ⊊ K°` the closed point `K⁺` of
  `Spa(K, K⁺)` is a closed subset that is not stable under generalization (`K°` generalizes it),
  so no immersion has image `{K⁺}`; the closed immersions into `Spa(K, K⁺)` are `∅` and the
  identity.

The Check `zariskiImmersion_annulus_negative` also tests the morphism predicate
`IsZariskiClosedImmersion` itself on this closed boundary inclusion.

### P4.18 the diagonal is an immersion

**The diagonal of a morphism of perfectoid spaces is an immersion.** (*diagonal-of-perfectoid-map*)
Let `f : Y → X` be any morphism of perfectoid spaces. Then `Δ_f : Y → Y ×_X Y` is an immersion
(P4.10). More precisely: (a) `Δ_f` is a split monomorphism, hence an injection;
(b) for open affinoids `V ⊆ Y` and `U ⊆ X` with `f(V) ⊆ U`, the restriction `V → V ×_U V` is strongly
Zariski closed: the multiplication map from the affinoid ring of `V ×_U V` onto `O_Y(V)` is
surjective and `O⁺_Y(V)` is the integral closure of the image of the plus ring; (c) the open subsets
`V ×_U V` cover an open `W ⊆ |Y ×_X Y|` containing `Δ_f(|Y|)`, and `Δ_f(|Y|)` is closed in `W`, with
`|Δ_f|` a homeomorphism onto its image ([ECD] §5, Proposition 5.9, p. 25). *Needs:*
*injection-of-perfectoid-spaces*; *perfectoid-immersion*; *strongly-zariski-closed-immersion*;
P2.16; Mathlib `CategoryTheory.IsSplitMono`.

`diagonalMultiplication` is the map from the actual P2 tensor pair, fixed by its two
identity composites in `diagonalMultiplication_inclusions`.
`diagonalMultiplication_strong` includes both ring surjectivity and the plus-ring clause.
Checks `diagonalMultiplication_zero`, `diagonalMultiplication_identity` and
`diagonalMultiplication_two_terms` compute a zero target, the tensor over the identity,
and the sum of a left and a right scalar.

### P4.19 separated morphisms

Separatedness is the closedness of the diagonal; this subsection defines it and the next gives its
valuative criterion.

**Separated morphisms of perfectoid spaces.** (*separated-perfectoid-map*) With fibre products as
in P2, a morphism `f : Y → X` of perfectoid spaces is separated if its diagonal `Δ_f : Y → Y ×_X Y`
is a closed immersion (P4.10) ([ECD] §5, Definition 5.10, p. 25); it is
quasiseparated if `Δ_f` is quasicompact (P4.2). In Mathlib's
vocabulary, `IsSeparated = MorphismProperty.diagonal IsClosedImmersion` on `Perfd`, the pattern of
`AlgebraicGeometry.IsSeparated`. Since `Δ_f` is always an immersion (P4.18),
`f` is separated if and only if `Δ_f(|Y|)` is closed. This is the notion that DiamondsAndVStacks
layer D3 (*immersions-separatedness-and-truncatedness*) restricts to on perfectoid spaces downstream.
The adic supplier is AdicSpacesPartII R0.10's `AdicSpace.IsSeparated`, and the comparison is made
only for `f` locally of weakly finite type (AdicSpacesPartII R0.6), the fibre product being that of
P2.16. P4's notion lives on `Perfd` and agrees with R0.10 through `YonedaAdicSpace.ofAdicSpace` when the
adic representative is locally of weakly finite type and its source has stably sheafy
charts. More precisely, `isSeparated_iff_adicDiagonal` identifies `IsSeparated p f`
with closedness of the diagonal of any compatible adic representative `g : A → B`
with a self-pullback, using the representative isomorphisms and commutative square
of P4.2. R0 includes the two additional hypotheses in its predicate; P4 imposes
neither on an arbitrary perfectoid morphism ([ECD] Definition 5.10 and Proposition
5.11, pp. 25–26). In `PerfectoidSpace`, define `IsSeparated`;
prove `isSeparated_iff_isClosed_range_diagonal`, `IsSeparated.quasiSeparated`,
`IsSeparated.of_affinoid`, `IsSeparated.of_isInjection`, `IsSeparated.comp`,
`IsSeparated.baseChange`. The identity `isSeparated_eq_diagonal` unfolds the
definition into Mathlib’s diagonal operation ([ECD] Definition 5.10, p. 25). *Needs:* *perfectoid-immersion*;
*quasicompact-and-quasiseparated-maps*; Mathlib `CategoryTheory.Limits.pullback.diagonal`,
`CategoryTheory.MorphismProperty.diagonal`, `AlgebraicGeometry.IsSeparated`; P2.16.

Analytic separatedness, a diagonal that is a Zariski-closed embedding, is a different condition,
owned by [PerfectoidTowers P8.15](../PerfectoidTowers/README.md#p815-analytically-separated-perfectoid-spaces);
it and its comparison with this subsection's notion belong to P8.15 and are not used here. That
comparison has two distinct conclusions there. `PerfectoidSpace.IsAnalyticallySeparated.isSeparated`
takes a morphism `f : X ⟶ Y` with both `X` and `Y` in `Perfd p` and concludes
this subsection's `IsSeparated p f`; in particular it applies over a perfectoid
field. `PerfectoidSpace.IsAnalyticallySeparated.closed_diagonal` takes
`f : X.obj ⟶ B`, analytic separation of `f`, and any chosen
`AnalyticSelfProduct p X B f`, and concludes that its diagonal is a closed
immersion. This is the representable closed-diagonal clause for v-sheaf
separatedness over an analytic base, including a nonperfectoid field
([ECD] Definition 10.7(ii)–(iii), p. 51). Its interpretation as separation of
`X^◇ → Spd(K, K⁺)` uses PerfectoidTowers' diamond comparison; the
perfectoid-base conclusion uses [ECD] Definition 5.10, p. 25.

**Checks.**

- `IsSeparated_affinoid` (computation) — every morphism `Spa(S, S⁺) → Spa(R, R⁺)` of affinoid
  perfectoid spaces is separated: its diagonal is strongly Zariski closed (P4.18 (b)), hence a
  Zariski closed immersion ([ECD] §7, after Lemma 7.19, p. 37).
- `IsSeparated.of_isInjection` (degenerate) — injections, in particular identities and open
  immersions, are separated: their diagonal is an isomorphism; so is the map from the empty space
  (`Δ : ∅ → ∅`).
- `isSeparated_sum_iff` (computation) — for `Y = Y₁ ⊔ Y₂` over `X`,
  `Y ×_X Y = ⊔_{i,j} Y_i ×_X Y_j` and `Δ_Y(|Y|) = Δ_{Y₁}(|Y₁|) ⊔ Δ_{Y₂}(|Y₂|)` lies in the two
  components `Y_i ×_X Y_i`, so `Y → X` is separated if and only if both `Y_i → X` are; two
  copies of `Spa(C, O_C)` over `Spa(C, O_C)` are separated although `Y → X` is not an injection.
- `not_isSeparated_gluedDisc` (non-example) — glue two copies of the perfectoid closed disc over a
  perfectoid field `C` along the rational subset `{|ϖ| ≤ |T|}`; the result `Y` is quasiseparated
  over `Spa(C, O_C)` but not separated: for the rank-two point `y` of the disc with
  `|T(y)| = |ϖ|⁻` (in the closure of the overlap, not in it), the point `Δ_D(y)` of the chart
  `D₁ ×_C D₂ = D ×_C D` lies in the closure of `Δ_Y(|Y|)` but not in `Δ_Y(|Y|)`.

Checks `IsSeparated_identity`, `IsSeparated_empty` and `IsSeparated_affinoid` test the
three basic cases. `sumSpaceEquiv` identifies the topological coproduct through its two
inclusions; `sumPullbackEquiv` orders the four pullback components as
`(1,1), (1,2), (2,1), (2,2)`. `sum_diagonal_four_components` places the diagonal in exactly
`(1,1)` and `(2,2)`, with the separate diagonals inside those components.
`sum_empty_left` computes the coproduct with the empty space;
`sum_disjoint_components` gives disjoint images of the two inclusions;
`sum_two_fields` gives a separated map which is not an injection.
The theorem `isSeparated_sum_iff` applies to arbitrary two maps over the same base.

`sumPullbackEquiv_coordinates` fixes both projections on each of the four components,
so the order is part of the comparison. `glueSelf X U` is the categorical pushout of
the two inclusions of the open subspace `U` in `X`; `glueSelf.fold` is induced by the
two identity maps. `glueSelf.openCharts` and `glueSelf.overlap` specify the two open
charts, their cover, and their intersection. Checks `glueSelf_whole`, `glueSelf_empty`
and `glueSelf_two_copies` compute gluing along the whole space, along the empty
space, and the two distinct copies of a point outside the overlap.
`discAnnulusOpen` is the rational overlap above, with Checks `discAnnulusOpen_gauss`,
`discAnnulusOpen_not_closed` and `discAnnulusOpen_compact`. The Check
`not_isSeparated_gluedDisc` applies quasiseparatedness and failure of separatedness to
the actual glued space and its structure map to the base field.

### P4.20 the valuative criterion for separatedness

Separatedness is detected on rank-one points of affinoid perfectoid fields.

**Valuative criterion for separatedness of perfectoid spaces.**
(*valuative-criterion-separatedness*) Let `f : Y → X` be a morphism of perfectoid spaces. The
following are equivalent: (i) `f` is separated; (ii) `|Δ_f| : |Y| → |Y ×_X Y|` is a closed
immersion of topological spaces; (iii) `f` is quasiseparated and for every affinoid perfectoid
field `(K, K⁺)` and every commutative square formed by `a : Spa(K, O_K) → Y` and
`b : Spa(K, K⁺) → X` with `f ∘ a = b ∘ j` (`j : Spa(K, O_K) → Spa(K, K⁺)` the inclusion of the
rank-one point) there is at most one `c : Spa(K, K⁺) → Y` with `c ∘ j = a` and `f ∘ c = b`
([ECD] §5, Proposition 5.11 (i)–(iii), p. 25). *Needs:* *separated-perfectoid-map*;
*diagonal-of-perfectoid-map*; *immersions-base-change-and-composition*.

Maps from `Spa(R, R⁺)` into a separated space are determined on `Spa(R, R°)`
(*separated-unique-extension-from-rank-one-locus*): for `f : Y → X` separated, `(R, R⁺)` any
perfectoid Tate ring with a ring of integral elements, and `a, b : Spa(R, R⁺) → Y` with
`f ∘ a = f ∘ b` and `a ∘ j = b ∘ j` for `j : Spa(R, R°) → Spa(R, R⁺)`, one has `a = b`; for
`(R, R⁺) = (K, K⁺)` this is the uniqueness in (iii) ([ECD] §10, Proposition 10.10, p. 52).

The declarations `valuativeCriterion_separated` and `separated_unique_extension` use the
actual rank-one inclusion and the identity-ring map from the smaller to the maximal plus
ring. The first retains quasiseparatedness; the second quantifies arbitrary perfectoid
Tate pairs. `valuativeCriterion_adicDiagonal` transports the first equivalence to
closedness of the actual diagonal of a compatible adic representative `g : A → B`
with a self-pullback; its right side is P4 quasiseparatedness and uniqueness for all
`AffinoidField p` tests. AdicSpacesPartII R0.10's `AdicSpace.IsSeparated.iff_valuativeCriterion`
is the corresponding uniqueness-of-centres criterion on adic spaces (Huber 1.3.7):
it assumes analytic source and target, stably sheafy source charts, local weak finite
type, and a quasicompact diagonal. On perfectoid representatives under those hypotheses,
`isSeparated_iff_adicDiagonal` and `valuativeCriterion_adicDiagonal` identify the two
criteria with the same closed diagonal (Huber 1996, Proposition 1.3.7, pp. 52–53).
The P4 field criterion itself retains the
generality of [ECD] Proposition 5.11, pp. 25–26.

### P4.21 tilting

**Tilting of injections, immersions, separatedness and strongly Zariski closed immersions.**
(*zariski-closed-tilting*) Let `X` be a perfectoid space with tilt `X♭` and `f : Y → X` a morphism
with tilt `f♭ : Y♭ → X♭` (P2.11; `|Y| = |Y♭|`, `|X| = |X♭|`). (a) `f` is an injection, an
immersion, an open or closed immersion, quasiseparated or separated if and only if `f♭` is. (b) For
`X = Spa(R, R⁺)` and `Z = Spa(S, S⁺)` affinoid: `f : Z → X` is strongly Zariski closed if and only
if `f♭` is ([Torsion] §II.2, Lemma II.2.7, p. 15). (c) Formulas for a strongly Zariski closed `f`
with `I := ker(R → S)`, choose `ϖ♭` with sharp `ϖ`, compatible roots,
and `ϖ^p | p` in `R⁺` (and use their images in `S`): `I♭ := ker(R♭ → S♭)` is a closed perfect ideal and `I` is the closure of the
ideal generated by `♯(I♭)`; `S♭⁺ = lim_{x↦x^p} S⁺` and `S♭⁺/ϖ♭ ≅ S⁺/ϖ` compatibly with
`R♭⁺/ϖ♭ ≅ R⁺/ϖ`; `S⁺ = {s ∈ S : |s(z)| ≤ 1 for z ∈ |Z|}`; and `|Z| = V(I) = V(I♭)` inside
`|X| = |X♭|`. *Needs:* *injection-of-perfectoid-spaces*; *perfectoid-immersion*;
*separated-perfectoid-map*; *strongly-zariski-closed-immersion*; P2.11.

Surjectivity and closed ideals with perfectoid quotient are detected on tilts
(*surjections-and-closed-ideals-under-tilting*): for a continuous homomorphism `f : A → B` of complete Hausdorff Tate
rings with `A` perfectoid, if `B` is uniform and `f` surjective then `B` is perfectoid; if `B` is
perfectoid then `f` is surjective iff `f♭` is; and `I ↦ I♭ := ker(A♭ → (A/I)♭)` is a bijection
between closed ideals with `A/I` perfectoid and closed perfect ideals of `A♭`, with inverse `J ↦`
the closure of the ideal generated by `♯(J)` ([AWS] Lecture 2, Theorem 2.4.4 and Corollaries 2.4.5–2.4.6, p. 58). Strong Zariski
closedness is characterised by almost surjectivity on plus rings
(*strongly-zariski-closed-criterion*): for `f : Spa(S, S⁺) → Spa(R, R⁺)` a morphism of affinoid
perfectoid spaces and `T⁺` the integral closure of the image of `R⁺` in `S`, `f` is strongly Zariski
closed iff `R⁺ → S⁺` (equivalently `R⁺/ϖ → S⁺/ϖ`) is almost surjective and `S⁺ = T⁺` (almost
surjectivity of `R⁺ → S⁺` already forces `R → S` surjective, `S = S⁺[1/ϖ]`, and the converse
is *surjective-perfectoid-map-almost-surjective-on-plus-rings*). Almost surjectivity without the
clause `S⁺ = T⁺` is strictly weaker: `Spa(R, R°) → Spa(R, R⁺)` with `R⁺ ⊊ R°` has `R → R` the
identity and `R⁺ → R°` almost surjective but `R° ≠ T⁺ = R⁺`, and it is not a closed immersion
(P4.13). If, however, `|Spa(S, S⁺)|` is in addition closed in `|X|`, the clause is automatic:
`Spa(S, S⁺) ⊆ Spa(S, T⁺) = V(I)` is closed, contains every rank-one point (`S⁺ ⊆ S°`), and the
rank-one points are dense in `Spa(S, T⁺)`, so `Spa(S, S⁺) = Spa(S, T⁺)` and
`S⁺ = {s : |s| ≤ 1 on Spa(S, S⁺)} = T⁺`. This is why [ECD] (after Definition 5.7) records, via
[KL15] Proposition 3.6.9 (c), that its definition agrees with [Torsion] Definition II.2.6, where
almost surjectivity is asked for a closed subset of `|X|` presented as `Spa(S, S⁺)`
([Torsion] §II.2, Definition II.2.6, p. 15; [ECD] §5, after Definition 5.7, p. 24).

The ideal correspondence is `tiltIdealEquiv`. `PerfectoidQuotientIdeal` consists of
closed ideals presented as kernels of continuous surjections onto perfectoid pairs;
the Tate open-mapping theorem gives the quotient topology. `ClosedPerfectTiltIdeal`
consists of closed ideals stable under taking pth roots in the perfect tilt.
`tiltIdealEquiv_kernel` fixes the forward map on a quotient presentation, and
`tiltIdealEquiv_symm` fixes the inverse as the closed ideal generated by sharp images.
`surjective_iff_tilt` and `stronglyZariskiClosed_tilt_iff` give the map predicates.
The complete uniform-target assertion is P2's `isPerfectoid_of_denseRange` (P2.14),
applied to a surjection. The integral quotient isomorphism is P1's
`Perfectoid.tilt.sharpQuotEquiv`, with its `_apply` characteristic equation;
its `sharp_quotient_two_terms` Check supplies both addition and multiplication conventions.

Checks `perfectoidIdeal_zero`, `perfectoidIdeal_unit`, `perfectoidIdeal_reduced_quotient`
test the zero and unit ideals and exclude a nonreduced quotient.
`tiltIdeal_zero`, `tiltIdeal_unit`, `tiltIdeal_characteristic_p` compute the correspondence
at the endpoints and under the characteristic-p untilt identification.
`tilt_surjection_identity`, `tilt_kernel_zero_target`, `tilt_kernel_two_roots` compute the
identity kernel, the whole kernel for a zero target, and the sharp-generated ideal's
closure under a two-term sum, together with sharp's multiplication equation.

### Examples

The worked examples of this layer are the checks above: `Spa(K, K°)` as a single point and the
two-point space `Spa(K, K⁺)` for `K⁺` of rank two; the open unit disc, quasiseparated but not
quasicompact over the perfectoid closed disc; the quadratic extension `Spa(L, O_L) → Spa(K, O_K)`,
bijective on points but not an injection; the closure of the annulus `{|T| = 1}` as a filtered
intersection of rational subsets and as the image of a closed immersion that is not Zariski
closed; the Gauss point `η` with `K(η)⁺ = K(η)°` and `i_η` an injection that is not an immersion;
the origin `Spa(C, O_C) → Spa(C⟨T^{1/p^∞}⟩, O_C⟨T^{1/p^∞}⟩)` as a closed, Zariski closed and
strongly Zariski closed immersion with kernel the closure of `(T^{1/pⁿ})` and not `(T)`;
`V((T − 1)) ≅ ℤ_p` with `S_I = C⁰(ℤ_p, ℂ_p)`; `Spa(R, R°) → Spa(R, R⁺)` as almost surjective on
plus rings but not strongly Zariski closed; and the glued disc, quasiseparated but not separated.

The unit-ideal Check follows from preservation of `1` by ring homomorphisms; the Lean targets
state the rooted-quotient construction and the universal surjective perfectoid quotient.

### Dependencies

Layer P1 (P1.3, P1.10, P1.20, P1.24, P1.28); layer P2 (P2.3, P2.5, P2.10, P2.11, P2.16); layer P3
(P3.7); P0.2 for the almost setup; P4.4 and the targets built on it use the completed direct limit
of P5.1–P5.3, which depends only on P1 and P2. Other roadmaps, by subsection: P4.2 uses
AdicSpacesPartII R0.6 (`AdicSpace.QuasiCompact`); P4.5 uses R0.5 (Banach tensors over a field);
P4.16 uses R0.8 (`AdicSpace.IsClosedImmersion`) and AdicSpaces 0.6 (the open mapping theorem); P4.19
uses R0.6 and R0.10 (`AdicSpace.IsSeparated`); P4.20 uses R0.10
(`AdicSpace.IsSeparated.iff_valuativeCriterion`); the carrier comparisons of P4.2, P4.11, P4.16 and
P4.19 use AdicEtaleGeometry A1.16 (`YonedaAdicSpace.ofAdicSpace` and its point and open-immersion
comparisons). P4.15.1 consumes DerivedModuleAlgebra §§0.1–0.4, 0.14 and 1.1–1.9
under the imported APIs listed there; P4.15.2–P4.15.5 use its finite-ideal descent and
transfinite adapters. That generic supplier prefix uses no P4.15, P6 or tower target.
Tau Ceti `TauCeti.Huber.Pair`, `TauCeti.Huber.Pair.quotient`,
`TauCeti.Huber.Pair.quotient_plus`, `TauCeti.Huber.Pair.Hom.spaComap`,
`TauCeti.Huber.Pair.Hom.range_spaComap_quotientHom`,
`TauCeti.Huber.Pair.Hom.isClosedEmbedding_spaComap_quotientHom`,
`TauCeti.ValuationSpectrum.isClosedEmbedding_spaComap_quotientMk`,
`TauCeti.PreAdicSpace.IsOpenImmersion`, `TauCeti.Huber.IsTateRing.isQuotientMap`,
`TauCeti.ValuationSpectrum.rationalSubset`; AdicSpacesPartII R0.6, R0.8 and R0.10's
`AdicSpace.QuasiCompact`, `.IsClosedImmersion`, `.IsSeparated` and
`.IsSeparated.iff_valuativeCriterion`; AdicEtaleGeometry A1.16's
`YonedaAdicSpace.ofAdicSpace` and `.ofAdicSpace_space` (*adic-spaces-in-yoneda-adic-spaces*);
Mathlib `CompactSpace`, `QuasiSeparatedSpace`,
`CategoryTheory.Mono`, `CategoryTheory.IsSplitMono`, `CategoryTheory.Limits.pullback.diagonal`,
`Topology.IsEmbedding`, `IsLocallyClosed`, `AlgebraicGeometry.IsSeparated`.

## Layer P5: cofiltered limits and finite-stage étale descent

The layer builds the completed direct limit `(R, R⁺)` of a cofiltered system of perfectoid Tate
pairs — the `ϖ`-adic completion of the colimit of the plus rings, with `R = R⁺[1/ϖ]` — proves that
it is again a perfectoid Tate pair, and shows that `Spa(R, R⁺)` is the limit of the affinoid
perfectoid spaces `Spa(R_i, R_i⁺)` in the category of perfectoid spaces, with underlying space the
inverse limit of the `|X_i|` and with every rational subset, quasicompact open and finite cover by quasicompact opens
coming from a finite stage. It then shows that no completion is needed when the index category is
`ω₁`-cofiltered, that every affinoid perfectoid space is an `ω₁`-cofiltered limit of topologically
countably generated ones, and that cofiltered limits respect cutoff cardinals. Finally it constructs
the filtered 2-colimit of a pseudofunctor of categories and proves finite-stage descent: finite
étale, quasicompact quasiseparated étale, and quasicompact separated étale objects over the limit
form the 2-colimit of those over the stages, with surjectivity, isomorphism and separatedness
recognised at a finite stage. No perfectoid base field is assumed anywhere.

### P5.1 the completed direct limit of a cofiltered system of perfectoid Tate pairs

The ring construction in P5.1–P5.3 depends on P1 and P2. The uses of P4 below concern
examples and the étale descent arguments in P5.10–P5.12.
Standing setting of the whole layer: `p` is a fixed prime; `I` is a small cofiltered category;
`X_i = Spa(R_i, R_i⁺)`, `i ∈ I`, is a cofiltered diagram of affinoid perfectoid spaces, given at ring
level by a functor `Iᵒᵖ → CAff` into complete Huber pairs (AdicEtaleGeometry A1.16),
`i ↦ (R_i, R_i⁺)`, with each `R_i` a perfectoid Tate ring in the sense of [ECD] Def. 3.1 (P1.1;
characteristic `0`, `p` or mixed, no perfectoid base field) and `R_i⁺ ⊆ R_i°` open and integrally
closed; for a morphism `a : j → i` of `I` the transition map `φ_a : R_i → R_j` is continuous with
`φ_a(R_i⁺) ⊆ R_j⁺` (Tau Ceti `Huber.Pair.Hom`). No injectivity, flatness or density of the
transition maps is assumed.

Prove first that every perfectoid Tate pair
has a pseudouniformiser in its plus ring with compatible roots and `ϖ^p | p` in that plus ring.
Using preservation of units, topological nilpotence and divisibility under morphisms of pairs,
the system has a compatible pseudouniformiser on a tail
(*compatible-pseudouniformizer*): for every object `i₀` of `I` there is `ϖ ∈ R_{i₀}⁺` such that
(a) `ϖ` is a pseudouniformiser of `R_{i₀}` (a topologically nilpotent unit, Tau Ceti
`Huber.IsPseudoUniformizer`), `ϖ^p` divides `p` in `R_{i₀}⁺`, and `ϖ` has a compatible system of
`p`-power roots `ϖ^{1/p^n} ∈ R_{i₀}⁺`, `(ϖ^{1/p^{n+1}})^p = ϖ^{1/p^n}`; (b) for every morphism
`a : j → i₀` of `I`, `φ_a(ϖ)` is a pseudouniformiser of `R_j`, `φ_a(ϖ)^p` divides `p` in `R_j⁺`,
and the `φ_a(ϖ^{1/p^n})` are compatible `p`-power roots of `φ_a(ϖ)` in `R_j⁺`. Such a pair
`(i₀, ϖ)` is a compatible pseudouniformiser of the system, with compatibility understood on that tail. The comma category `I/i₀` (Mathlib
`CategoryTheory.Over i₀`) is initial in `I`, so every limit over `I` may be computed on the tail
`I/i₀`; for parallel `a, b : j ⇉ i₀` the elements `φ_a(ϖ)`, `φ_b(ϖ)` need not agree in `R_j` but
have the same image in `colim_i R_i⁺`, the only place where compatibility is used ([ECD] §6,
Prop. 6.4, statement, p. 27; [ECD] §3, Lem. 3.10, p. 16; [ECD] §3, Ex. 3.22(ii), p. 19). For
`R_{i₀} = ℂ_p⟨T^{1/p^∞}⟩` the element `ϖ = p^{1/p}` with roots `p^{1/p^{n+1}}` works and its image
under `T^{1/p^m} ↦ p^{1/p^{m+1}}T^{1/p^m}` is again such a pseudouniformiser; in characteristic `p`
the condition `ϖ^p | p = 0` is empty and `ϖ = t` with roots `t^{1/p^n}` works in `𝔽_p((t^{1/p^∞}))^`
([ECD] §3, Prop. 3.5). *Needs:* P1.1, P1.3; Tau Ceti
`TauCeti.Huber.IsRingOfIntegralElements.mem_of_isTopologicallyNilpotent`; Mathlib
`CategoryTheory.IsCofiltered`, `CategoryTheory.Over.initial_forget`.

**The completed direct limit `(R, R⁺)` of a filtered system of perfectoid Tate pairs.**
(*completed-colimit-of-perfectoid-pairs*) In the standing setting fix a compatible pseudouniformiser
`(i₀, ϖ)`. Put `A := colim_{i ∈ Iᵒᵖ} R_i⁺`, the filtered colimit of rings (Mathlib
`CommRingCat.FilteredColimits.colimit`), and write `ϖ` for its image in `A`. Define
`R⁺ := Â = lim_n A/ϖ^nA`, the `ϖ`-adic completion of `A`, and `R := R⁺[1/ϖ]`, topologised so that
`R⁺` is an open subring with its `ϖ`-adic topology; thus `(R⁺, ϖR⁺)` is a pair of definition and `R`
is a complete Tate ring with pseudouniformiser `ϖ`. For `i` in the tail `I/i₀` the canonical map
`ι_i : R_i = R_i⁺[1/ϖ] → R` is continuous with `ι_i(R_i⁺) ⊆ R⁺` and `ι_j ∘ φ_a = ι_i`; `(R, R⁺)` with
the `ι_i` is the completed direct limit of the system. The completion is the `ϖ`-adic completion of
the uncompleted colimit of the plus rings, never a completion of `colim R_i` for another topology;
`A` need not be `ϖ`-adically separated, so the `ι_i` need not be injective; `R⁺` is integrally
closed in `R` and `(R, R⁺)` is a perfectoid Tate pair (P5.2); the
construction is independent of `(i₀, ϖ)` and of the tail (see *limit-independence-of-tail-and-choice*).
Universal property: for every complete Tate pair `(S, S⁺)` with `S⁺` bounded in `S` (for instance
`S` uniform; the plus ring is then `ϖ`-adically complete — the property is not claimed for arbitrary
complete Huber pairs) and every compatible family of morphisms of pairs
`ψ_i : (R_i, R_i⁺) → (S, S⁺)`, `i ∈ I/i₀`, there is a unique morphism of pairs
`ψ : (R, R⁺) → (S, S⁺)` with `ψ ∘ ι_i = ψ_i`. In `Perfectoid`, define `completedColimit` (the
topological ring `R = R⁺[1/ϖ]` of a functor `D : Iᵒᵖ ⥤ CAff` with perfectoid Tate values and a
compatible pseudouniformiser), `completedColimitPlus` (`R⁺`, the Mathlib `AdicCompletion` of the
`CommRingCat` filtered colimit, as an open subring of `R`), `completedColimitPair` (`(R, R⁺)` as a
Tau Ceti `Huber.Pair`, using that `R⁺` is open, integrally closed and power-bounded) and
`completedColimit.ι` (the morphism of Huber pairs `ι_i` for `i` in the tail); prove
`completedColimit.hasBasis_nhds_zero` (powers of `ϖ` times `R⁺` form a neighbourhood
basis). `completedColimit.instIsTateRing` exposes the Tate field supplied by
`completedFilteredColimit.isTateRing`; completeness and Hausdorffness are P1 instances. Prove `completedColimit.ι_comp` (`ι_j ∘ φ_a = ι_i` for every
`a : j → i` of the tail), `completedColimit.lift` (the induced morphism `(R, R⁺) → (S, S⁺)` for
`S⁺` bounded), `completedColimit.lift_comp_ι` (`lift ψ ∘ ι_i = ψ_i`), `completedColimit.hom_ext`
(two morphisms into a complete Hausdorff pair agreeing after every `ι_i` are equal),
`completedColimit.plusQuotientEquiv` (`R⁺/ϖ^nR⁺ ≅ colim_i R_i⁺/ϖ^nR_i⁺` for `n ≥ 1`, compatibly
with the `ι_i`), `completedColimit.denseRange_ι` (`⋃ ι_i(R_i⁺)` is dense in `R⁺` and `⋃ ι_i(R_i)`
in `R`), and
`completedColimit.equivOfInitial` (restricting along an initial functor `I' → I` with `I'` small cofiltered, or changing
`(i₀, ϖ)`, gives a canonically isomorphic pair) and
`completedColimit.colimitEquiv_of_isCardinalFiltered` (if `Iᵒᵖ` is `ℵ₁`-filtered then
`colim R_i⁺ ≅ R⁺` and `colim R_i ≅ R`, P5.5) ([ECD] §6, Prop. 6.4, statement, p. 27; [Sch12] §6,
proof of Lem. 6.13(i), p. 37 (characteristic-`p` field case); [KL15] §2.6, Def. 2.6.1, p. 53; [Bhatt] §6.2, Rem. 6.2.9, printed
p. 48). *Needs:* *compatible-pseudouniformizer*; P1.1; AdicEtaleGeometry A1.16 (`CAff`); Mathlib
`CommRingCat.FilteredColimits.colimitCoconeIsColimit`, `AdicCompletion`,
`AdicCompletion.isAdicComplete`; Tau Ceti `TauCeti.Huber.Pair`, `TauCeti.Huber.Pair.Hom`,
`TauCeti.Huber.IsTateRing`, `TauCeti.Huber.PairOfDefinition`,
`TauCeti.Huber.IsPseudoUniformizer.hasBasis_nhds_zero`.

**Checks.**

- `completedColimit_test_initial` (degenerate) — if `I` has an initial object `i_*` (so the diagram
  of rings has a terminal index), `ι_{i_*} : (R_{i_*}, R_{i_*}⁺) → (R, R⁺)` is an isomorphism of
  Huber pairs.
- `completedColimit_test_const` (degenerate) — for the constant system `R_i = R₀`, `R_i⁺ = R₀⁺`
  with identity transition maps (`Iᵒᵖ` filtered, hence connected), `colim R_i⁺ = R₀⁺` is already
  `ϖ`-adically complete and every `ι_i` is the identity of `(R₀, R₀⁺)`.
- `completedColimit_test_profinite` (computation) — for `S = lim_n {0,1}^n` and `R_n = K^{S_n}`,
  `R_n⁺ = O_K^{S_n}` over a perfectoid field `K`: `R⁺ = C^0(S, O_K)` and `R = C^0(S, K)`; the function
  `s ↦ Σ_k s_k ϖ^k` lies in `R⁺` but in no `ι_n(R_n⁺)`.
  The Check evaluates the first-coordinate idempotent at the all-zero and all-one
  sequences as `0` and `1`; it is neither zero nor one. `binaryFunctionEquiv`
  and `binaryFunctionEquiv_ι` give the general comparison; the non-cylinder
  continuous functions are tested by `completedColimit_test_uncompleted`
  ([ECD] Proposition 6.5, p. 29).
- `completedColimit_test_plusQuotient` (characterisation) — in the profinite example
  `R⁺/ϖ^n = C^0(S, O_K/ϖ^n)` equals the ring of locally constant functions `S → O_K/ϖ^n`, which is
  `colim_n (O_K/ϖ^n)^{S_n}`.
- `completedColimit_test_uncompleted` (non-example) — in the profinite example the uncompleted
  colimit `colim R_n = LC(S, K)`, with the `ϖ`-adic topology on `LC(S, O_K)`, is not complete, so it
  is not the ring of functions of the limit.
- `completedColimit_test_compatibility_sch12` (compatibility) — for a system of perfectoid affinoid
  `K`-algebras over a characteristic-`p` perfectoid field `K`, with each `R_i⁺` a
  `K°`-algebra, `(R, R⁺)` uses the completion construction of [Sch12] Lem. 6.13. For a system with a common choice of `ϖ`, `R_i⁺` as norm unit balls and
  submetric transition maps, the infimum seminorm induces the same `ϖ`-adic topology.
  [KL15] Def. 2.6.1 uses this seminorm construction for affinoid systems over `ℤ((z))`; it
  does not itself state the comparison for arbitrary norms on perfectoid algebras.
- `completedColimit_test_not_injective` (non-example) — for `R_n = K⟨T^{1/p^∞}⟩` (all `n`) with
  `R_n⁺ = R_n°` and transition maps `T^{1/p^m} ↦ ϖ^{1/p^m}T^{1/p^m}`, the nonzero element
  `T ∈ R_0⁺` is divisible by every `ϖ^n` in `colim R_n⁺`, so `ι_0(T) = 0` in `R⁺ = O_K`:
  the `ι_i` need not be injective. The completed pair is `(K, K°)`, represented by
  `(AffinoidField.ofRankOne p K).toPairCat p`, since the closure of the integral images is
  `K°`, independently of the specified `K⁺` ([ECD] Proposition 6.4, p. 27).
- `completedColimit_proper_plus` (non-example) — if `K⁺ ⊊ K°`, this completed disc pair is
  not isomorphic to `(K, K⁺)`: every topological field automorphism preserves `K°`, whereas
  a pair isomorphism must carry the plus ring onto the plus ring. This includes the
  two-point case of `AffinoidField.card_spa_eq_two`, with residue field `𝔽_p(t)^perf` and
  `K⁺` the inverse image of its `t`-adic valuation ring `𝔽_p[t]^perf_(t)`.

The completion step rests on a ring lemma (*completion-preserves-integral-closedness*): let `A` be a
commutative ring and `ϖ ∈ A` a nonzerodivisor such that `A` is integrally closed in `A[1/ϖ]` (Mathlib
`IsIntegrallyClosedIn A (Localization.Away ϖ)`), and `Â := lim_n A/ϖ^nA` its `ϖ`-adic completion
(Mathlib `AdicCompletion (Ideal.span {ϖ}) A`); no noetherian or separatedness hypothesis on `A`,
since filtered colimits of separated rings need not be separated. Use Mathlib `AdicCompletion.isAdicComplete` for completeness and separatedness and
`AdicCompletion.pow_smul_top_eq_ker_eval` with `AdicCompletion.evalₐ` for the quotient
comparison `A/ϖ^nA ≅ Â/ϖ^nÂ`, `n ≥ 1` ([Stacks] Tag 05GG, Lem. 10.96.3). Prove
(a) `ϖ` is a nonzerodivisor on `Â`; (b) the kernel of `A → Â` is `J := ⋂_n ϖ^nA`, which
satisfies `J = ϖJ`, and `A/J` is again `ϖ`-torsion free and integrally closed in `(A/J)[1/ϖ]`;
(c) `Â` is integrally closed in `Â[1/ϖ]` (unlike Tau Ceti’s topological closure theorem, this identifies the algebraic `AdicCompletion`
and its localization without a given ambient topology) ([Bhatt] §5.1, Lem. 5.1.2, printed p. 34); (d) if `A` is
`p`-root closed in `A[1/ϖ]` (`g ∈ A[1/ϖ]` and `g^p ∈ A` imply `g ∈ A`), so is `Â` in `Â[1/ϖ]`
([Bhatt] §5.1, Lem. 5.1.1 and its proof, printed p. 33); (e) for the topology on `Â[1/ϖ]` in which
`Â` is open with its `ϖ`-adic topology, the image of `A[1/ϖ]` is dense. For
`A = ℤ_p[p^{1/p^∞}]` and `ϖ = p`, `Â` is the valuation ring of the completion of `ℚ_p(p^{1/p^∞})`;
the hypothesis of (c) is needed, since for `p` odd `A = ℤ_p[pζ_p] ⊆ ℚ_p(ζ_p)` is `p`-adically
complete (a finite `ℤ_p`-module) and `ζ_p ∈ A[1/p]` is integral over `A` but not in `A`
(`|ζ_p − a| ≥ p^{−1/(p−1)} > |p|` for every `a ∈ ℤ_p`); for `p = 2` one has `ζ_2 = −1 ∈ ℤ_2`, and
`A = ℤ_2[2i] ⊆ ℚ_2(i)` with `i ∉ A` (`|i − a| ≥ |2|^{1/2} > |2|`) is the example instead. *Needs:*
Mathlib `AdicCompletion`,
`AdicCompletion.isAdicComplete`, `IsAdicComplete`, `IsIntegrallyClosedIn`.

**Checks for the algebraic completion and indexing conventions.**

- `completion_unit` — for `A = ℤ`, `ϖ = 1`, the completion is the zero ring; regularity
  does not imply that the completion is nonzero.
- `completion_zero` — for `A = ℤ`, `ϖ = 0`, the canonical map to the completion is
  bijective. The regular-element hypothesis fails, so this is a Check of the completion
  construction, not an instance of the regularity theorem.
- `completion_two_terms` — in the `2`-adic completion of `ℤ`, reduction modulo `2`
  sends the images of `1` and `2` to `[1]` and `0`. This pins the exponent convention
  `n = 1`; at `n = 0` the quotient by `(2)^0 = (1)` is zero.
- `cofiltered_two_terms` — for an affinoid perfectoid diagram on `(Fin 2)ᵒᵖ`,
  `cofilteredLimit.π` at `1` is an isomorphism, and its composite with `X₁ → X₀`
  equals the projection at `0`. This computes the actual limit cone
  ([ECD] Prop. 6.5, p. 29).
- `profinite_two_terms` — send the two-digit cylinder function `s₀ + s₁ϖ`
  through `completedColimit.ι` and `binaryFunctionEquiv`. Its values on the
  zero sequence and the sequences supported at `0` and `1` are `0`, `1`, `ϖ`.
  The Check applies to every perfectoid pair and retains these values in
  characteristic `2`.

<a id="p5-2"></a>

### P5.2 the completed direct limit is a perfectoid Tate pair

**The completed direct limit of perfectoid Tate pairs is a perfectoid Tate pair.**
(*completed-colimit-is-perfectoid*) In the situation of *completed-colimit-of-perfectoid-pairs*,
with `(i₀, ϖ)` a compatible pseudouniformiser with `ϖ^p | p` in `R_{i₀}⁺`, prove: (a) `R⁺` is open
and bounded in `R` and integrally closed in `R`, so `R⁺` is a ring of integral elements and
`ϖR° ⊆ R⁺ ⊆ R°`; (b) `R` is uniform; (c) `ϖ^p` divides `p` in `R⁺`, `ϖ` is a nonzerodivisor on
`R⁺`, and Frobenius `R⁺/ϖR⁺ → R⁺/ϖ^pR⁺`, `x ↦ x^p`, is an isomorphism, so `R⁺` is an integral
perfectoid ring ([BMS] Def. 3.5; [BMS] §3, Lem. 3.20, p. 26; [BMS] §3, Lem. 3.10(ii), p. 22);
(d) `R` is a perfectoid Tate ring in the sense of [ECD] Def. 3.1, by (c) and
*perfectoid-tate-ring-from-integral-perfectoid* (P1.20: `R⁺` integral perfectoid with `ϖ` a
nonzerodivisor and `R = R⁺[1/ϖ]`), so `(R, R⁺)` is a perfectoid Tate pair and the `ι_i` are
morphisms of perfectoid Tate pairs; its tilt `R♭` is the completed direct limit of the `R_i♭`
(P1.29 (c), tilting commutes with completed colimits); (e) if all `R_i` have characteristic
`p` then `R` is a perfect complete Tate ring of characteristic `p`, and if the system lives over a
perfectoid Tate ring `B` (compatible maps `B → R_i`) then `R` is a perfectoid `B`-algebra. No
perfectoid base field is assumed. P1.29 supplies the underlying ring
`Perfectoid.completedFilteredColimit`; P5 adds the chosen plus ring through
`Perfectoid.completedColimitPlus` and `Perfectoid.completedColimitPair`.
Prove `completedColimitPlus_isRingOfIntegralElements` on the specified closure subring
of `completedFilteredColimit`: it is open, integrally closed in the ambient ring,
and contained in `powerBoundedSubring`. Prove `completedColimitPlus_isBounded` on
that same subring. Both assume a small filtered diagram of perfectoid pairs and
compatible roots of a pseudouniformizer at the chosen stage; no injectivity of the
transition maps is assumed ([ECD] Proposition 6.5, p. 29). The constructor
`completedColimitPair` uses the first theorem as its integral-elements proof.
`completedColimit.isPerfectoidTateRing` exposes P1’s
`completedFilteredColimit.isPerfectoidTateRing` on the bundled pair.
`Perfectoid.completedColimit.plusCompletionEquiv` identifies that plus ring with the completion
of `colim_i R_i⁺` ([ECD] Proposition 6.5, p. 29). P4.4 gives the pro-rational application;
P7.6 treats Frobenius-controlled towers. A completed colimit of
non-perfectoid rings can be perfectoid — the tower `ℚ_p(p^{1/p^n})` of discretely valued fields
([ECD] Rem. 3.3) has perfectoid completed colimit — but that is a tilde-limit (layer P7), not an
instance of this theorem. For every profinite `S` and perfectoid field `K`, `C^0(S, K)` is a
perfectoid Tate ring with `C^0(S, K)° = C^0(S, O_K)`; over a perfectoid field this is the end of the
proof of [Sch13] §4, Lem. 4.6, p. 25, here proved without a base field ([ECD] §6, proof of Prop. 6.5,
p. 29). *Needs:* *completed-colimit-of-perfectoid-pairs*; *completion-preserves-integral-closedness*;
P1.1; P1.20; P1.29 (for the ring and its tilt); Tau Ceti
`TauCeti.Huber.IsRingOfIntegralElements`,
`TauCeti.Huber.IsRingOfIntegralElements.mem_of_isTopologicallyNilpotent`,
`TauCeti.Huber.powerBoundedSubring`.

<a id="p5-3"></a>

### P5.3 cofiltered limits of affinoid perfectoid spaces

The second group passes to spaces: the limit in perfectoid spaces, its underlying space, and the
`ω₁`-cofiltered case in which no completion is needed.

**Cofiltered limits of affinoid perfectoid spaces.** (*cofiltered-limits-of-affinoid-perfectoid-spaces*)
In the standing setting let `(R, R⁺)` be the completed direct limit for a compatible
pseudouniformiser (P5.1), a perfectoid Tate pair
(P5.2), and put `X := Spa(R, R⁺)`, an affinoid perfectoid space (P2.9),
with projections `p_i : X → X_i` induced by `ι_i` (`i` in the tail; for general `i` through any
`j → i`, `i₀`). Then `(X, (p_i))` is a limit of the diagram `(X_i)` in the category of perfectoid
spaces: for every perfectoid space `Y`, `Hom(Y, X) → lim_i Hom(Y, X_i)` is bijective. In particular
`X` is the limit in the full subcategory of affinoid perfectoid spaces, `O_X(X) = R` and
`O⁺_X(X) = R⁺`, and `X` is independent of the choices up to unique isomorphism
(see *limit-independence-of-tail-and-choice*). Morphisms into affinoid perfectoid spaces are described
by the adjunction of AdicEtaleGeometry A1.16 for sheafy pairs, and perfectoid pairs are sheafy
(P2.6). The limit is claimed in perfectoid spaces only: for a non-uniform adic space `Y` a
compatible family `Y → X_i` need not induce a map continuous for the `ϖ`-adic topology of
`colim R_i⁺`; and the `X_i` are affinoid perfectoid — towers of non-perfectoid spaces are
tilde-limits (layer P7). Companion statements: `|X| → lim |X_i|` is a homeomorphism of spectral
spaces (P5.4); rational subsets and their rings are computed at finite stages
(see *limit-restriction-to-rational-subsets*); for `ω₁`-cofiltered `I` no completion is needed (P5.5);
cardinal bounds (P5.7). In `PerfectoidSpace`, define `cofilteredLimit` (for a cofiltered diagram
`X : I ⥤ PerfectoidSpace` of affinoid perfectoid spaces, the affinoid perfectoid space `Spa(R, R⁺)`
of the completed direct limit) and `cofilteredLimit.π` (the projections `p_i`, with `π_comp`:
`X(a) ∘ p_j = p_i` for `a : j → i`); prove `cofilteredLimit.isLimit` (the cone `(cofilteredLimit X, p)`
is a limit cone in perfectoid spaces, with `lift` for a compatible family `Y → X_i`),
`cofilteredLimit.lift_π` (`p_i ∘ lift f = f_i`), `cofilteredLimit.hom_ext` (two morphisms
`Y → cofilteredLimit X` agreeing after every `p_i` are equal), `cofilteredLimit.isAffinoid`
(`cofilteredLimit X` is affinoid perfectoid with `O(X) ≅ Perfectoid.completedColimit` and
`O⁺(X) ≅ Perfectoid.completedColimitPlus`), `cofilteredLimit.homeomorph` (`|cofilteredLimit X| ≃ₜ lim |X_i|`
as spectral spaces, P5.4), `cofilteredLimit.restrictRational` (for a rational `U_i ⊆ X_i` the open
subspace `p_i⁻¹(U_i)` is the cofiltered limit of the preimages `U_j`), `cofilteredLimit.baseChange`
(for a qcqs `Y_i → X_i`, `Y_i ×_{X_i} cofilteredLimit X = lim_j Y_i ×_{X_i} X_j`,
*limit-of-qcqs-base-changes*), `cofilteredLimit.isoOfInitial` (for an initial functor `u : I' ⥤ I` with `I'` small cofiltered,
`cofilteredLimit (u ⋙ X) ≅ cofilteredLimit X` over the `X_i`), `cofilteredLimit.plusQuotient`
(`O⁺(X)/ϖ^n ≅ colim_i O⁺(X_i)/ϖ^n`), `cofilteredLimit.globalSections_of_isCardinalFiltered` (for
`ℵ₁`-cofiltered `I`, `O(X) = colim O(X_i)` and `O⁺(X) = colim O⁺(X_i)` without completion, P5.5),
`cofilteredLimit.card_lt` (the cardinal bounds of P5.7) and `cofilteredLimit_prodProfinite` (for an
affinoid perfectoid `Z` and a profinite `S = lim S_n`, `Z × S := lim (Z × S_n)`, with
`Z × S = Spa(C^0(S, K), C^0(S, K⁺))` for `Z = Spa(K, K⁺)`, the `S × Spa(K, O_K)` of DiamondsAndVStacks
layer D4; [Berkeley] Lecture 8, Ex. 8.2.2, printed p. 57) ([ECD] §6, Prop. 6.5, statement, p. 29;
[ECD] §6, proof of Prop. 6.5, p. 29; [ECD] §5, Ex. 5.2, p. 21). *Needs:*
*completed-colimit-of-perfectoid-pairs*; *completed-colimit-is-perfectoid*; P2.9, P2.10, P2.6;
AdicEtaleGeometry A1.16 (`CAff` and its adjunction for sheafy pairs); Mathlib
`CategoryTheory.IsCofiltered`, `CategoryTheory.Limits.IsLimit`.

**Checks.**

- `cofilteredLimit_test_profinite` (computation) — for a perfectoid field `K` and
  `S = lim_n {0,1}^n`: `cofilteredLimit (n ↦ {0,1}^n × Spa(K, O_K)) ≅ Spa(C^0(S, K), C^0(S, O_K))`,
  with underlying space `S × |Spa(K, O_K)|`.
- `cofilteredLimit_test_initial` (degenerate) — if `I` has an initial object `i_*`, then
  `p_{i_*} : cofilteredLimit X → X_{i_*}` is an isomorphism.
- `cofilteredLimit_test_residueField` (characterisation) — for `x ∈ X₀` affinoid perfectoid, the
  limit over the rational neighbourhoods of `x` is `Spa(K(x), K(x)⁺)`, and the induced map to `X₀` is
  the injection `i_x` of [ECD] Ex. 5.2 (P4.5); for a filtered family of rational subsets the limit is
  the pro-rational subspace of P4.4.
- `cofilteredLimit_test_fibreProduct` (compatibility) — for an affinoid perfectoid `Y_i → X_i`,
  `Y_i ×_{X_i} cofilteredLimit X ≅ cofilteredLimit (j ↦ Y_i ×_{X_i} X_j)` over the tail, compatibly
  with the fibre products of layer P2.
- `cofilteredLimit_test_notPerfectoidTower` (non-example) — the system
  `n ↦ Spa(ℚ_p(p^{1/p^n}), ℤ_p[p^{1/p^n}])` is not a diagram of perfectoid spaces ([ECD] Rem. 3.3),
  since no diagram `ℕᵒᵖ ⥤ Perfd p` can have its zeroth object isomorphic,
  as a Yoneda adic space, to `Spa(ℚ_p, ℤ_p)`. The Check uses the actual input
  category of `cofilteredLimit`. Identifying `Spa` of the completion of
  `ℚ_p(p^{1/p^∞})` with its limit is a tilde-limit statement of layer P7.
- `cofilteredLimit_test_uncompletedPair` (non-example) — in the profinite example the Huber pair
  `(LC(S, K), LC(S, O_K))` (uncompleted colimit) has the same adic spectrum as the limit (Tau Ceti
  `TauCeti.ValuationSpectrum.spaCompletionHomeomorph`; as adic spaces `Spa(A, A⁺) = Spa(Â, Â⁺)`),
  but it is not complete and is not the pair of global sections of the limit:
  `O(X) = C^0(S, K) ≠ LC(S, K)`.
- `cofiltered_empty_index` (non-example) — the empty category is not cofiltered.
  The localization defining `FilteredTwoColimit` of the empty diagram with
  one-point fibre is empty and admits no equivalence to that fibre. This tests
  why `FilteredTwoColimit.equivConst` needs a nonempty filtered index; the
  completed-pair construction likewise needs an object supplying `ϖ`.

Two lemmas support the construction. Independence of the choices
(*limit-independence-of-tail-and-choice*): in the situation of
*completed-colimit-of-perfectoid-pairs*, (a) if `(i₀, ϖ)` and `(i₀', ϖ')` are two compatible
pseudouniformisers, the `ϖ`-adic and `ϖ'`-adic topologies on `A = colim R_i⁺` coincide and the
identity of `A` induces an isomorphism of Huber pairs `(R_ϖ, R_ϖ⁺) ≅ (R_{ϖ'}, R_{ϖ'}⁺)` compatible
with all `ι_i`; (b) for an initial functor `u : I' → I` with `I'` small cofiltered (the inclusion of a tail `I/i₁`, or of a
cofinal subposet) the completed direct limit of the restricted system is canonically isomorphic to
that of the original one, compatibly with the `ι`; (c) these isomorphisms compose (for three choices
the composite of two comparison isomorphisms is the third) and commute with `lift`; (d) consequently
`Spa(R, R⁺)` with its projections to the `X_i` is well defined up to unique isomorphism. In the
profinite example `ϖ`, `ϖ^{1/p}` and `uϖ` (`u ∈ O_K^×`) give the same `C^0(S, O_K)`, and restricting
to `S_n` with `n ≥ 5` gives the same pair; if `I` has a terminal object `t` the tail `I/t` is all of
`I` and (b) is the identity ([ECD] §6, Prop. 6.5, statement, p. 29; [ECD] §3, Rem. 3.2, p. 14).
*Needs:* *completed-colimit-of-perfectoid-pairs*; *completed-colimit-is-perfectoid*; Mathlib
`CategoryTheory.Over.initial_forget`, `CategoryTheory.Functor.Final.colimitIso`.

Uniformity of perfectoid spaces (*integral-sections-are-varpi-complete*): let `Y` be a perfectoid
space (P2.10) and `ϖ ∈ O_Y(Y)` a unit whose restriction to every affinoid perfectoid open `U ⊆ Y` is
a pseudouniformiser of `O_Y(U)` (for instance the pullback of a pseudouniformiser along a morphism
`Y → Spa(R_{i₀}, R_{i₀}⁺)`). Prove that `O_Y⁺(Y)` is `ϖ`-torsion free and `ϖ`-adically complete and
separated, and that the restriction maps `O_Y⁺(Y) → O_Y⁺(U)` to affinoid perfectoid opens are
continuous for the `ϖ`-adic topologies, the `ϖ`-adic topology of `O_Y⁺(U)` being its subspace
topology in `O_Y(U)`; consequently, for every ring `A` with a specified element `t` and every ring map
`f : A → O_Y⁺(Y)` satisfying `f(t) = ϖ`, `f` extends uniquely to the `t`-adic completion `Â`, and the induced maps
`Â[1/t] → O_Y(U)` are continuous. This integral completeness gives the
universal property; for `Y = S × Spa(K, O_K)` with `S` profinite, `O⁺(Y) = C^0(S, O_K)` is
`ϖ`-adically complete although it is a limit over the finite covers of `S`; for a complete Tate
ring `S` with `S⁺` unbounded the `ϖ`-adic topology of `S⁺` is strictly coarser than its subspace
topology (the subspace neighbourhoods `ϖⁿS_0`, `S_0 ⊆ S⁺` a ring of definition, lie in the `ϖⁿS⁺`,
and no `ϖⁿS⁺` lies in `ϖS_0` when `S⁺` is unbounded; for `S = K[ε]/ε²` with `S⁺ = S° = O_K ⊕ Kε`
one has `⋂_n ϖⁿS° = Kε ≠ 0`, so the `ϖ`-adic topology is not even Hausdorff), so a compatible
family into such an `S` need not extend to the completion, which is why the limit is taken in
perfectoid spaces ([ECD] §6, proof of Prop. 6.5, p. 29). *Needs:* P2.10, P2.9, P2.8, P2.6; Mathlib
`IsAdicComplete`, `AdicCompletion`.

Lean: `PerfectoidSpace.exists_affinoidPerfectoid_limit` states the affinoid limit and its
universal property on the pinned category `Perfd`, together with the projection-compatible
homeomorphism on point spaces of P5.4.

### P5.4 the underlying space of the limit

**The underlying space of a cofiltered limit of affinoid perfectoid spaces is the inverse limit.**
(*limit-underlying-space-homeomorphism*) Let `X = lim_i X_i` be the limit of a cofiltered diagram of
affinoid perfectoid spaces `X_i = Spa(R_i, R_i⁺)` (P5.3).
Then the map `|X| → lim_i |X_i|` induced by the projections is a homeomorphism. That `lim_i |X_i|`
(inverse limit in topological spaces) is spectral with spectral projections, for a cofiltered
system of spectral spaces along spectral maps, is AdicSpacesPartII R0.19.5’s
spectral-limit lemma ([ECD] Lemma 2.11, p. 13), applied to
the `|X_i|` and the spectral transition maps
(Tau Ceti `isSpectralMap_spaComap_of_isTateRing`); what is proved here is the homeomorphism, and
that every rational subset `U ⊆ X` equals `p_i⁻¹(U_i)` for some `i` and some rational subset
`U_i ⊆ X_i`. The proof route is completion (Huber) followed by the direct-limit computation. For profinite `S`,
`|S × Spa(K, O_K)| = S × |Spa(K, O_K)|` with `S` in its profinite topology; for `x` in an affinoid
perfectoid `X₀`, `|Spa(K(x), K(x)⁺)| = ⋂_{U ∋ x} U` with the subspace topology, `U` ranging over the
rational neighbourhoods of `x` ([ECD] §6, Prop. 6.4(o), p. 27; [ECD] §6, proof of Prop. 6.4, p. 27;
[Sch12] §6, Lem. 6.13(ii), p. 37). *Needs:* *cofiltered-limits-of-affinoid-perfectoid-spaces*;
AdicSpacesPartII R0.19.7 (*spa-of-filtered-colimit-of-tate-pairs*); AdicSpacesPartII R0.19.8 (*quasicompact-opens-in-limits-of-spectral-spaces*);
*completed-colimit-of-perfectoid-pairs*; P1.1, P2.9; AdicSpacesPartII R0.19.5 (the spectral-limit lemma); Tau Ceti `TauCeti.Huber.Pair.Hom.isSpectralMap_spaComap_of_isTateRing`.

The generic Tate-pair spectrum comparison and finite-stage detection of opens are supplied by
AdicSpacesPartII R0.19.7–R0.19.8. The following consequences concern perfectoid base changes,
open subspaces and rational restrictions.

Base changes of a qcqs perfectoid space along a cofiltered limit form a limit
(*limit-of-qcqs-base-changes*): in the situation of
*cofiltered-limits-of-affinoid-perfectoid-spaces* let `g_i : Y_i → X_i` be a morphism of perfectoid
spaces with `Y_i` quasicompact and quasiseparated (for instance a qcqs étale `Y_i`), and put
`Y_j := Y_i ×_{X_i} X_j` (`a : j → i`) and `Y := Y_i ×_{X_i} X`, fibre products of perfectoid spaces
(P2.16; fibre products of affinoid perfectoid spaces are affinoid perfectoid). Prove that
`Y = lim_{j ∈ I/i} Y_j` in perfectoid spaces, that `|Y| → lim_j |Y_j|` is a homeomorphism of
spectral spaces, and that for every affinoid perfectoid open `V_i ⊆ Y_i` the `V_j := V_i ×_{X_i} X_j`
are affinoid perfectoid with `V := V_i ×_{X_i} X = lim V_j`, so `O(V)` is the completed direct limit
of the `O(V_j)`. This gives `|Z| = lim |Z_j|` for `Z_i = Y_i' ×_{X_i} Y_i''` with `Y_i', Y_i''` qcqs
étale, and `Y = S × Y_0`, `|Y| = S × |Y_0|` for a finite étale
`Y_0 → Spa(K, O_K)` in the profinite example ([ECD] §6, proof of Prop. 6.4(ii), pp. 27–28).
*Needs:* *cofiltered-limits-of-affinoid-perfectoid-spaces*; *limit-underlying-space-homeomorphism*;
P2.16, P2.15; AdicSpacesPartII R0.19.5 (the spectral-limit lemma); Mathlib `QuasiSeparatedSpace`.

Quasicompact opens and rational subsets of a cofiltered limit descend to a finite stage
(*descent-of-quasicompact-opens*): let `X = lim X_i` be as in
*cofiltered-limits-of-affinoid-perfectoid-spaces*, or more generally `Y = lim_j Y_j` for a qcqs
`Y_i → X_i` and its base changes (see *limit-of-qcqs-base-changes*). Prove that (a) every quasicompact
open `U ⊆ |Y|` is `p_j⁻¹(U_j)` for some `j` and some quasicompact open `U_j ⊆ |Y_j|`, and the open
perfectoid subspace `U` is `U_j ×_{Y_j} Y`; (b) two quasicompact opens of `|Y_j|` with the same
preimage in `Y` have the same preimage in some `Y_k`, and a quasicompact open containing the image
of `Y` contains the image of `Y_k` for some `k` ([ECD] §6, proof of Prop. 6.4(ii), p. 28); (c) a
quasicompact open of `|Y_j|` whose preimage in `Y` is open and closed has open and closed preimage
in `Y_k` for some `k`; (d) every finite cover of `Y` by quasicompact opens is the pullback of a
finite cover of some `Y_j` by quasicompact opens ([KL15] §2.6, Rem. 2.6.7, p. 54); (e) for `Y = X`,
every rational subset of `X` is `p_i⁻¹(U_i)` for a rational subset `U_i ⊆ X_i`: for `T ⊆ R` finite
generating `R` and `s ∈ R` there are `i`, `T_i ⊆ R_i` finite generating `R_i` and `s_i ∈ R_i` with
`R(T/s) = R(ι_i(T_i)/ι_i(s_i))` ([Sch12] §6, Lem. 6.13(ii), p. 37). In the source's proof of 6.4(ii)
the image of `f_i : Y_i → Y_i'` is a quasicompact open over which `Y' → Y_i'` factors, hence
`Y_j' → Y_i'` factors over it for large `j`. *Needs:* *limit-underlying-space-homeomorphism*;
AdicSpacesPartII R0.19.8 (*quasicompact-opens-in-limits-of-spectral-spaces*); *limit-of-qcqs-base-changes*;
*limit-restriction-to-rational-subsets*; P2.16; layer P2.

Restriction of a cofiltered limit to a rational subset of a finite stage
(*limit-restriction-to-rational-subsets*): in the situation of
*cofiltered-limits-of-affinoid-perfectoid-spaces* let `U_i ⊆ X_i` be a quasicompact open subset,
`U_j := X(a)⁻¹(U_i)` for `a : j → i`, and `U := p_i⁻¹(U_i) ⊆ X`. Prove that `U = lim_{j ∈ I/i} U_j` in
perfectoid spaces, and that if `U_i` is rational then each `U_j` is a rational subset of `X_j`, hence
affinoid perfectoid (P2.3), `U` is a rational subset of `X`, and `(O_X(U), O_X⁺(U))` is the completed
direct limit of the `(O_{X_j}(U_j), O⁺_{X_j}(U_j))`: the canonical map from the `ϖ`-adic completion
of `colim_j O⁺_{X_j}(U_j)` to `O⁺_X(U)` is an isomorphism ([Sch12] §6, Lem. 6.13(iii), p. 37; [KL15]
§2.6, Lem. 2.6.5(a), p. 54). For a clopen `C_n ⊆ S_n` the restriction of `S × Spa(K, O_K)` to
`C × Spa(K, O_K)` (`C` the preimage of `C_n`) is `Spa(C^0(C, K), C^0(C, O_K))`; for `x ∈ U_0` rational
in an affinoid perfectoid `X₀`, restricting the rational neighbourhoods of `x` to those inside `U_0`
does not change the limit `Spa(K(x), K(x)⁺)`. *Needs:*
*cofiltered-limits-of-affinoid-perfectoid-spaces*; *limit-underlying-space-homeomorphism*; P2.3,
P2.4; layer P2; Tau Ceti `TauCeti.Huber.Pair.Hom.spaComap_preimage_rationalSubset`.

### P5.5 `ω₁`-cofiltered limits need no completion

**`ω₁`-cofiltered limits of affinoid perfectoid spaces need no completion.**
(*omega1-cofiltered-uncompleted-colimit*) In the situation of
*cofiltered-limits-of-affinoid-perfectoid-spaces* assume that `I` is `ω₁`-cofiltered: every
functor `J → I` from a category `J` with countably many morphisms extends to a cone `J^◁ → I`
(equivalently Mathlib `CategoryTheory.IsCardinalFiltered Iᵒᵖ ℵ₁`, `ℵ₁` regular by
`Cardinal.isRegular_aleph_one`; [ECD] §6, Prop. 6.5, definition following the statement, p. 29).
The supporting ring theorem `Perfectoid.isAdicComplete_colimit_of_isCardinalFiltered` takes
a small `ℵ₁`-filtered diagram of commutative rings `A_j`, compatible elements `t_j`, and
`(t_j)`-adic completeness of every `A_j`; it asserts `(t)`-adic completeness of `colim A_j`,
where `t` is their common image. Neither injectivity nor a nonzerodivisor hypothesis is needed
for this statement. Apply it on the tail of the perfectoid diagram.
Then `A = colim_i R_i⁺` is `ϖ`-adically separated and complete, so the canonical maps
`colim_i R_i⁺ → R⁺` and `colim_i R_i → R` are ring isomorphisms (bijective). Equivalently, for an
`ℵ₁`-filtered diagram of perfectoid Tate pairs the uncompleted colimit `(colim R_i, colim R_i⁺)`,
with `colim R_i⁺` carrying its `ϖ`-adic topology, is a perfectoid Tate pair and
`O(lim X_i) = colim O(X_i)`. For countable index categories such as `I = ℕᵒᵖ` both properties fail
in general: for the profinite system `colim R_n⁺ = LC(S, O_K) ≠ C^0(S, O_K) = R⁺` (not surjective),
and for `R_n = K⟨T^{1/p^∞}⟩` with `T^{1/p^m} ↦ ϖ^{1/p^m}T^{1/p^m}` the element `T` lies in
`⋂ ϖ^n colim R_n⁺`, so `colim R_n⁺ → R⁺ = O_K` is not injective. That `O` and `O⁺` commute with
`ω₁`-filtered colimits of affinoid perfectoid pairs ([ECD] §8, end of the proof of Prop. 8.5, p. 42,
and Cor. 8.6) is this theorem applied to `X ↦ O(X)` ([ECD] §6, Prop. 6.5, statement, p. 29; [ECD]
§6, proof of Prop. 6.5, p. 29). *Needs:* *completed-colimit-of-perfectoid-pairs*;
*completed-colimit-is-perfectoid*; *cofiltered-limits-of-affinoid-perfectoid-spaces*; Mathlib
`CategoryTheory.IsCardinalFiltered`, `Cardinal.isRegular_aleph_one`, `IsAdicComplete`.

Lean: `isAdicComplete_colimit_of_isCardinalFiltered`.

### P5.6 every affinoid perfectoid space is an `ω₁`-cofiltered limit of countably generated ones

The third group makes the `ω₁`-cofiltered case the general one, bounds cardinalities, and sets up
the 2-colimits of categories used for finite-stage descent.

Every countable subset of a perfectoid Tate ring lies in a closed, topologically countably
generated perfectoid subring (*countable-perfectoid-hull*): let `(R, R⁺)` be a perfectoid Tate pair
([ECD] Def. 3.1) and `ϖ ∈ R⁺` a pseudouniformiser with `ϖ^p | p` in `R⁺` and compatible `p`-power
roots `ϖ^{1/p^n}` (*compatible-pseudouniformizer* for the constant system). Prove that for every
countable subset `C ⊆ R` there is a closed subring `R' ⊆ R` such that
`C ∪ {ϖ, ϖ⁻¹} ∪ {ϖ^{1/p^n}} ⊆ R'`; `R'` has a countable dense subset (equivalently `R'` is the
closure of a countably generated subring, i.e. topologically countably generated); `R'` with the
subspace topology is a perfectoid Tate ring with `R'° = R' ∩ R°`; `R'⁺ := R' ∩ R⁺` is a ring of
integral elements of `R'`, and the inclusion `(R', R'⁺) → (R, R⁺)` is a morphism of perfectoid Tate
pairs; moreover `|R'| ≤ 2^{ℵ₀}` ([ECD] §6, proof of Prop. 6.5, p. 29; [ECD] §4, Rem. 4.3, p. 20). For
`R = C^0(S, K)` with `S = {0,1}^{ℵ₁}`, `K = ℂ_p`, a constant pseudouniformiser with constant chosen roots, and `C` a countable set
of functions depending only on the coordinates in a countable `J ⊆ ℵ₁`, one may take `R' = C^0({0,1}^J, ℂ_p)` pulled back to
`S`; if `R` already has a countable dense subset (for example `ℂ_p⟨T^{1/p^∞}⟩`), one may take
`R' = R`. *Needs:* *compatible-pseudouniformizer*; P1.1; layer P1; Tau Ceti
`TauCeti.Huber.IsTateRing`, `TauCeti.Huber.IsRingOfIntegralElements`,
`TauCeti.Huber.powerBoundedSubring`; `completion_cardinality_bound` (P6.2); Mathlib `Cardinal.power_le_power_left`.

**Every affinoid perfectoid space is an `ω₁`-cofiltered limit of topologically countably generated
affinoid perfectoid spaces.** (*countably-generated-decomposition*) Let `X = Spa(R, R⁺)` be an
affinoid perfectoid space, with no countability assumption on `R`, and `ϖ` as in
*countable-perfectoid-hull*. Let `𝒮(R)` be the set of closed subrings `R' ⊆ R` containing `ϖ^{±1}`
that are perfectoid Tate rings for the subspace topology and admit a countable dense subset,
ordered by inclusion, and put `R'⁺ := R' ∩ R⁺`. Then: (a) `𝒮(R)` is `ω₁`-directed (every countable
subset of `𝒮(R)` has an upper bound in `𝒮(R)`) and `⋃_{R' ∈ 𝒮(R)} R' = R`;
(b) `colim_{R' ∈ 𝒮(R)} (R', R'⁺) → (R, R⁺)` is an isomorphism of Huber pairs, without completion
(P5.5 for the `ω₁`-directed `𝒮(R)`);
(c) `X = lim_{R' ∈ 𝒮(R)} Spa(R', R'⁺)` in perfectoid spaces and `|X| = lim |Spa(R', R'⁺)|`: `X` is
an `ω₁`-cofiltered limit of affinoid perfectoid spaces with topologically countably generated rings
of functions; (d) every `R' ∈ 𝒮(R)` has cardinality `≤ 2^{ℵ₀}`, hence for every uncountable strong
limit cardinal `κ` (in particular every cutoff cardinal of [ECD] Lem. 4.1) `X` is an `ω₁`-cofiltered
limit of affinoid perfectoid spaces whose rings have cardinality `< κ`. This `ω₁`-presentation is
the form in which the cutoff-independence argument of [ECD] §8, proof of Prop. 8.2, p. 39, is
carried out for singular cutoff cardinals. For `X = Spa(C^0(S, ℂ_p), C^0(S, O_{ℂ_p}))` with
`S = {0,1}^{ℵ₁}` and constant `ϖ` the subrings `C^0({0,1}^J, ℂ_p)`, `J ⊆ ℵ₁` countable, are cofinal in `𝒮(R)`, since
a continuous function on `S` depends on countably many coordinates; if `R` has a countable dense
subset, `R` is the largest element of `𝒮(R)` and the presentation is trivial; if `R` has no
countable dense subset, `X` is not the limit of an `ℕᵒᵖ`-indexed tower of topologically countably
generated affinoid perfectoid spaces (the union of the countable dense subsets of the images would
be a countable dense subset of `R`), so an uncountable `ω₁`-cofiltered index category is necessary
([ECD] §6, Prop. 6.5, statement, p. 29; [ECD] §6, proof of Prop. 6.5, p. 29). *Needs:*
*countable-perfectoid-hull*; *omega1-cofiltered-uncompleted-colimit*;
*cofiltered-limits-of-affinoid-perfectoid-spaces*; *limit-underlying-space-homeomorphism*;
Mathlib `Cardinal.IsStrongLimit`, `CategoryTheory.IsCardinalFiltered`,
`Cardinal.isRegular_aleph_one`.

### P5.7 cardinal bounds

**Cardinal bounds for cofiltered limits of affinoid perfectoid spaces.** (*limit-cardinal-bounds*)
Let `κ` be a cutoff cardinal as in [ECD] §4, Lem. 4.1(iii), p. 19, `λ < κ` a
cardinal, `I` a cofiltered category with fewer than `λ` morphisms, and `X_i = Spa(R_i, R_i⁺)`
(`i ∈ I`) a cofiltered diagram of affinoid perfectoid spaces such that `|R_i| < κ'` for all `i`, for
some strong limit cardinal `κ' < κ` (this holds when all `X_i` are `κ'`-small in the sense of [ECD]
Def. 4.2). Let `X = Spa(R, R⁺) = lim X_i`. Then `|colim R_i⁺| < κ`, `|R⁺| < κ`, `|R| < κ`, `|X| < κ`
(by `|X| ≤ 2^{|R|}`, [ECD] §4, proof of Prop. 4.4, p. 20), and `|O_X(U)| < κ` for every open affinoid
`U ⊆ X`. These are the two conditions of [ECD] Def. 4.2, so `X` is `κ`-small; layer P6 defines
`κ`-smallness by exactly these conditions. For `I = ℕᵒᵖ`, `K = ℂ_p`, `S = {0,1}^ℕ`:
`|C^0(S, ℂ_p)| = 2^{ℵ₀} < κ` for every cutoff cardinal `κ`; and `X_∞ = lim_J X_J` over fewer than `κ`
finite subsets `J` with all `X_J` bounded by one strong limit `κ' < κ` is `κ`-small, as in the proof of [ECD] Lem. 7.18
(which DiamondsAndVStacks D1 uses downstream) ([ECD] §6, Prop. 6.4, final statement, p. 27; [ECD] §6, proof of
Prop. 6.4, p. 27). *Needs:* *cofiltered-limits-of-affinoid-perfectoid-spaces*;
*completed-colimit-of-perfectoid-pairs*; P2.3, P2.6; `completion_cardinality_bound` (P6.2); Tau Ceti `TauCeti.ValuationSpectrum`; Mathlib `Cardinal.IsStrongLimit`,
`Cardinal.mk_set`, `Cardinal.power_le_power_left`.

### P5.8 the 2-colimit of a filtered diagram of categories

**The 2-colimit of a filtered diagram of categories.** (*filtered-two-colimit-of-categories*) Let
`J` be a small filtered category and `F` a pseudofunctor from `J` (as a locally discrete
bicategory) to categories (Mathlib `CategoryTheory.Pseudofunctor`; base change is only defined up to
canonical isomorphism, hence a pseudofunctor): categories `C_j`, functors `F_a : C_i → C_j` for
`a : i → j`, and coherent natural isomorphisms `F_{ba} ≅ F_b ∘ F_a` and `F_{id} ≅ id`. The filtered
2-colimit `2-colim_j C_j` is the category whose objects are pairs `(i, x)` with `x` an object of
`C_i`, whose morphisms are

`Hom((i, x), (j, y)) := colim_{(k, a: i → k, b: j → k)} Hom_{C_k}(F_a x, F_b y)`

(a filtered colimit of sets over the category of cospans under `i` and `j`), with composition and
identities induced from the `C_k`. It carries functors `ι_i : C_i → 2-colim C_j` and isomorphisms
`ι_j ∘ F_a ≅ ι_i`, and is the pseudo-colimit of `F`: pseudo-cocones (`G_i : C_i → D`,
`G_j ∘ F_a ≅ G_i` coherently) correspond, up to unique isomorphism, to functors
`G : 2-colim C_j → D`, together with specified comparison isomorphisms; uniqueness is
among natural isomorphisms respecting those comparisons. For a pseudo-cocone the induced `G` is faithful iff morphisms in some `C_k`
with equal images in `D` become equal in some further `C_{k'}`; full iff every morphism
`G_i x → G_j y` of `D` comes from a morphism in some `C_k`; essentially surjective iff every object
of `D` is isomorphic to some `G_i x`. In this layer `J = Iᵒᵖ`, `C_i` is a category of objects over
`X_i` and `F_a` is base change; "the base change functors induce an equivalence
`2-colim (X_i)_? → X_?`" means that the induced `G` is an equivalence. In `CategoryTheory`, define
`FilteredTwoColimit` (the category `2-colim F` with objects `Σ j, F.obj j`), `FilteredTwoColimit.ι`
(the canonical functor `ι_j : F.obj j ⥤ FilteredTwoColimit F`), `FilteredTwoColimit.ιCompIso` (for
`a : i → j`, the isomorphism `F_a ⋙ ι_j ≅ ι_i`, compatible with the coherence isomorphisms of `F`)
and `FilteredTwoColimit.desc` (the functor `FilteredTwoColimit F ⥤ D` induced by a pseudo-cocone);
prove `FilteredTwoColimit.homEquiv` (`Hom((i, x), (j, y)) ≃` the colimit over cospans `(k, a, b)` of
`Hom_{F k}(F_a x, F_b y)`), `FilteredTwoColimit.ιCompDescIso` (`ι_i ⋙ desc G ≅ G_i`),
`FilteredTwoColimit.descUniqueIso` (any `H` with isomorphisms `ι_i ⋙ H ≅ G_i` compatible with the
cocone has a unique isomorphism to `desc G` respecting those specified comparisons), `FilteredTwoColimit.isEquivalence_desc_iff` (`desc G`
is an equivalence iff (a) `colim_k Hom(F_a x, F_b y) → Hom_D(G_i x, G_j y)` is bijective for all
`(i, x)`, `(j, y)` and (b) every object of `D` is isomorphic to some `G_i x`),
`FilteredTwoColimit.equivOfFinal` (for a final functor `u : J' ⥤ J` between small filtered categories,
`FilteredTwoColimit (F ∘ u) ⥤ FilteredTwoColimit F` is an equivalence: restriction to a tail),
`FilteredTwoColimit.fullSubcategory` (if `C'_j ⊆ C_j` are full subcategories stable under the `F_a`,
then `2-colim C'_j` is equivalent to the full subcategory of `2-colim C_j` on the `(i, x)` with
`x ∈ C'_i`) and `FilteredTwoColimit.equivConst` (for the constant pseudofunctor at a category `C`,
with identity transition functors, `FilteredTwoColimit ≌ C`). Use Mathlib
`((Functor.const J).obj (Cat.of C)).toPseudofunctor'` for that constant pseudofunctor. For the filtered system of categories
of finite étale algebras over complete Tate rings `A_i`, `Hom` in the 2-colimit is the colimit of
`Hom` over common targets, and the comparison functor to finite étale algebras over the completed
colimit is an equivalence (layer P3, read through `isEquivalence_desc_iff`) ([Berkeley] Lecture 7,
Rem. 7.4.7, printed p. 54; [KL15] §1.2, Rem. 1.2.9, p. 15). *Needs:* Mathlib
`CategoryTheory.Pseudofunctor`, `CategoryTheory.IsFiltered`, `CategoryTheory.Functor.IsEquivalence`,
`CategoryTheory.Grothendieck`.

**Checks.**

- `filteredTwoColimit_test_const` (computation) — for a one-object index with discrete
  Boolean fibre, each included object has exactly one endomorphism and there is no
  morphism from `false` to `true`. For a general constant fibre `C`, `equivConst`
  supplies `FilteredTwoColimit F ≌ C`.
- `filteredTwoColimit_test_discrete` (computation) — for `J = ℕ` and `C_n` the discrete category on
  a set `E_n` with maps `E_n → E_{n+1}`, `FilteredTwoColimit F` is equivalent to the discrete
  category on `colim_n E_n`.
- `filteredTwoColimit_test_finitelyPresentedModules` (compatibility) — for a filtered diagram of
  rings `B_j` with colimit `B`, base change `2-colim_j Mod^{fp}(B_j) → Mod^{fp}(B)` on finitely
  presented modules is an equivalence ([Stacks] [Tag 05N7](https://stacks.math.columbia.edu/tag/05N7),
  Lem. 10.127.6; filtered categories reduce to directed sets by §4.21, Lem. 4.21.5).
- `filteredTwoColimit_test_notNaive` (computation) — for the two-stage diagram from
  the discrete Boolean category to the one-object category, the initial cross-Hom
  is empty and the cross-Hom in the 2-colimit has exactly one element, an isomorphism.
  More generally `eventually_isomorphic` applies when `C_0` has two objects with no
  morphisms between them and `C_n` (`n ≥ 1`) has two distinct isomorphic objects:
  the category with object set `colim Ob(C_n)` but morphisms taken in `C_0` misses
  their isomorphism in the 2-colimit. These computations use the cospan description
  of morphisms in P5.8. The discrete Boolean and one-object carriers are Mathlib's
  `CategoryTheory.Discrete`.

**Checks for constant pseudofunctors and their 2-colimits.** Each uses the
roadmap's `FilteredTwoColimit` and its canonical inclusions.

- `constantPseudofunctor_empty_fibre` — the 2-colimit over a one-object index
  with empty discrete fibre has no objects.
- `constantPseudofunctor_point` — every object of the 2-colimit of the one-point
  fibre is isomorphic to the included point, and its endomorphisms are unique.
- `constantPseudofunctor_two_terms` — over `0 → 1` in `Fin 2`, the inclusions
  of `false` at the two stages are isomorphic, as are the inclusions of `true`.
  There is no arrow from the included `false` to the included `true`.

### P5.9 finite étale objects come from a finite stage

Throughout P5.9–P5.12, for a perfectoid space `Z`, `Z_ét` is the category of perfectoid spaces
étale over `Z` ([ECD] Def. 6.2; P3.18), and `Z_fét`, `Z_ét,qcqs` and `Z_ét,qc,sep` are its full
subcategories of finite étale, of quasicompact quasiseparated (P4.2), and of quasicompact
separated étale spaces; base change along `X_j → X_i` or `X → X_i` is the fibre product of
perfectoid spaces (P2.16), which preserves these classes (layer P3; P3.18).

**Finite étale objects over a cofiltered limit of affinoid perfectoid spaces come from a finite
stage.** (*finite-etale-finite-stage-approximation*) Let `X = Spa(R, R⁺) = lim_i X_i` be a
cofiltered limit of affinoid perfectoid spaces `X_i = Spa(R_i, R_i⁺)`
(P5.3), with `(i₀, ϖ)` a compatible pseudouniformiser.
Then the base change functors `(X_i)_fét → X_fét` induce an equivalence of categories
`2-colim_i (X_i)_fét → X_fét` (P5.8). Ring form: base change
`S_i ↦ S_i ⊗_{R_i} R` (no completion; the tensor product is finite projective over `R`) induces an
equivalence `2-colim_i FÉt(R_i) → FÉt(R)`, compatible with tensor products and ranks — for the
uncompleted colimit `colim_i R_i` this is AdicEtaleGeometry A1.0.3
(*finite-etale-algebras-filtered-colimit*: `2-colim FÉt(A_i) ≃ FÉt(colim A_i)` for any filtered
colimit of rings), and invariance under completion `FÉt(colim R_i) ≃ FÉt(R)` is
AdicEtaleGeometry A1.0.5, `finiteEtale_completedColimit` (*finite-etale-completed-direct-limit*). AdicEtaleGeometry A4.5's
*finite-etale-invariance-along-tower* is its specialization to finite étale torsor towers, downstream
of AdicEtaleGeometry A1.0.5; it does not supply the arbitrary-transition case here and is not an input. What is
proved here is the perfectoid cofiltered index with its compatible pseudouniformiser and the plus
rings: the plus rings of the corresponding perfectoid spaces are the integral closures of `R_i⁺`,
`R⁺`. Explicitly, the cited equivalences say: (lifting) every finite étale `R`-algebra is
`S_i ⊗_{R_i} R` for some `i` and some finite étale `R_i`-algebra `S_i`; (uniqueness) for finite
étale `R_i`-algebras `S_i`, `S_i'`, every `R`-algebra map `S_i ⊗ R → S_i' ⊗ R` is the base change
of an `R_j`-algebra map for some `j → i`, and two `R_j`-algebra maps with the same base change to
`R` agree over some `R_k`. The theorem concerns
finite étale algebras; finitely presented algebras over `R` do not in general come from a finite
stage: for `S = {0,1}^ℕ`, `0 < |ϖ| < 1` and `f(s) = Σ_k s_k ϖ^k`, the algebra
`B = C^0(S, K)/(f)` is finitely presented but not flat. Indeed `f` vanishes precisely at the
all-zero sequence; this singleton is not open, so `(f)` is not generated by an idempotent.
A flat quotient by a finitely generated ideal would have an idempotent kernel. Every algebra
over the finite product of fields `K^{S_n}` is flat as a module, and base change preserves
flatness, so `B` cannot descend. The two-term calculation `f(0) = 0`, `f(e_n) = ϖ^n ≠ 0`, where
`e_n` has just its `n`-th digit equal to one, also shows that `f` lies in no finite stage. The algebraic input — finite étale algebras over a completed direct
limit of complete Tate rings, through the henselian approximation theorem ([Berkeley] Lecture 7, Thm 7.4.8, printed p. 54)
— is supplied by AdicEtaleGeometry A1.0.5 (the invariance of finite étale algebras under completed colimits); here the
completion is identified with the limit's ring and the statement is carried to spaces. Over `C^0(S, K)`
finite étale algebras are locally on `S` constant: each is an `S_n`-indexed family of finite étale
`K`-algebras tensored with `C^0(S, K)` for some `n`; and a finite étale cover of the limit is the
pullback of one at a finite stage. A specified isomorphism between two pullbacks descends,
and two lifts of that specified isomorphism agree after enlarging the stage (used
in [ECD] proof of Prop. 7.16: a splitting over a connected component extends to a neighbourhood)
([ECD] §6, Prop. 6.4(i), p. 27; [ECD] §6, proof of Prop. 6.4(i), p. 27; [Sch12] §7, proof of
Lem. 7.5(i), p. 41). *Needs:* *completed-colimit-of-perfectoid-pairs*;
*completed-colimit-is-perfectoid*; *cofiltered-limits-of-affinoid-perfectoid-spaces*;
*filtered-two-colimit-of-categories*; AdicEtaleGeometry A1.0.5, P3.17, P3.14, P3.10, P2.16; layer P3;
AdicEtaleGeometry A1.0.3 (*finite-etale-algebras-filtered-colimit*); Mathlib `CommAlgCat.FiniteEtale`.

**Checks for ranks and uniqueness.**

For the constant diagram on `Spa(R,R⁺)` indexed by `ℕᵒᵖ`, these Checks pass
from finite étale algebras through `finiteEtaleAffinoidEquiv.inverse`, the
canonical `FilteredTwoColimit.ι`, and `finiteStageFiniteEtaleEquiv`.

- `finiteStage_rank_zero` — the image of `R^{Fin 0}` has empty underlying space.
- `finiteStage_rank_one` — the image of `R^{Fin 1}` has an isomorphism as its
  structural arrow to the limit.
- `finiteStage_iso_not_unique` — when `R ≠ 0`, the coordinate swap of `R × R`
  still differs from the identity after these functors, and its square is the
  identity. Thus uniqueness of finite-stage descent concerns a specified
  isomorphism over the limit ([ECD] Prop. 6.4(i), p. 27).

### P5.10 quasicompact quasiseparated étale objects form the 2-colimit

**Quasicompact quasiseparated étale objects over a cofiltered limit form the 2-colimit.**
(*qcqs-etale-finite-stage-descent*) Let `X = lim_i X_i` be a cofiltered limit of affinoid
perfectoid spaces (P5.3). Then the base change
functors `(X_i)_ét,qcqs → X_ét,qcqs` induce an equivalence of categories
`2-colim_i (X_i)_ét,qcqs → X_ét,qcqs` (the 2-colimit of *filtered-two-colimit-of-categories* for
the base-change pseudofunctor on `Iᵒᵖ`). Explicitly: for `Y_i, Y_i' ∈ (X_i)_ét,qcqs`,
`colim_{j → i} Hom_{X_j}(Y_j, Y_j') → Hom_X(Y, Y')` is bijective, and every qcqs étale `Y → X` is
isomorphic to the base change of some qcqs étale `Y_i → X_i`. Quasicompactness is necessary: the
non-quasicompact open `X ∖ ({s₀} × Spa(K, O_K))` of `S × Spa(K, O_K)` does not descend. Uses: for
`Y = lim Y_i` affinoid pro-étale over `X`, `Y_ét,qcqs = 2-colim (Y_i)_ét,qcqs` ([ECD] §7, proof of
Prop. 7.10, p. 33); a qcqs étale cover of `X` that splits over a connected component
`Spa(C, C⁺) = lim` of its clopen neighbourhoods splits over a neighbourhood ([ECD] proof of
Prop. 7.16, which DiamondsAndVStacks D1 uses downstream) ([ECD] §6, Prop. 6.4(ii), p. 27). *Needs:*
*finite-stage-faithfulness*; *finite-stage-fullness*; *qcqs-etale-essential-surjectivity*;
*filtered-two-colimit-of-categories*; *cofiltered-limits-of-affinoid-perfectoid-spaces*; P3.18,
P2.16; layer P3.

Three lemmas give the equivalence. Faithfulness by the graph argument
(*finite-stage-faithfulness*): in the situation of *cofiltered-limits-of-affinoid-perfectoid-spaces*
let `Y_i, Y_i' ∈ (X_i)_ét,qcqs` and `f_i, g_i : Y_i → Y_i'` morphisms over `X_i`. Prove that if their
base changes `f, g : Y → Y'` over `X` are equal, then `f_j = g_j` for some `a : j → i` (`f_j`, `g_j`
the base changes to `X_j`); equivalently `2-colim_i (X_i)_ét,qcqs → X_ét,qcqs` is faithful. Two
`X_0`-endomorphisms of a finite étale `Y_0 → X_0` that agree over the limit agree at a finite
stage; faithfulness fails for non-quasicompact `Y_i`, since the argument needs the images of the
graphs to be quasicompact: for `X_n = {|T| ≤ |ϖ|ⁿ} ⊆ X_0 = Spa(K⟨T^{1/p^∞}⟩, O_K⟨T^{1/p^∞}⟩)`
with limit the origin `Spa(K, O_K)`, `Y_0 := ⊔_{m ≥ 1} X_0` and `Y_0' := ⊔_{m ≥ 1} (X_0 ⊔_{X_m} X_0)`
(two copies of the disc glued along `X_m`), the maps `f_0, g_0 : Y_0 → Y_0'` that are the first,
resp. second, inclusion on the `m`-th component differ on the components `m > n` over every
`X_n`, but agree over the origin, which lies in every `X_m` ([ECD] §6, proof of Prop. 6.4(ii),
pp. 27–28; [ECD] §5, Prop. 5.3, p. 21, injections are determined by their image). *Needs:*
*limit-of-qcqs-base-changes*;
*descent-of-quasicompact-opens*; P3.15, P3.17, P4.3, P4.7, P2.16; layer P3; AdicEtaleGeometry
A1.3.

Fullness by graph approximation (*finite-stage-fullness*): in the same situation let
`Y_i, Y_i' ∈ (X_i)_ét,qcqs` and `f : Y → Y'` a morphism over `X` between their base changes. Prove
that there are `a : j → i` and a morphism `f_j : Y_j → Y_j'` over `X_j` whose base change to `X` is
`f`; with *finite-stage-faithfulness*, `2-colim_i (X_i)_ét,qcqs → X_ét,qcqs` is fully faithful. For
an affinoid pro-étale `Y = lim Y_i` and an étale `Z`, `Hom_X(Y, Z) = colim_i Hom_X(Y_i, Z)` ([ECD]
proof of Prop. 7.10) is this fullness applied to `Y_i ×_X Z → Y_i`; a section over the limit of an
étale cover descends to a section over a finite stage ([ECD] §6, proof of Prop. 6.4(ii), p. 28).
*Needs:* *finite-stage-faithfulness*; *limit-of-qcqs-base-changes*; *descent-of-quasicompact-opens*;
*finite-stage-isomorphism-detection*; P3.17; layer P3; layer P2; AdicEtaleGeometry A1.3.

Every qcqs étale space over the limit descends to a finite stage
(*qcqs-etale-essential-surjectivity*): in the same situation prove that every `Y ∈ X_ét,qcqs` is
isomorphic over `X` to `Y_i ×_{X_i} X` for some `i` and some `Y_i ∈ (X_i)_ét,qcqs`, by the local
structure of étale maps ([ECD] §6, Def. 6.2(ii), p. 26) and gluing along quasicompact overlaps. A
rational subset of a finite étale cover of a rational subset of `X` descends, and a finite union of
such opens glued along quasicompact overlaps descends; for `X = S × Spa(K, O_K)` with `S` profinite
without isolated points, the open subspace `Y = X ∖ ({s₀} × Spa(K, O_K))` is étale and
quasiseparated but not quasicompact, and is not the base change of an open subspace of any finite
stage ([ECD] §6, proof of Prop. 6.4(ii), p. 28). *Needs:* *finite-etale-finite-stage-approximation*;
*finite-stage-fullness*; *finite-stage-faithfulness*; *descent-of-quasicompact-opens*;
*limit-restriction-to-rational-subsets*; *limit-of-qcqs-base-changes*; P3.15, P3.17;
AdicEtaleGeometry A1.3 (étale morphisms by the local open-immersion/finite-étale description);
layer P3; layer P2.

### P5.11 quasicompact separated étale objects form the 2-colimit

**Quasicompact separated étale objects over a cofiltered limit form the 2-colimit.**
(*quasicompact-separated-etale-finite-stage-descent*) Let `X = lim_i X_i` be a cofiltered limit of
affinoid perfectoid spaces. Then the base change functors `(X_i)_ét,qc,sep → X_ét,qc,sep` induce an
equivalence of categories `2-colim_i (X_i)_ét,qc,sep → X_ét,qc,sep`, where
`X_ét,qc,sep ⊆ X_ét,qcqs` is the full subcategory of quasicompact separated étale perfectoid spaces
over `X` (a separated étale space over an affinoid is quasiseparated). A quasicompact separated
étale space with descent data over an affinoid pro-étale limit comes from a finite stage together
with its descent datum ([ECD] §9, proof of Prop. 9.7, p. 48; used by DiamondsAndVStacks D3 downstream) ([ECD]
§6, Prop. 6.4(iii), p. 27; [ECD] §6, proof of Prop. 6.4(iii), p. 28). *Needs:*
*qcqs-etale-finite-stage-descent*; *separatedness-recognition-at-finite-stage*;
*filtered-two-colimit-of-categories*; P4.19.

Separatedness is recognised at a finite stage (*separatedness-recognition-at-finite-stage*): in
the situation of *cofiltered-limits-of-affinoid-perfectoid-spaces* let `f_i : Y_i → X_i` be qcqs
étale. Prove that if the base change `f : Y → X` is separated ([ECD] Def. 5.10: `Δ_f` is a closed
immersion; P4.19), then `f_j : Y_j → X_j` is separated for some `a : j → i`, by the topological
criterion for separatedness ([ECD] §5, Prop. 5.11(ii), p. 25) and the clopen descent of
AdicSpacesPartII R0.19.8 (*quasicompact-opens-in-limits-of-spectral-spaces*). The stage must be enlarged in general: for
`X_n := {|T| ≤ |ϖ|^n} ⊆ X_0 = Spa(K⟨T^{1/p^∞}⟩, O_K⟨T^{1/p^∞}⟩)`, with limit `Spa(K, K⁺)` (the
origin), `Y_1 :=` two copies of `X_1` glued along `V_1 = {|T| ≤ |ϖ|^2}` is qcqs étale over `X_1` and
not separated, while its base change to `X_2` (two copies of `X_2` glued along `X_2`) is `X_2`,
which is separated ([ECD] §6, proof of Prop. 6.4(iii), p. 28). *Needs:* *limit-of-qcqs-base-changes*;
*descent-of-quasicompact-opens*; P4.19, P4.20, P4.18; layer P3.

### P5.12 finite-stage descent along cofiltered limits

**Finite-stage descent along cofiltered limits of affinoid perfectoid spaces.**
(*finite-stage-descent-of-qcqs-etale-objects*) [ECD] §6, Prop. 6.4(i)–(iii), p. 27 — for a
cofiltered system of affinoid perfectoid spaces with limit `X = Spa(R, R⁺)` (P5.3), base change is
an equivalence `2-colim_i (X_i)_ét,qcqs → X_ét,qcqs` restricting to equivalences on the full
subcategories `X_fét ⊆ X_ét,qc,sep ⊆ X_ét,qcqs` — is the conjunction of P5.9, P5.10 and P5.11 and
is not stated again; affinoid étale objects (part (iv)) are P5.13.4, which uses P5 and
is not an input here (P6.5), the
cardinal bound is *limit-cardinal-bounds*, and the recognition of surjectivity and isomorphism at a
finite stage is the supporting result below. *Needs:*
*finite-etale-finite-stage-approximation*; *qcqs-etale-finite-stage-descent*;
*quasicompact-separated-etale-finite-stage-descent*; *finite-stage-isomorphism-detection*;
*separatedness-recognition-at-finite-stage*; *cofiltered-limits-of-affinoid-perfectoid-spaces*;
*filtered-two-colimit-of-categories*; P3.18, AdicEtaleGeometry A1.0.5; layer P3.

Surjectivity and isomorphism of qcqs étale maps are detected at a finite stage
(*finite-stage-isomorphism-detection*): in the situation of
*cofiltered-limits-of-affinoid-perfectoid-spaces* let `h_i : Y_i → Y_i'` be a morphism in
`(X_i)_ét,qcqs` (automatically étale by cancellation, layer P3) with base changes `h_j` over `X_j`
and `h` over `X`. Prove that (a) if `|h|` is surjective, then `|h_j|` is surjective for some
`a : j → i`; in particular, if a qcqs étale `Y_i → X_i` has surjective base change to `X`, then
`Y_j → X_j` is surjective for some `j`; (b) if `h` is an isomorphism, then `h_j` is an isomorphism
for some `j`, by the diagonal step and the injection criterion of [ECD] §5, Prop. 5.3(iv), p. 21
([ECD] §6, proof of Prop. 6.4(ii), p. 28). Statement (b) is the intermediate step to fullness and
agrees with P5.13.4’s finite-stage comparison on analytic carriers;
statement (a) is used by DiamondsAndVStacks layers D1 and D2
(an étale surjection over the limit is an étale surjection at a finite stage). *Needs:*
*limit-of-qcqs-base-changes*; *descent-of-quasicompact-opens*; P4.3, P4.7, P4.2, P2.16; layer P3;
layer P4.

<a id="p5-13-1"></a>

### P5.13.1 tubular neighbourhoods and sousperfectoid specializations

**Tubular neighbourhoods of standard étale loci: a nonarchimedean implicit function theorem.**
(*tubular-neighbourhood*) Let `(A, A⁺)` be a complete Tate Huber pair with pseudouniformiser `ϖ`
and `A` sousperfectoid (P3.17a), so that `A⟨T₁, …, T_N⟩` and its rational
localisations are sousperfectoid, hence uniform, and `O⁺(W')` is a `ϖ`-adically complete ring of
definition of `O(W')` for every rational `W'`; the Newton arguments below use only this
uniformity, but the targets are stated, and used, over sousperfectoid bases. Let
`X = Spa(A, A⁺)`, `e = (e₁, …, e_N) ∈ A⁺⟨T₁, …, T_N⟩^N`, `k ≥ 0` and `m ≥ 2k + 1`, and let
`R ⊆ B^N_X` be a finite intersection of rational subsets `R(g_j/h_j)` with
`g_{j,1}, …, g_{j,r_j}, h_j ∈ A⁺⟨T⟩` and `ϖ^{c_j} ∈ (g_{j,1}, …, g_{j,r_j}, h_j)·A⁺⟨T⟩` for some
`c_j < m − k` (the empty intersection being `B^N_X`), so that `R` is unchanged when a point is
moved by `ϖ^{m−k}·A⁺⟨T⟩` (AdicSpaces layer 3, perturbation invariance; `tubeSet` is the tube in
the whole ball, and `R` enters the proposed Lean statements through the hypothesis on the rational
subset `W`). Let `Δ := det(∂eᵢ/∂T_j) ∈ A⁺⟨T⟩` and
`W := R ∩ {|eᵢ| ≤ |ϖ^m| for all i} ∩ {|ϖᵏ| ≤ |Δ|}`, a rational subset, the [*tube*](../AdicSpacesPartIII/README.md) (AdicSpacesPartIII R2.4) of `e` of
radius `|ϖ|^m`, on which the Jacobian is bounded below by `|ϖᵏ|`. Put
`C := O(W)/closure(e·O(W))` with `C⁺` the integral closure of the image of `O⁺(W)`, and
`V(e) ∩ W := Spa(C, C⁺)`. Prove (a) `tube_product_equiv_sousperfectoid` (its section clause alone
is `exists_section_tube_sousperfectoid`, for a single `R(g/h)`), (b) `tube_product_restrict` and
(c) `tube_quotient_baseChange`: (a) there is a unique isomorphism of complete Huber pairs over
`(A, A⁺)`

`Φ : (C⟨u₁, …, u_N⟩, C⁺⟨u⟩) → (O(W), O⁺(W)),   Φ(uᵢ) = ϖ^{−m}eᵢ,`

such that `Φ` restricted to `C` is a ring section `ρ` of the quotient map `q : O(W) → C` with
`ρ(q(T)) − T ∈ ϖ^{m−k}O⁺(W)^N`; geometrically `W ≅ (V(e) ∩ W) ×_X B^N_X(|ϖ|^m)` with `e`
corresponding to `ϖ^m u` (the section `ρ` is the Newton solution `t*` of `e(t*) = 0` starting
from `t = T`, and the point `T ↦ t*` lies in `W` because `|Δ(t*) − Δ| ≤ |ϖ|^{k+1} < |ϖᵏ| ≤ |Δ|`
and because `R` is stable under the move; the inverse of `Φ` sends `T` to the Newton solution
of `e(τ) = ϖ^m u` over `C⟨u⟩`); (b) the same holds, with `C` replaced by
`C_{W'} := O(W')/closure(e·O(W'))`, for every rational subset `W' ⊆ W` which is the preimage of a
rational subset `V' ⊆ V(e) ∩ W` under the projection `W → V(e) ∩ W` of (a), and for
`W' = W ∩ R'` with `R'` subject to the same condition as `R` (this second case is (a) for the
family `R ∩ R'`, and is covered by `tube_product_equiv_sousperfectoid`); it does not hold for an
arbitrary rational `W' ⊆ W` (*test_small_ball_not_product*, *test_section_fails_off_locus*); (c) for
every rational `W' ⊆ W` the ideal `e·O(W')` is closed (`isWeaklyRegular_tube_sousperfectoid`),
`O(W')/(e) = C_{W'} = C ⊗̂_{O(W)} O(W')`, and `V(e) ∩ W'` is the rational subset of `V(e) ∩ W` cut
out by the same inequalities (for `W' ∩ V(e) = ∅` both sides are the zero ring).
Integrality of `e` as a power series in `T` is essential: the Newton step needs integral Taylor
coefficients, and for `e ∈ O⁺(R)` involving the rational coordinates of `R` the statement fails;
equations with rational coordinates are handled by adjoining those coordinates as variables
([*standard-presentation-basic-pieces*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2) (c)). This is the analytic-algebra replacement of
Fargues–Scholze's local statement that smooth spaces are balls near sections ([FS] §IV.4.1, before
Lem. IV.4.14, p. 137; proof of Lem. IV.4.16, p. 139, which establishes the local product
structure after reduction modulo `ϖ`, here obtained directly by Newton iteration); the
perturbation argument of [ECD] §6, proof of Prop. 6.4(iv), p. 28, is carried out inside such a
tube. Prove clauses (a)–(b) first from Newton's theorem, independently of
`isWeaklyRegular_tube_sousperfectoid`. Then prove [*fs-iv-4-13-regular-sequence*](#p5-13-1) (P5.13.1) using only those
two tube clauses and its listed inputs. Finally prove clause (c) from that regular-sequence
theorem. *Needs for (a)–(b):* [*restricted-power-series-newton*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2); P3.17a
(sousperfectoid rings, stably uniform); AdicSpacesPartII layer R0 (completed tensor products, rational localisation,
continuous differentials); Tau Ceti
`TauCeti.ValuationSpectrum.exists_mem_nhds_forall_rationalSubset_eq_of_sub_mem`,
`TauCeti.ValuationSpectrum.existsUnique_continuous_ringHom_of_forall_comap_mem_rationalSubset`,
`TauCeti.Huber.existsUnique_continuous_ringHom_completion_weightedRestrictedSubring`,
`Huber.PairOfDefinition.completionLocalization`, `TauCeti.ValuationSpectrum.mem_iff_forall_vle_one`,
`ValuationSpectrum.rationalSubset`. *Needs for (c):* clauses (a)–(b);
[*fs-iv-4-13-regular-sequence*](#p5-13-1) (P5.13.1); AdicSpacesPartII layer R0 (rational base change of completed
quotients). README only (no Lean declaration).

**Checks.**

- *tube_integral_certificate* — For `F = 4T` over `Q₂`, `tubeSet` with `k = 0` is empty, while `k = 2` gives a nonempty tube; the integral Jacobian bound detects the difference.
- *test_translation* — `N = 1`, `R = B¹_X`, `e = T − a` with `a ∈ A⁺`, `k = 0`, `m = 1`:
  `W = {|T − a| ≤ |ϖ|}` (the check *tube_linear*, for every radius `|ϖ|^m`), `C = A` and `Φ` is
  the translation `u ↦ ϖ^{−1}(T − a)`.
- *test_unit_jacobian_product* — for `R = B^N_X`, `k = 0` and `e` with `Δ ≡ 1 modulo (e₁, …, e_N)·A⁺⟨T⟩` (so `|Δ| = 1` on `{|e| ≤ |ϖ|}`), the tube is `W = {|e| ≤ |ϖ|}` and
  `W ≅ V(e) ×_X B^N(|ϖ|)`.
- *test_small_ball_not_product* (non-example for (b) with arbitrary `W'`) — `A = ℚ_p`, `N = 1`,
  `e = T`, `k = 0`, `m = 1`, `W = {|T| ≤ |p|}`, `C = ℚ_p`, and `W' = {|T| ≤ |p|²} ⊆ W`:
  `C_{W'} = ℚ_p`, but `O(W') = ℚ_p⟨T/p²⟩` is not `ℚ_p⟨u⟩` with `u ↦ T/p`, since the image of
  `ℚ_p⟨u⟩` is `ℚ_p⟨p·(T/p²)⟩ ⊊ ℚ_p⟨T/p²⟩`. Writing `v = T/p²`, the series
  `∑ p^n v^n` has no restricted preimage, whereas `v` itself is the image of `p⁻¹u`; equally, with `R = {|T| ≤ |p|^{10}}` (here
  `c = 10 ≥ m − k = 1`) the claim (a) fails for `W = R`.
- *test_section_fails_off_locus* (non-example) — same data, `W' = {|T − p| ≤ |p|²} ⊆ W`: `T` is a
  unit on `W'` (`|T| = |p|` there), so `C_{W'} = 0` while `O(W') ≠ 0`, and no ring section
  `C_{W'} → O(W')` of the quotient map exists; the Newton point `t* = 0` of `e = T` does not lie
  in `W'`.
- *test_integral_coefficients_needed* (non-example) — `A = ℚ_p`, `p` odd, `ϖ = p`, `N = 1`,
  `R = {|T| ≤ |p|}`: `e = (T/p)²` lies in `O⁺(R)` but not in the image of `ℤ_p⟨T⟩`,
  `Δ = 2T/p²`, and for `k = 0`, `m = 2` the set `W = {|T| = |p|²}` is a nonempty rational subset on
  which `e` is a unit, so `C = 0` while `O(W) ≠ 0` and no isomorphism `C⟨u⟩ ≅ O(W)` exists
  (when formalised, the check presents `W` as `R({T², p²T, p⁴}/p²T)` and records that `T` is a unit of the
  nonzero ring `O(W)`).

The proposed theorem `tube_product_restrict` applies to the product isomorphism `Φ` and a rational
localisation `C → C'`. Its numerators in `O(W)` are exactly the images under the coefficient
section `C → O(W)`. It produces `C'⟨u⟩ ≅ O(W')`, commuting with coefficients and labelled
coordinates and preserving plus rings. The proposed theorem `tube_quotient_baseChange` allows any
rational inclusion `W' ⊆ W` inside the tube. It states closedness of both equation ideals,
identifies their images under `ratLocRestriction`, and identifies the completed tensor
quotient with `O(W')/(e)`, including the plus ring. Its formula on pure tensors fixes the
comparison map. Together with rational base change, these statements identify the
restricted zero locus and its quotient pair.

**Stability of standard étale presentations under perturbation.**
(*standard-presentation-perturbation*) Let `(N, F, 2k, π)` be a standard étale presentation of
`(C, C⁺)` in explicit-inverse form with exponent `k`, `F = (F', S·J_{F'} − ϖᵏ)`, over a complete
Tate pair `(A, A⁺)` with `A` sousperfectoid (as in [*tubular-neighbourhood*](#p5-13-1) (P5.13.1): the argument uses the
uniformity of the rational localisations of the Tate algebras over `A`), and put
`m₀ := 4k + 2`. For every
`G ∈ A⁺⟨T⟩^N` with `G − F ∈ ϖ^{m₀}A⁺⟨T⟩^N` coefficientwise, prove: (a) the quotient pair
`(C_G, C_G⁺) := A⟨T⟩/closure(G)` has the standard étale presentation `(N, G, 2k, π_G)`, the
Jacobian of `G` being invertible on `V(G)` with inverse bounded by `|ϖ|^{−2k}`; (b) `V(G)` and
`V(F)` both lie in the tube `W` of `F` with Jacobian exponent `2k` and radius `|ϖ|^{4k+1}`, and
there is a unique isomorphism of complete Huber pairs `α_{F,G} : (C, C⁺) → (C_G, C_G⁺)` over
`(A, A⁺)` with `α_{F,G}(π(T)) − π_G(T) ∈ ϖ^{m₀−2k}(C_G⁺)^N`, and it is already the only continuous
`A`-algebra map `C → C_G` with this congruence; (c)
if `G = (G', S·J_{G'} − ϖᵏ)` is itself in
explicit-inverse form with exponent `k` (which is the case when `G' − F' ∈ ϖ^{m₀}A⁺⟨T'⟩^{N−1}` and
the last equation is rebuilt from `G'`), then `α_{G,F}` is defined and `α_{G,F} = α_{F,G}^{−1}`, the
comparison maps being mutually inverse on the permitted tube with no further shrinking,
`α_{G,H}∘α_{F,G} = α_{F,H}` whenever all three are defined, and `α` is compatible with base change
along morphisms of complete Tate pairs, all by the uniqueness in (b); (d) plus rings correspond,
`α(C⁺) = C_G⁺`. In the proposed Lean form, for `F = P.withInverse.F` the explicit-inverse form of
a presentation `P`, `α_{F,G}` is `StandardEtalePresentation.perturb`, and (a), (b) and (d), with the
uniqueness, are its specification over a sousperfectoid base; a perturbed system that is itself in
explicit-inverse form presents its quotient pair by `nonempty_ofExplicitInverse`.
The inverse is characterised by the reverse coordinate congruence: every continuous reverse
comparison satisfying it is that inverse. The composite
with any continuous, plus-preserving nearby comparison `G → H` is the chosen comparison
`F → H`. Taking the reverse or intermediate comparison supplied by `perturb` on an
explicit-inverse system gives `α_{G,F} = α_{F,G}⁻¹` and the cocycle identity without shrinking.
The base-change square on rings commutes for
compatible presentations over sousperfectoid source and target bases, fixing the maps
on coefficients and labelled coordinates; the coordinate formula of `baseChange` supplies these
equations for completed base change. Hence
`Spa(C_G, C_G⁺) ≅ Spa(C, C⁺)` over `X`, and every property of `V(F)` over `X` (étale,
sousperfectoid, perfectoid) holds for `V(G)`. If `A` has a noetherian ring of definition this
agrees with Huber's 1.7.2 (AdicSpacesPartII layer R0, [*etale-presentation-perturbation*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.17); the
closure-of-the-ideal form over a non-noetherian base is the content here) ([ECD] §6,
proof of Prop. 6.4(iv), p. 28: slightly perturbed equations still define an affinoid
sousperfectoid space étale over `X`, and close approximations are isomorphic to `Y`, here through
`α_{F,G}`; [Huber96] §1.7, Cor. 1.7.2(i), p. 85, gives neighbourhood-wise perturbation
stability for Huber’s affinoid rings; the explicit exponent here is additional). *Needs:* [*standard-etale-presentation*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2);
[*perturbation-of-generators*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2); [*tubular-neighbourhood*](#p5-13-1) (P5.13.1); AdicSpacesPartII layer R0; Tau Ceti
`TauCeti.ValuationSpectrum.existsUnique_continuous_ringHom_of_forall_comap_mem_rationalSubset`,
`TauCeti.ValuationSpectrum.exists_mem_nhds_forall_rationalSubset_eq_of_sub_mem`. README only (no Lean declaration).

**Checks.**

- *test_perturb_linear_identity* — `F = (T − a, S − 1)` with `a ∈ A⁺` (explicit-inverse form with
  `k = 0`) and `G = (T − a', S − 1)` with `a − a' ∈ ϖ²A⁺`: `C = C_G = A` and `α_{F,G}` is the
  identity of `A`; its coordinate error `α(π(T)) − π_G(T) = a − a'` is not zero and uses the full
  allowance `ϖ^{m₀−2k} = ϖ²` (when formalised: for any presentation with an equation `T − a`).
- *perturb_zero* (degenerate) — `G = F` gives `α_{F,F} = quotientEquiv⁻¹`, the identification of
  `C` with its own quotient presentation.
- *perturb_plus* — For `Fᵢ = Tᵢ² − Tᵢ` and `Gᵢ = (Tᵢ − δ)² − (Tᵢ − δ)`, where `δ = ϖ^(4k+2)`, the perturbation equivalence sends the old idempotent to the integral idempotent `Tᵢ − δ` and excludes its complement over a nonzero target.
- *perturb_coordinate* — For the same shifted idempotent equation, the coordinate error is `−δ` and differs from `1 − 2Tᵢ + δ`, the error obtained by exchanging the two roots.
- *test_perturb_rational* — perturbing `s, t` in `(tT − s)` inside `ϖ^{m₀}` does not change
  `R(s/t)` (Tau Ceti `exists_mem_nhds_forall_rationalSubset_eq_of_sub_mem`), consistent with (b).

**Standard étale loci over sousperfectoid bases are sousperfectoid affinoids.**
(*standard-etale-locus-sousperfectoid*) Let `(A, A⁺)` be a complete Tate Huber pair with `A`
sousperfectoid (P3.17a; for instance perfectoid, or a rational localisation of
`A'⟨T⟩` with `A'` perfectoid) and `(N, F, k, π)` a standard étale presentation of `(C, C⁺)` over
`(A, A⁺)`. Prove that `C` is sousperfectoid (`StandardEtalePresentation.isSousperfectoid`): if
`W` is the tube of `F` (*tubular-neighbourhood*),
`ρ : C → O(W)` the section and `q : O(W) → C` the quotient, so that `C` is a `C`-linear direct
summand of the sousperfectoid `O(W)`, and `(P̃, ι, σ)` is a perfectoid frame of `O(W)`, then
`(P̃, ι∘ρ, q∘σ)` is a perfectoid frame of `C`. Consequently `(C, C⁺)` is sheafy and stably
uniform, `Spa(C, C⁺)` is a sousperfectoid affinoid adic space (P3.17a,
sousperfectoid sheafy, stably uniform, sousperfectoid adic space), and for every rational
`U ⊆ Spa(C, C⁺)` the ring `O(U)` is sousperfectoid with a standard étale presentation over
`(A, A⁺)` ([*standard-presentation-basic-pieces*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2) (d)). This is the étale case of the
sousperfectoid clause of [FS] §IV.4.1, Prop. IV.4.17, p. 139 (vanishing loci of functions with
independent differentials are sous-perfectoid spaces smooth over `X`), using the stability
properties of sous-perfectoid rings of [FS] §IV.4.1, p. 135. Whenever the locus is already known
to be étale over `Spa(A, A⁺)` — in characteristic `p` by [*jacobian-criterion-char-p*](#p5-13-2) (P5.13.2) — its
sousperfectoidness is P3.17a [*sousperfectoid-etale-stability*](#p3-17a-4) (P3.17a.4) (P3.17a); the direct
tube argument here is valid in every characteristic. *Needs:*
[*standard-etale-presentation*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2); [*standard-presentation-basic-pieces*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2); [*tubular-neighbourhood*](#p5-13-1) (P5.13.1);
P3.17a (sousperfectoid rings, sheafiness, stable uniformity, frames, adic
spaces). README only (no Lean declaration).

**Checks.**

- *test_field_trivial* — `A = K` a perfectoid field, `F = T` (`N = 1`): `C = K`, sousperfectoid
  trivially.
- *test_field_quadratic* — `F = T² − a` with `a ∈ (K⁺)ˣ` and residue characteristic `p ≠ 2`: `C` is a finite étale
  `K`-algebra, `J = 2T` a unit of `C` with `k = 0`. For residue characteristic `p = 2` and `char K = 0`, the same `F` is a presentation
  only with `k ≥ 1` such that `|ϖ|^k ≤ |2|`, since `ϖᵏ/(2T) ∈ C⁺` requires it; `k = 0` is
  rejected (check *test_field_quadratic_p2*). In characteristic two its derivative is zero, so no exponent works
  (the check *characteristic_two_jacobian* evaluates the Jacobian `tateSeriesJacobian` on this
  restricted-series equation).


**Checks.**

- In zero variables the locus is the original sousperfectoid base with its identity frame.

**The equations of a standard étale locus form a regular sequence with closed ideal on its tube.**
(*fs-iv-4-13-regular-sequence*) In the situation of [*tubular-neighbourhood*](#p5-13-1) (P5.13.1) (`(A, A⁺)` complete
Tate with `A` sousperfectoid, `e ∈ A⁺⟨T⟩^N`, tube `W` of `e` in `B^N_X`), prove for every rational
subset `W' ⊆ W` (`isWeaklyRegular_tube_sousperfectoid`): (a)
`e₁, …, e_N` is a weakly regular sequence on `O(W')` (Mathlib
`RingTheory.Sequence.IsWeaklyRegular`) and the Koszul complex `K(e; O(W'))` is a finite free
resolution of `O(W')/(e)`; (b) the ideal `e·O(W')` is closed and `O(W')/(e) ≅ O(V(e) ∩ W')`; (c)
`(e)/(e)²` is a free `O(V(e) ∩ W')`-module with basis the classes of the `eᵢ`; (d) consequently
`O(W)/(e)` is a stably N-fpd, hence stably pseudocoherent, `O(W)`-module and, for `N ≥ 1`, the ideal `(e)·O(W)`
is stably `(N−1)`-fpd; for `N = 0` that ideal is the zero module. The proposed Lean form states (a)–(d)
for every rational `W' ⊆ W` except the Koszul complex of (a), which has no Mathlib carrier at the
pin (the stably `N`-fpd clause is its consequence). This is the étale case of [FS]
§IV.4.1, Lem. IV.4.13, p. 137 (the coordinates of an étale map to a relative ball over a
sous-perfectoid base are regular with closed ideal), whose printed proof uses the spreading of
étale maps, a diamond-level input replaced here by tube clauses (a)–(b). This proof uses no
part of tube clause (c). *Needs:* [*koszul-regular-sequence-fpd*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1);
[*stably-pseudocoherent-module*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1); [*tubular-neighbourhood*](#p5-13-1) (P5.13.1) (a)–(b) only; Mathlib
`RingTheory.Sequence.IsWeaklyRegular`. README only (no Lean declaration).

**Checks.**

- *test_linear_regular* — `N = 1`, `e = T − a` on `B¹_X`: `T − a` is a non-zero-divisor with
  closed ideal on every rational subset of `{|T − a| ≤ |ϖ|}`, and the quotient is `A`-restricted:
  `O(W')/(T − a) = O(W' ∩ {T = a})`.


**Checks.**

- For `N = 0` the equation ideal is zero and the conormal module has rank zero.
- On a rational subset disjoint from the zero locus the quotient is zero; the asserted closed ideal is the unit ideal.


#### Pseudocoherent and finite étale specializations

Finite étale algebras over perfectoid rings admit the standard étale presentations of
AdicEtaleGeometry A3.2 and are perfectoid by P3.10 ([ECD] §6, Theorem 6.1(i), p. 26).
The sousperfectoid specializations of AdicEtaleGeometry A3.1 are
`cech_injective_of_isStablyPseudoCoherent_sousperfectoid`,
`exact_tensor_ratLoc_sousperfectoid` and `isStrictMap_mul_X_sub_sousperfectoid`:
apply its sheafy-base Čech injectivity, tensor exactness and strict Laurent multiplication
theorems with the sheafiness and uniformity of P3.17a. They retain AdicEtaleGeometry A3.1's stably
pseudocoherent hypotheses, ordered restriction maps and four strict multiplication maps.
*Needs:* P3.17a.1–P3.17a.3; AdicEtaleGeometry A3.1
([*stably-pseudocoherent-module*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1), [*rational-localisation-pseudoflat*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1),
[*simple-laurent-strict-multiplication*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1)) and AdicEtaleGeometry A3.2 (*standard-etale-presentation*).

**Checks.**

- Over a perfectoid field, the regular module has the supplied augmented Čech exactness.
- Over `ℚ_p⟨T⟩`, these specializations recover the strongly noetherian flat restriction maps.
- The zero module gives zero maps and an exact zero Čech complex.
- A nonzero constant sequence does not tend to zero in a Hausdorff coefficient ring,
  so it cannot be used as the null coefficient sequence in the strict Laurent argument.
<a id="p5-13-2"></a>

- *test_valuationRing_residue* (non-example; the degree shifts are sharp) — for a valuation ring
  `O` whose maximal ideal `𝔪` is not finitely generated, such as the valuation ring of `ℂ_p` or of
  any perfectoid field, `O/𝔪` is 0- but not 1-pseudocoherent, and in `0 → 𝔪 → O → O/𝔪 → 0` the
  middle term is pseudocoherent while `𝔪` is not even 0-pseudocoherent. So (c) at `m = 0` fails
  when `M₂` is only 0-pseudocoherent, and (b) at `m = 1` fails without the hypothesis on `M₁`. For
  `0 → ℤ →(×2) ℤ → ℤ/2 → 0` the projective dimensions are `0, 0, 1` (check *fpd_not_projective*), so
  (e) cannot conclude that `M₂` is m-fpd.
- *test_not_stably* (non-example) — [KL16] §2.4, Ex. 2.4.2, pp. 35–36: an affinoid perfectoid
  `(A, A⁺)` and a non-zero-divisor `f` with `fA` closed, so `A/fA` strictly pseudocoherent, but
  `fB` not closed for some rational localisation `B`, so `A/fA` is not stably pseudocoherent. A
  definition using only completeness of `M` itself fails this test.
- *test_disc_localisation* — for `A = ℚ_p⟨X⟩` and `f = X/p`: `B₁ = ℚ_p⟨X, T⟩/(T − X/p) = ℚ_p⟨X/p⟩`, the ring of `U₁ = {|X| ≤ |p|}`, and `B₂ = ℚ_p⟨X, T⟩/(1 − XT/p) = ℚ_p⟨X, p/X⟩`, the
  ring of `U₂ = {|X| ≥ |p|}`: on `U₂` the element `T = p/X` is the power-bounded inverse of `f`.
  For `f = 1/p ∈ ℚ_p` (not power-bounded), `T − 1/p = −p^{−1}(1 − pT)` is a unit of `ℚ_p⟨T⟩`, so
  `B₁ = 0 = O(∅)`, as `U₁ = {|1/p| ≤ 1} = ∅`. When formalised, the checks identify `B₁` with the rational
  localisation at `R({X, p}/p)` and `B₂` with that at `R({X, p}/X)`.

### P5.13.2 the Fargues–Scholze specialisations by analytic algebra

The étale specialisations of [FS] §IV.4.1, pp. 135–141, are reconstructed by analytic algebra.
The order is: tube clauses (a)–(b) give regular sequences and finite projective dimension,
which then give tube clause (c); the characteristic-`p` Jacobian criterion supplies the étale parts of IV.4.15 and IV.4.17;
pseudocoherent gluing and the integral cut-off then give the closed-image and conormal statements
used in IV.4.19. The arguments of IV.4.14, IV.4.16 and IV.4.18 involving diamonds are not inputs.
The specialisations are exactly those [ECD] 6.4(iv) uses: étale (relative dimension `0`) loci and
closed immersions of affinoid étale perfectoid spaces into rational subsets of relative balls over
affinoid perfectoid bases; the smooth cases over general sousperfectoid bases are outside this
construction.

**A perfectoid Tate ring of characteristic `p` is an algebra over a perfectoid field and a
completed colimit of `p`-finite algebras.** (*char-p-perfectoid-base-field*) Let `(R, R⁺)` be a
perfectoid Tate pair of characteristic `p` (layer P1: `R` a perfect complete
Tate ring) and `ϖ ∈ R⁺` a pseudouniformiser (Tau Ceti `Huber.IsPseudoUniformizer`); no base
field is assumed, it is constructed. The compatible roots `ϖ^{1/pⁿ}` of the given `ϖ`, each again
a pseudouniformiser, are `exists_pow_roots_pseudoUniformizer_charP`. Let `K := F_p((t^{1/p^∞}))^∧` be the `t`-adic completion of
the perfection of `F_p((t))`, a perfectoid field (layer P1). Prove: (a) there is
a unique continuous ring homomorphism `K → R` with `t^{1/pⁿ} ↦ ϖ^{1/pⁿ}`, the unique `pⁿ`-th
roots in the perfect ring `R`; it is injective when `R ≠ 0` and maps `K° = F_p[[t^{1/p^∞}]]^∧`
into `R⁺`, so `(R, R⁺)` is a perfectoid affinoid `K`-algebra with `R⁺` a `K°`-algebra; (b)
consequently `(R, R⁺)` is the completion of a filtered colimit of `p`-finite perfectoid affinoid
`K`-algebras `(R_α, R_α⁺)`, each the completed perfection of a reduced affinoid `K`-algebra
`(S_α, S_α⁺)` of topologically finite type ([Sch12] §6, Lem. 6.13(i), p. 37, under the hypothesis
supplied by (a); `p`-finite is [Sch12] §6, Def. 6.9, p. 35; layer P2, completed
direct limits of `p`-finite affinoids), and `R⁺/ϖ^M = colim_α colim_r S_α^{+,1/p^r}/ϖ^M` for
every `M`, so finitely many elements of `R⁺` are approximated modulo `ϖ^M` by elements of one
`S_α^{1/p^r} ≅ S_α`, again a reduced `K`-affinoid algebra of topologically finite type, hence
strongly noetherian. *Needs:* layers P1 and P2; Mathlib `PerfectRing`,
`frobenius`; Tau Ceti `Huber.IsPseudoUniformizer`.

**Checks.**

- *test_base_field_identity* — `R = K` itself: the map is the identity.
- *test_perfectoid_disc_pfinite* — `R = K⟨T^{1/p^∞}⟩` is `p`-finite, the completed perfection of
  `K⟨T⟩`. The inclusion has the Gauss topology, roots of integral elements are dense in the
  integral subring, and each positive root `T^{1/pⁿ}` lies outside the original disc algebra.


**Checks.**

- A constant system of characteristic-p perfectoid fields recovers its coefficient field and identity completed colimit.

**Jacobian criterion over characteristic-`p` perfectoid bases.** (*jacobian-criterion-char-p*)
Let `(A, A⁺)` be a perfectoid Tate pair of characteristic `p`, `X = Spa(A, A⁺)`, and
`(B, B⁺) := (O(Q), O⁺(Q))` for a rational subset `Q ⊆ B^m_X` (`m ≥ 0`; `m = 0` and `Q = X` give
`(B, B⁺) = (A, A⁺)`; `B` is sousperfectoid, P3.17a). Let `(N, F, k, π)` be a
standard étale presentation of `(C, C⁺)` over `(B, B⁺)`, put in explicit-inverse form first.
Prove: (a) `Spa(C, C⁺) → Spa(B, B⁺)` is étale in the local-description sense
([*etale-morphisms-local-description-and-comparison*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.4) of AdicEtaleGeometry layer A1: every point has an open
neighbourhood on which the map factors as an open immersion, a finite étale morphism and an open
immersion); (b) if `m = 0`, then `C` is a perfectoid Tate ring (`C` is perfect), `Spa(C, C⁺)` is
an affinoid perfectoid space and `Spa(C, C⁺) → X` is étale in the sense of [ECD] Def. 6.2
(layer P3); the ring part is that `C` is perfect and a perfectoid Tate ring
(PerfectoidSpaces' `Perfectoid.IsPerfectoidTateRing`); (c) a morphism `g : U → V` over `X` between rational subsets
`U ⊆ B^n_X` and `V ⊆ B^n_X` such that `det(∂gᵢ/∂T_j)` is a unit of `O(U)` with bounded inverse is
étale in the local-description sense (the étale case of [FS] §IV.4.1, Prop. IV.4.15(ii), p. 138:
an isomorphism on differentials gives étaleness; Prop. IV.4.17, p. 139: vanishing loci with
independent differentials, étale when `r = n`). The proof approximates the finitely many
coefficients by a strongly noetherian `K`-affinoid algebra (*char-p-perfectoid-base-field*), uses
Huber's noetherian theory (AdicSpacesPartII layer R0: étale local structure, the finite étale
embedding of an étale affinoid, the local open–finite étale factorisation), base changes along
the sousperfectoid `Spa(B)` (P3.17a, sousperfectoid étale base change) and
perturbs back (*standard-presentation-perturbation*) — the strategy of [Sch12] §7, Prop. 7.7, p. 41
(in characteristic `p` étale maps of perfectoid spaces are locally base changes of étale maps of
noetherian adic spaces); [ECD] §6, proof of Prop. 6.4(iv), p. 28, uses that the perturbed locus is
étale over `X` and thus perfectoid. No diamond, no [ECD] 11.30 or 15.6 and no v-descent is used:
the printed proofs of IV.4.15 and IV.4.17 rest on [ECD] Lem. 12.17 (the deformation step, [FS]
§IV.4.1, proof of Prop. IV.4.15, p. 138), on Lem. IV.4.14 and on Lem. IV.4.16 (descent to
strictly totally disconnected bases). *Needs:* [*standard-etale-presentation*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2);
[*standard-presentation-perturbation*](#p5-13-1) (P5.13.1); [*standard-etale-locus-sousperfectoid*](#p5-13-1) (P5.13.1);
[*char-p-perfectoid-base-field*](#p5-13-2) (P5.13.2); [*etale-morphisms-local-description-and-comparison*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.4) (AdicEtaleGeometry layer A1);
AdicSpacesPartII layer R0 (Huber 1.7.1, étale local structure, completed tensor products);
P3.17a (sousperfectoid étale base change, sheafiness);
layers P1–P3; Tau
Ceti `TauCeti.ValuationSpectrum.exists_mem_nhds_forall_rationalSubset_eq_of_sub_mem`; Mathlib
`PerfectRing`. README only (no Lean declaration).

**Checks.**

- *test_rational_finite_etale_pieces* — rational subsets of `X` and finite étale algebras over
  rational subsets (*standard-presentation-basic-pieces*) give étale affinoid perfectoid spaces,
  as they must ([ECD] §3, Thm. 3.18 (rational subsets) and §6, Thm. 6.1(i) (finite étale
  algebras)).
- *test_frobenius_locus_not_etale* (non-example) — `F = T^p − a` has `J_F = 0` and is not a
  standard étale presentation; when the quotient is nonzero, its locus `Spa(A⟨T⟩/(T^p − a))` is not étale over `X`.


**Checks.**

- In zero variables the Jacobian is the empty determinant 1 and the locus is the original base.

**Comparison of a graph neighbourhood with the original target and its quotient modules.**
(*graph-neighbourhood-target-comparison*) Let `(A, A⁺)` be a complete sousperfectoid Tate pair
with pseudouniformiser `ϖ`, `X = Spa(A, A⁺)`, `R ⊆ B^N_X` rational, `B = O(R)`,
`B⁺ = O⁺(R)`, and `q : (B, B⁺) → (C, C⁺)` a quotient mapping to a complete Huber pair
over `(A, A⁺)`. Suppose `(C, C⁺)` has a standard
étale presentation `(n, H, k, π_c)` with **exact** lifts `w ∈ (B⁺)^n`,
`π_c(v_l) = q(w_l)`. Write `Y = Spa(C, C⁺)`, `g : Y → R`,
`E = B ⊗̂_A C`, and `p : Spa(E, E⁺) = R ×_X Y → R`; `E⁺` is the completed tensor pair's
plus ring. The scalar map is `B → E`, `b ↦ b ⊗ 1`, and the graph map on rings is
`E → C`, `b ⊗ c ↦ q(b)c`. Its equations are
`h_i = T_i ⊗ 1 − 1 ⊗ q(T_i)` for `1 ≤ i ≤ N`. Prove:

(a) There is a finite rational cover `{U_j}` of `R` such that each member is either disjoint
from `g(Y)`, or admits a rational `W_j ⊆ Spa(E, E⁺)` containing the graph over
`Y_j := g⁻¹U_j`, for which `p|_{W_j} : W_j → U_j` is an isomorphism of adic spaces,
inducing an isomorphism of complete Huber pairs. In particular the induced scalar map
`B_j := O(U_j) → D_j := O(W_j)` is an isomorphism; this is the scalar map used to transfer
the graph resolution.
The neighbourhood `W_j` lies in a graph tube over `C` for the integral equations `h_i`,
whose Jacobian in the `T` coordinates is the identity. Choose its radius small enough that
the rational inequalities defining `R ×_X Y` are preserved by translation from the graph.

Construct these comparisons by quantitative Newton iteration, not by restriction of scalars
along `B → E`. On sufficiently small rational neighbourhoods of `g(Y)`, impose
`H(w) ∈ ϖ^m O⁺(U_j)^n` and
`ϖ^k/det J_H(w) ∈ O⁺(U_j)`, with `m ≥ 2k + 1`; take `m` larger as needed for the finitely
many rational inequalities. These conditions hold on `Y` since `H(qw) = 0` and the standard
presentation bounds the inverse Jacobian. Newton supplies `r ∈ O⁺(U_j)^n` with `H(r) = 0`
and `r − w ∈ ϖ^{m−k}O⁺(U_j)^n`, hence a pair map `C → B_j`, `π_c(v) ↦ r`.
In `p⁻¹U_j`, the rational inequalities
`|v_l − w_l| ≤ |ϖ^{k+1}|` select this branch: both `v` and the image of `r` solve `H = 0`
in that congruence class, so [*restricted-power-series-newton*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2) identifies them. Thus this
branch projects isomorphically to `U_j`. Shrink it inside a graph tube, carrying the
shrinking across this isomorphism to `U_j`. The graph is retained because Newton
functoriality along the ambient pair map to `O(Y_j)` gives `q(r) = q(w)`.
The quotient-pair hypothesis identifies `g(Y)` with the closed vanishing set of `ker q`,
so rational subsets disjoint from it complete the cover. Compactness gives a finite cover.

(b) On every rational `V ⊆ U_j`, transfer the Koszul resolution of the graph equations
through `O(V) ≅ O((p|_{W_j})⁻¹V)`. Its quotient is
`C_V := O(g⁻¹V)`, with plus ring the integral closure of the image of `O⁺(V)`.
The equation ideal is closed, the quotient has its natural Hausdorff complete topology,
and the transferred resolution and its kernel give stably pseudocoherent modules over
`B_j`. These identifications commute with rational restriction: restrict `W_j` by the
inverse image under its isomorphism with `U_j`, use [*fs-iv-4-13-regular-sequence*](#p5-13-1) (P5.13.1) on that
rational subchart, and identify both quotient maps with the given map to `C_V`.
For members disjoint from `g(Y)`, the quotient module is zero and the kernel is `B_j`.

(c) Glue these quotient modules and their kernels as sheaves on `R`. Their identifications
with `g_*O_Y` and `ker(O_R → g_*O_Y)` fix the overlap maps. Apply
[*pseudocoherent-kiehl-gluing*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1) to these locally stably pseudocoherent modules, rather than
assuming that an algebraic tensor product of `C` is already complete. Global sections
are `C` and `I := ker q`, so both are stably pseudocoherent **B-modules**. For every rational
`V ⊆ R`, `B' = O(V)`, prove the algebraic comparisons

`0 → I ⊗_B B' → B' → C ⊗_B B' → 0, C ⊗_B B' ≅ C_V = B'/I B' ≅ C ⊗̂_B B'.`

The maps are multiplication and restriction, `I ⊗_B B'` identifies with the closed ideal
`ker(B' → C_V)`, and both algebraic tensor modules are Hausdorff complete for their natural
`B'`-module topologies. The quotient comparison is a homeomorphism; its plus ring is the
integral closure of the image of `O⁺(V)`. These assertions follow from gluing and rational
pseudoflatness after stable pseudocoherence has been obtained, and include the empty preimage.

This target supplies the analytic graph comparison underlying the étale-source case of
[FS] §IV.4.1, Lem. IV.4.14, p. 137, and Prop. IV.4.19, p. 140, using Newton iteration in place
of their spreading and differential arguments; the module passage is [KL16] §2.5,
Thm. 2.5.5 and Cor. 2.5.6, pp. 42–43. It does not require `E` to be finite or pseudocoherent
over `B`. *Needs:* [*restricted-power-series-newton*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2); [*standard-etale-presentation*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2);
[*standard-etale-locus-sousperfectoid*](#p5-13-1) (P5.13.1); [*tubular-neighbourhood*](#p5-13-1) (P5.13.1);
[*fs-iv-4-13-regular-sequence*](#p5-13-1) (P5.13.1); [*pseudocoherent-sheaf*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1); [*pseudocoherent-kiehl-gluing*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1);
[*rational-localisation-pseudoflat*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1); AdicSpacesPartII layer R0 (completed tensor pairs and
quotient pairs); P3.17a (rational sousperfectoid algebras and balls);
Tau Ceti `Pair.quotient`, `TauCeti.Huber.Pair.Hom.isClosedEmbedding_spaComap_quotientHom` and rational
perturbation invariance. README only (no Lean declaration).

**Checks.**

- *test_graph_comparison_nonfinite_ambient* — `K` a perfectoid field, `A = K⟨s⟩`,
  `C = K⟨s,s⁻¹⟩ = A⟨v⟩/(sv−1)`, `B = A⟨T⟩`, `q(T) = v`, `w = T`:
  `E = B⟨1/s⟩` is not finite over `B`. Indeed `B → E` is injective and `s` is a nonunit
  of `B`; integrality of `1/s` would make it a unit. The graph equation `T−v` over `E`
  has quotient `C`, whereas over `B` the resolution is
  `0 → B → B → C → 0`, with differential `sT−1`. On
  `U = {|sT−1| ≤ |ϖ|}`, both `s` and `T` have absolute value one; Newton gives the integral
  root `r = 1/s` and the branch `|v−T| ≤ |ϖ|` projects isomorphically onto `U`.
- *test_graph_comparison_section* — `C = A`, `R = B^N_X`, and `q(T_i) = a_i ∈ A⁺`:
  `E = B`, the projection comparison is the identity on a tube around the section,
  and the graph Koszul resolution is the one for `T_i−a_i`; it gives the original-base
  quotient and kernel after every rational restriction.
- *test_graph_comparison_empty* — on a rational `V` disjoint from `g(Y)`,
  `C ⊗_B O(V) = C_V = 0` and `I ⊗_B O(V) → O(V)` is an isomorphism;
  when `N = n = 0`, `C = A`, `R = X` and `q = id`, the comparison instead has zero kernel.

**Local presentations of a closed subspace from integral cut-offs and graphs; pseudocoherence of
its ideal.** (*integral-cutoff-graph-presentation*) Let `(A, A⁺)` be a complete Tate Huber pair
with `A` sousperfectoid, so that rational subsets of relative balls are sheafy, `X = Spa(A, A⁺)`,
`R ⊆ B^N_X` a rational subset with coordinates `T₁, …, T_N`, and `(C, C⁺)` a complete Huber pair
with a standard étale presentation `(n, F, k, π_v)` over `(A, A⁺)` in variables `v₁, …, vₙ`. Let
`g : Spa(C, C⁺) → R` be a morphism over `X`, i.e. a continuous `A`-algebra map `g* : O(R) → C`
with `g*(O⁺(R)) ⊆ C⁺`. Suppose given `w₁, …, wₙ ∈ O(R)`, not necessarily power-bounded, monic
polynomials `P_l(Z) ∈ O⁺(R)[Z]` and `M ≥ 1` such that (i) `c_l := g*(w_l)` satisfies
`c_l − π_v(v_l) ∈ ϖ·π_v(A₀⟨v⟩)` for a ring of definition `A₀ ⊆ A⁺`, and (ii)
`g*(P_l(w_l)) ∈ ϖ^{M+1}C⁺` for all `l`. Put `R'' := R ∩ ⋂_l {|P_l(w_l)| ≤ |ϖ^M|}` and
`Q_l := R ∩ {|P_l(w_l)| ≥ |ϖ^M|}`. Prove: (a) `{R''} ∪ {Q_l}` is a finite rational covering of
`R`, `g` factors through `R''` and `g⁻¹(Q_l) = ∅`; (b) `w_l ∈ O⁺(R'')` for all `l` — a monic
polynomial with bounded coefficients is large where its argument is; (c) `g* : O(R'') → C` is a
quotient mapping of complete Huber pairs, and for every rational `R''' ⊆ R''` the induced
`O(R''') → O(g⁻¹R''') = C ⊗̂_{O(R'')} O(R''')` is again one. Before substituting the lifts,
use [*perturbation-of-generators*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2) to replace `π_v` by `π_c = π_v ∘ θ`, `π_c(v_l) = c_l`,
and replace the equations by `H_l = θ⁻¹(F_l)`. The inverse coordinate substitution preserves
the inverse-Jacobian bound up to an integral unit, so the exponent `k` still works.
The exact lifts `w_l ∈ O⁺(R'')` of `c_l` then give the quotient mapping in (c):
`π_c` factors through `O(R'') → C`, and its plus image makes `C⁺` integral over the image
of `O⁺(R'')`. The transported equations satisfy `g*(H(w)) = 0`.
(d) Put `B = O(R'')`, `I = ker(B → C)`. Apply [*graph-neighbourhood-target-comparison*](#p5-13-2) (P5.13.2)
to this quotient and exact-lift presentation. It gives stable pseudocoherence of `C` and `I`
over `B`, and, for every rational `B → B' = O(R''')`, an exact sequence
`0 → I ⊗_B B' → B' → C ⊗_B B' → 0` of naturally Hausdorff complete modules,
with `C ⊗_B B' ≅ O(g⁻¹R''')` and `I ⊗_B B' ≅ ker(B' → O(g⁻¹R'''))`.
The intermediate quotient `B/closure(H(w))` admits a pair map from `C` by `v ↦ w` and
a retraction to `C`, but may be larger: for `n = 0`, `C = A` and `g` a section of `R → X`,
the equation list is empty. The graph comparison supplies the full ideal, including the
remaining ambient coordinates. (e) Hence the ideal sheaf
`I := ker(O_R → g_*O_{Spa C})` is a pseudocoherent sheaf on `R`, equal to `O_R` on each `Q_l`.
The integral cut-off, possible because plus rings of rational localisations are completed integral
closures ([KL15] §2.4, Lem. 2.4.13(a), p. 42), removes the boundary specialisations (points with
`|w| = 1⁺`) that otherwise prevent the image of `Y` from being closed; the graph argument replaces
Fargues–Scholze's local identification of an étale closed immersion ([FS] §IV.4.1, proof
of Prop. IV.4.19, p. 140). Quotient mappings of Huber pairs are `Pair.Hom.IsQuotientMapping`:
surjective and open, with target plus ring the integral closure of the image of the source plus
ring (Huber's quotient mappings of affinoid rings; between complete Tate rings openness follows
from surjectivity by the open mapping theorem, and the notion is the affinoid condition of [FS]
§IV.4.1, Prop. IV.4.19(i), p. 140). *Needs:* [*pseudocoherent-module*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1); [*stably-pseudocoherent-module*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1);
[*pseudocoherent-sheaf*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1); [*pseudocoherent-kiehl-gluing*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1); [*perturbation-of-generators*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2);
[*standard-etale-presentation*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2); [*standard-presentation-basic-pieces*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2);
[*graph-neighbourhood-target-comparison*](#p5-13-2) (P5.13.2); AdicSpacesPartII layer R0; P3.17a; Tau Ceti
`TauCeti.ValuationSpectrum.mem_iff_forall_vle_one`, `ValuationSpectrum.rationalSubset`.

**Checks.**

- *test_identity_cutoff* (degenerate) — `N = n = 0`, `R = X`, `C = A` and `g = id`:
  the presentation and the list of cutoff equations are empty, `R'' = R`, the identity is a
  quotient mapping and the ideal is `0`.
  A positive-dimensional ball `C = A⟨T₁,…,T_N⟩` is not étale over nonzero `A` for `N > 0`,
  so it cannot supply the standard étale presentation required by this theorem.
- *test_surjective_application* — in [*fs-iv-4-19-zariski-closed-char-p*](#p5-13-2) (P5.13.2): `g* : O(R) → C`
  surjective, `w` lifts of the presentation variables, `P_l` monic equations of `π_v(v_l)` over
  the image of `O⁺(R)`, over which `C⁺` is integral.
- *test_cutoff_transport_equations* — `A = C = K` a perfectoid field, `F(v) = v`,
  `π_v(v) = 0`, `R = B¹_K`, `g*(T) = 0`, `w = T+ϖ`, `P(Z) = Z−ϖ`, `M = 2`:
  `R'' = {|T| ≤ |ϖ|²}` and the original `F(w) = T+ϖ` is a unit there, so its quotient
  is zero. The exact-coordinate presentation has `π_c(v) = ϖ` and `H(v) = v−ϖ`;
  `H(w) = T` cuts out `K` with its quotient plus ring and closed principal ideal.
- *test_cutoff_nonfinite_graph_ambient* — take the data of
  *test_graph_comparison_nonfinite_ambient*, with `P(Z) = Z−T` and any `M ≥ 1`:
  `P(w) = 0`, `R'' = R`, and the complementary piece is empty. The conclusion uses the
  resolution by `sT−1` over `B`, although `B ⊗̂_A C` is not finite over `B`.
- *quotient_identity*, *quotient_top*, *quotient_ideal* — the identity of a pair, the quotient by
  the unit ideal (zero ring, full plus ring) and every Tau Ceti quotient pair `Pair.quotientHom`
  are quotient mappings; the last is the agreement with Tau Ceti's `Huber.Pair.quotient`.
- *quotient_plus_too_big* (non-example) — the identity of a field `K` from `(K, K⁺₁)` to
  `(K, K⁺₂)` with `K⁺₁ ⊊ K⁺₂` is surjective and open but not a quotient mapping, since a ring of
  integral elements is integrally closed. The identity of `ℤ_p` from the discrete to the `p`-adic
  topology is continuous and bijective but not open, which the openness clause excludes.
  When formalised, the check specifies both topologies explicitly on the identity function and asserts
  continuity, bijectivity and failure of `IsOpenMap` for every prime `p`.

**Zariski closed immersions of affinoid étale perfectoid spaces into relative balls.**
(*fs-iv-4-19-zariski-closed-char-p*) Let `(A, A⁺)` be a perfectoid Tate pair, `X = Spa(A, A⁺)`,
`Y' ⊆ B^N_X` an open subspace which is a finite union of rational subsets (for instance a
rational subset or `B^N_X` itself; `Y'` is sousperfectoid and smooth over `X`), `Y` an affinoid
perfectoid space with an étale morphism `Y → X` ([ECD] Def. 6.2), and `g : Y → Y'` a morphism
over `X`. Prove that the following are equivalent: (i) `Y'` is covered by rational subsets
`V' ⊆ B^N_X` such that `V := g⁻¹(V')` is affinoid and `O(V') → O(V)` is a quotient mapping
(surjective, `O⁺(V)` the integral closure of the image of `O⁺(V')`); (ii) for every rational
subset `V' ⊆ Y'` the same holds ([FS] §IV.4.1, Prop. IV.4.19(i), p. 140; the passage (i) ⇒ (ii)
is Kedlaya–Liu's gluing, proof of Prop. IV.4.19, p. 140). If they hold, `g` is a Zariski closed
immersion ([FS] §IV.4.1, Def. IV.4.20, p. 141), and: (a) the ideal sheaf
`I_{Y⊂Y'} := ker(O_{Y'} → g_*O_Y)` is pseudocoherent ([FS] §IV.4.1, Prop. IV.4.19, p. 140); (b)
for every rational `V' ⊆ Y'`, `I(V')` is a stably pseudocoherent `O(V')`-module and
`O(V')/I(V') ≅ O(g⁻¹V')`, the *affine quotient description*, with plus ring the integral closure of
the image of `O⁺(V')`, the *plus-ring description*. The statement holds for
perfectoid `(A, A⁺)` of any characteristic; characteristic `p` is the case used in P5.13.4.
Fargues–Scholze prove IV.4.19 for smooth maps of sous-perfectoid spaces using the
diamond-dependent IV.4.15 and IV.4.17 and Kedlaya–Liu's gluing; here only the gluing is used,
together with [*integral-cutoff-graph-presentation*](#p5-13-2) (P5.13.2). *Needs:* [*pseudocoherent-sheaf*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1);
[*pseudocoherent-kiehl-gluing*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1); [*standard-etale-presentation*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2); [*integral-cutoff-graph-presentation*](#p5-13-2) (P5.13.2);
P3.17a; layer P3
(étale morphisms of perfectoid spaces); Tau Ceti `TauCeti.ValuationSpectrum.mem_iff_forall_vle_one`,
`ValuationSpectrum.rationalSubset`.

The proposed Lean form states the ring-level equivalence on each
rational chart of the ball, and a companion gives every rational
restriction its quotient map, stably pseudocoherent kernel and homeomorphic quotient-ring
comparison; the quotient-map predicate includes the integral-closure condition on plus rings.
These are the affine assertions of [FS] IV.4.19, printed p. 140 (PDF page 140). README only (no Lean declaration).

**Checks.**

- *test_rational_closed* — `Y = X(s/t)` with `sᵢ, t ∈ A⁺` and `g : Y → B^n_X`, `Tᵢ ↦ sᵢ/t`: (i)
  holds with the single member `V' = B^n_X`, since `A⟨T⟩ → A⟨s/t⟩` is surjective with kernel the
  closure of `(tTᵢ − sᵢ)` and plus ring the completed integral closure ([KL15] §2.4,
  Lem. 2.4.13(a)).


**Checks.**

- The identity embedding into the zero-dimensional relative ball has zero closed ideal.
- For a rational localization the equations `tT_i−s_i` keep their coordinate labels and closed equation ideal.

**Affinoid étale perfectoid spaces have standard rational coverings by standard étale pieces.**
(*etale-standard-piece-cover*) Let `X = Spa(A, A⁺)` be an affinoid perfectoid space and
`f : Y = Spa(B, B⁺) → X` an étale morphism ([ECD] §6, Def. 6.2(ii), p. 26: locally an open
immersion into a space finite étale over an open of `X`) with `Y` affinoid perfectoid. Prove that
there are `σ₁, …, σ_m ∈ B` generating the unit ideal such that every
`V_j := Y(σ/σ_j) = {|σᵢ| ≤ |σ_j| for all i}` is `X`-isomorphic to a rational subset of an affinoid
perfectoid space `W_j` finite étale over a rational subset `U_j ⊆ X` (`O(W_j)` finite étale over
`O(U_j)`, `O⁺(W_j)` the integral closure of `O⁺(U_j)`); consequently each `(O(V_j), O⁺(V_j))` has
a standard étale presentation over `(A, A⁺)` (*standard-presentation-basic-pieces*), and so does
every rational subset of `Y` contained in some `V_j` ([ECD] §6, proof of Prop. 6.4(ii), p. 28:
the reduction of qcqs étale maps to composites of rational subsets and finite étale maps; remark
after Def. 6.2, p. 26: composites and base changes of (finite) étale maps are (finite) étale).
*Needs:* [*standard-presentation-basic-pieces*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2); [*finite-etale-affinoid-algebra*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.2) (P3.17a) (AdicEtaleGeometry layer A1);
layer P3; Tau Ceti
`TauCeti.ValuationSpectrum.exists_span_eq_top_forall_rationalSubset_subset_of_isTateRing`,
`ValuationSpectrum.rationalSubset`.

The proposed Lean form supplies a finite unit-ideal
family and, for each standard rational member, the rational–finite étale–rational
factorization over the base, together with a standard presentation. Further rational
restriction uses (d) of [*standard-presentation-basic-pieces*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2). README only (no Lean declaration).

**Checks.**

- *test_rational_one_piece* — `Y = X(s/t)`: one piece, `V₁ = Y`.
- *test_disjoint_union_idempotent* — `Y = X' ⊔ X''` a disjoint union of two rational subsets:
  `σ = (e, 1 − e)` with `e` the idempotent.
<a id="p5-13-3"></a>

### P5.13.3 the closed embedding of an affinoid étale object


**Checks.**

- For a constant identity étale object the unit numerator gives one standard piece.

**Local plus-generators are approximated by elements integral over global plus elements and
ratios.** (*integral-approximation-of-generators*) Let `(B, B⁺)` be a complete Tate Huber pair
with pseudouniformiser `ϖ` (in P5.13.3 `B` is perfectoid or sousperfectoid); let
`σ₁, …, σ_m ∈ B` generate the unit ideal, `V := Y(σ/σ_j) ⊆ Y = Spa(B, B⁺)` and `ū ∈ O⁺(V)`.
Prove that for every `M ≥ 0` there are finitely many `β₁, …, β_r ∈ B⁺`; an element `z ∈ O⁺(V)`
with `z − ū ∈ ϖ^M O⁺(V)`; a monic polynomial `P(Z) = Z^d + Σ_{e<d} a_e Z^e` with `P(z) = 0` whose
coefficients `a_e` are polynomials in the ratios `σ₁/σ_j, …, σ_m/σ_j` with coefficients in
`ℤ[β₁, …, β_r]`; and `b ∈ B`, `s ≥ 0` with `b/σ_j^s − z ∈ ϖ^M O⁺(V)`
(`exists_integral_approx`). Since `O⁺(V)` is the closure of the integral closure of
`B⁺[σ/σ_j]` in `B[1/σ_j]` ([KL15] §2.4, Lem. 2.4.13(a), p. 42; Tau Ceti
`Huber.PairOfDefinition.completedPlusSubring`) and `ϖ^M O⁺(V)` is an open neighbourhood of `0`, an
element `z = b/σ_j^s` of that integral closure close to `ū` does it, with `P` its integral
equation; no uniformity of `B` is used. *Needs:* Tau Ceti
`Huber.PairOfDefinition.completedPlusSubring`, `Huber.PairOfDefinition.completionLocalization`,
`ValuationSpectrum.rationalSubset`.

**Checks.**

- *test_ratio_is_generator* — `B = K⟨X⟩` (`K` a perfectoid field of characteristic `p`),
  `σ = (X, ϖ)`, `j = 1`, `V = {|ϖ| ≤ |X|}`: `ū = ϖ/X` is itself the ratio `σ₂/σ₁`, with
  `P(Z) = Z − σ₂/σ₁` and `r = 0`.


**Checks.**

- For a global plus element, take that element itself and zero approximation error.
- For `ū = 0`, take `z = b = 0`, `s = 0`, `P(Z) = Z` and no auxiliary `β`-coefficients.
  The chart ratios stay fixed; in particular `σ_j/σ_j = 1` on a nonempty chart.

**An affinoid étale perfectoid space is Zariski closed in a relative ball, with stably
pseudocoherent ideal.** (*affinoid-etale-zariski-closed-embedding*) Let `(A, A⁺)` be a perfectoid
Tate pair of any characteristic, `X = Spa(A, A⁺)`, and `f : Y = Spa(B, B⁺) → X` an étale
morphism of affinoid perfectoid spaces ([ECD] Def. 6.2). Prove that there are `N ≥ 0` and
`b = (b₁, …, b_N) ∈ (B⁺)^N` such that the `X`-morphism `g_b : Y → B^N_X`, `Tᵢ ↦ bᵢ`, is a Zariski
closed immersion in the sense of [*fs-iv-4-19-zariski-closed-char-p*](#p5-13-2) (P5.13.2) with `Y' = B^N_X`;
equivalently, `π_b : A⟨T₁, …, T_N⟩ → B`, `T ↦ b`, is surjective and `B⁺` is the integral closure
of `π_b(A⁺⟨T⟩)`, a quotient mapping; the ideal `I := ker π_b` is a stably pseudocoherent
`A⟨T⟩`-module; the ideal sheaf `Ĩ` on `B^N_X` is pseudocoherent; and for every rational
`V' ⊆ B^N_X`, `O(V')/I·O(V') ≅ O(g_b⁻¹V')` with plus ring the integral closure of the image. Any
enlargement `b' = (b, b'')` with `b'' ∈ (B⁺)^{N'}` has the same properties. When `(B, B⁺)` is
given with a standard étale presentation over a sousperfectoid base, the stable pseudocoherence
of its ideal `closure (F)` in `A⟨T⟩` is part of this target.
[ECD] §6, proof of
Prop. 6.4(iv), p. 28, asserts this closed immersion without constructing it, and affinoid
subdomains of non-noetherian adic spaces are not controlled in general ([KL15] §2.4, after
Def. 2.4.12, p. 41); the construction takes global coordinates — the `σᵢ` of
[*etale-standard-piece-cover*](#p5-13-2) (P5.13.2), a unit-ideal certificate `aᵢ`, the coefficients `β` of the monic
equations and the numerators `b` of the approximations of
[*integral-approximation-of-generators*](#p5-13-3) (P5.13.3) — forms the rational covering
`R₀ = {|h| ≥ |ϖ^{2c+1}|} ∪ {R''_j} ∪ {Q_{jl}}` of the whole ball from the integral cut-off, and
verifies the local-on-the-target condition of [FS] §IV.4.1, Prop. IV.4.19(i), p. 140, for it,
applying [*fs-iv-4-19-zariski-closed-char-p*](#p5-13-2) (P5.13.2) with `Y' = B^N_X` (Zariski closed immersions in the
sense of [FS] §IV.4.1, Def. IV.4.20, p. 141). *Needs:* [*stably-pseudocoherent-module*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1);
[*pseudocoherent-sheaf*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1); [*integral-cutoff-graph-presentation*](#p5-13-2) (P5.13.2);
[*fs-iv-4-19-zariski-closed-char-p*](#p5-13-2) (P5.13.2); [*etale-standard-piece-cover*](#p5-13-2) (P5.13.2);
[*integral-approximation-of-generators*](#p5-13-3) (P5.13.3); P3.17a; Tau Ceti
`ValuationSpectrum.rationalSubset`, `Huber.restrictedMvPowerSeriesCompletion`.

The proposed Lean form quantifies the integer `N`, integral
coordinates `b`, and pair map `π`; it fixes both the coefficients and coordinates of `π`
and asserts that it is a quotient mapping with stably pseudocoherent kernel.

**Checks.**

- *test_rational_embedding* — `Y = X(s/t)` rational with `s, t ∈ A⁺`: `b = (sᵢ/t)` works
  directly, `I = closure(tTᵢ − sᵢ)`.
- *test_finite_etale_embedding* — `Y` finite étale over `X` with `B = A[x]/(P)` and
  `det(∂P/∂x)` a unit: `b = ϖ^c x` works, and `I` is the closure of the ideal generated by the
  equations `P(ϖ^{−c}T)`, multiplied by a power of `ϖ` that puts them into `A⁺⟨T⟩`.


**Checks.**

- For the identity object use the zero-dimensional ball and zero ideal.

**Continuous differentials on relative balls and their closed étale loci over perfectoid bases.**
(*continuous-differentials-perfectoid-balls*) Let `(A,A⁺)` be a perfectoid Tate pair, `X = Spa(A,A⁺)`,
`V'` a rational subset of `B^N_X`, and `V = Spa(C,C⁺)` an affinoid perfectoid space étale over
`X` with a Zariski closed immersion into `V'` and quotient mapping `B := O(V') → C`.
Write `I` for its kernel. Construct continuous relative differentials on the smooth spaces
`V'/X` and `V/X` as the conormal sheaves of their diagonals, and their continuous derivations,
with convention `db = 1 ⊗ b − b ⊗ 1`. On affinoid charts they represent continuous
`A`-derivations into complete Hausdorff finite modules. Prove that
`Ωᶜ_{B/A} ≅ ⊕_{i=1}^N B·dTᵢ`, including rational restriction and termwise coordinate
derivations, and that `Ωᶜ_{C/A} = 0`. Construct the right exact sequence
`I/I² → C ⊗_B Ωᶜ_{B/A} → Ωᶜ_{C/A} → 0`, whose first map is `e ↦ 1 ⊗ de`.
This sequence alone asserts no injectivity on the left; [*conormal-description*](#p5-13-3) (P5.13.3) supplies that
additional conclusion for the closed étale locus.
[FS] §IV.4.1, Def. IV.4.11 and Prop. IV.4.12, pp. 136–137, construct the diagonal
conormal sheaf and its ball basis over sousperfectoid bases; their étale case has rank zero.
The quotient derivation argument gives the displayed sequence, the analytic input used in
[ECD] §6, proof of Prop. 6.4(iv), p. 28. These constructions have no noetherian-type assumption.
*Needs:* [*relative-closed-polydisc*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A2.2) (AdicEtaleGeometry A2); [*fs-iv-4-19-zariski-closed-char-p*](#p5-13-2) (P5.13.2);
[*fs-iv-4-13-regular-sequence*](#p5-13-1) (P5.13.1); [*pseudocoherent-kiehl-gluing*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1);
P3.17a (sousperfectoid balls, rational restrictions and étale base change);
AdicSpacesPartII R0 (completed tensor products).
README only (no Lean declaration).

**Checks.**

- *perfectoid_differentials_ball* — over the nonnoetherian perfectoid algebra
  `A = K⟨U^{1/p^∞}⟩` for `K = (𝔽_p((t^{1/p^∞})))^`, the one-variable ball has basis `dT`;
  the continuous derivation `∂/∂T` sends `T` to `1` and constants in `A` to `0`.
- *perfectoid_differentials_etale* — in characteristic `p`, the finite étale algebra
  `C = A[Z]/(Z^p − Z − a)` has zero relative differentials, since the derivative is `−1`.
- *perfectoid_differentials_rational* — for `V = X(s/t)` embedded by `Tᵢ = sᵢ/t`,
  the conormal map sends `tTᵢ − sᵢ` to `t·dTᵢ`; `t` is invertible on `V`.
- *perfectoid_differentials_zero* — for `N = 0` and the identity closed locus, both
  the ball differential module and the conormal module are zero.

**Conormal description of an affinoid étale closed subspace of a relative ball.**
(*conormal-description*) Let `(A, A⁺)` be a perfectoid Tate pair, `X = Spa(A, A⁺)`, `U ⊆ B^N_X`
a finite union of rational subsets, `Y` affinoid perfectoid étale over `X` and `g : Y → U` a
Zariski closed immersion (*fs-iv-4-19-zariski-closed-char-p*). For a rational `V' ⊆ U` with
`V := g⁻¹(V')` and `I := ker(O(V') → O(V))`, prove: (a) the continuous differentials vanish,
`Ω^c_{O(V)/A} = 0` (*continuous-differentials-perfectoid-balls*); (b) the `O(V)`-linear
map `δ : I/I² → ⊕_{i=1}^N O(V)·dTᵢ`, `e ↦ Σᵢ (∂e/∂Tᵢ)|_V dTᵢ`, is an isomorphism; in particular
`I/I²` is free of rank `N` and every basis of it lifts to elements of `I` whose Jacobian with
respect to `T` is invertible on `V`. The vanishing of relative differentials gives this conormal identification in [ECD] §6, proof of Prop. 6.4(iv), p. 28. *Needs:*
[*fs-iv-4-19-zariski-closed-char-p*](#p5-13-2) (P5.13.2); [*affinoid-etale-zariski-closed-embedding*](#p5-13-3) (P5.13.3);
[*pseudocoherent-kiehl-gluing*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1); [*fs-iv-4-13-regular-sequence*](#p5-13-1) (P5.13.1); [*standard-etale-presentation*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2);
[*integral-cutoff-graph-presentation*](#p5-13-2) (P5.13.2); [*continuous-differentials-perfectoid-balls*](#p5-13-3) (P5.13.3);
AdicSpacesPartIII layer R3 (glueing squares). AdicSpacesPartII R0's [*continuous-differentials*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.15)
and [*conormal-sequence*](../AdicSpacesPartII/README.md) (AdicSpacesPartII R0.15) apply when the base is of noetherian type.

The proposed Lean forms supply the continuous termwise derivations on the Tate algebra
`Huber.tateSeries`, state vanishing of continuous differentials and identify the conormal map on
the classes of ideal elements by partial differentiation, and give the same statement on a
rational chart, using the continuous coordinate derivations there. The scalar ring `O(V')/I` is identified with
`O(V)` by the quotient map; the displayed linear equivalence has target `(O(V')/I)^N`. README only (no Lean declaration).

**Checks.**

- *test_rational_conormal_basis* — `Y = X(s/t)`, `I = closure(tTᵢ − sᵢ)`: the classes of
  `tTᵢ − sᵢ` map to `t·dTᵢ`, a basis since `t` is a unit on `Y`.
- *test_no_tor_independence* (non-example) — the global step must not use Tor-independent base
  change: for `R = ℂ_p⟨T₁, T₂⟩`, `I = (T₁, T₂)`, the resolution `0 → R → R² → I → 0`
  (`r ↦ (−T₂r, T₁r)`) becomes zero after `⊗ R/I`, so `Tor₁^R(R/I, I) ≅ R/I ≠ 0`, although `I/I²`
  is free of rank `2` and `δ` is an isomorphism (`e ↦ (∂e/∂T₁, ∂e/∂T₂)` at the origin).
- *conormal_zero_variables* — for `N = 0` and `Y = X`, `I = 0` and `δ` identifies two zero modules.


**Checks.**

- The identity embedding into the zero-dimensional relative ball has zero closed ideal.
- For a rational localization the equations `tT_i−s_i` keep their coordinate labels and closed equation ideal.

**The ideal of an affinoid étale `Y` in a relative ball is generated by `N` equations with unit
Jacobian on a rational neighbourhood of `Y`.** (*local-equations-near-closed-subspace*) In the
situation of [*conormal-description*](#p5-13-3) (P5.13.3) with `U = V' = B^N_X` (`Y` affinoid perfectoid étale over
the affinoid perfectoid `X`, `g : Y → B^N_X` a Zariski closed immersion, `V = Y`,
`I = ker(A⟨T⟩ → O(Y))`; the case used by [*global-standard-etale-presentation*](#p5-13-3) (P5.13.3)), prove that there
are `e₁, …, e_N ∈ I` with `δ(eᵢ) = dTᵢ`, so that `Δ := det(∂eᵢ/∂T_j)` is `≡ 1` on `V`. For each `a ≥ 0` with `ϖ^a eᵢ ∈ A⁺⟨T⟩`, there is a
rational subset `W ⊆ V'` containing `g(V)` such that `e·O(W)` is a closed ideal,
`O(W)/(e) ≅ O(V)` compatibly with plus rings, and `W` lies in the tube of `ϖ^a e`, where
the tube is taken for these scaled equations ([*tubular-neighbourhood*](#p5-13-1) (P5.13.1) with `k = aN` and radius `|ϖ|^{2k+1}`).
In the tube of `e` the kernel `K` of `O(W₀)/(e) → O(Y)` satisfies `K = K²` and is finitely
generated, hence generated by an idempotent (Mathlib `Ideal.isIdempotentElem_iff_of_fg`), and
cutting the other component off by a rational condition gives `W`. For a proper rational `V'` the
tube step needs the rational coordinates of `V'` as extra variables, since [*tubular-neighbourhood*](#p5-13-1) (P5.13.1)
requires equations with integral Taylor coefficients. Together with the rational presentation of
`W` this gives a standard étale presentation of `(O(V), O⁺(V))` over `(A, A⁺)`
([*standard-presentation-basic-pieces*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2) (c)). This supplies the neighbourhood and its equations required in [ECD] §6, proof of Prop. 6.4(iv), p. 28, and the
idempotent argument replaces Fargues–Scholze's use of étaleness to identify a closed immersion locally ([FS] §IV.4.1, proof of Prop. IV.4.19, p. 140). *Needs:* [*conormal-description*](#p5-13-3) (P5.13.3);
[*affinoid-etale-zariski-closed-embedding*](#p5-13-3) (P5.13.3); [*tubular-neighbourhood*](#p5-13-1) (P5.13.1);
[*fs-iv-4-13-regular-sequence*](#p5-13-1) (P5.13.1); [*standard-presentation-basic-pieces*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2); Mathlib
`Ideal.isIdempotentElem_iff_of_fg`; Tau Ceti `ValuationSpectrum.rationalSubset`.

The proposed Lean form chooses equations whose derivative
matrix restricts to the identity on `Y`. For every integral scaling it supplies a rational
neighbourhood containing `Y`, contained in the specified tube, and a quotient mapping
whose kernel is the closed ideal generated by those equations. Its factorization equation
identifies the quotient with the original pair, including its plus ring.

**Checks.**

- *test_rational_local_equations* — `Y = X(s/t)` embedded in `B^n_X` by `Tᵢ ↦ sᵢ/t`, `I` the
  closure of `(tTᵢ − sᵢ)`: `δ(tTᵢ − sᵢ) = t·dTᵢ`, so `eᵢ := λ·(tTᵢ − sᵢ)` with `λ ∈ A⟨T⟩` any
  lift of `t^{−1} ∈ B` satisfies `δ(eᵢ) = dTᵢ`, and the tube `W` is a rational neighbourhood of
  the graph of `s/t` over `R(s/t)`.


**Checks.**

- With no equations the ideal is zero and the determinant is one.
- For a rational presentation `tT−s`, the derivative is `t` and its inverse on the locus fixes the conormal basis.

**Huber's standard étale presentation for affinoid étale maps of affinoid perfectoid spaces.**
(*global-standard-etale-presentation*) Let `(A, A⁺)` be a perfectoid Tate pair,
`X = Spa(A, A⁺)`, and `f : Y = Spa(B, B⁺) → X` an étale morphism of affinoid perfectoid spaces
([ECD] Def. 6.2). Prove that `(B, B⁺)` admits a standard étale presentation over `(A, A⁺)`:
`B ≅ A⟨T₁, …, T_N⟩/closure(F₁, …, F_N)` with `det(∂Fᵢ/∂T_j)` a unit of `B` with bounded inverse
and `B⁺` the integral closure of the image of `A⁺⟨T⟩`, and that it can be chosen in
explicit-inverse form; conversely, if `A` has characteristic `p`, every standard étale
presentation over `(A, A⁺)` defines an affinoid perfectoid space étale over `X`
(*jacobian-criterion-char-p*). Hence, in characteristic `p`, an affinoid perfectoid space `Y`
over `X` is étale if and only if `Y ≅ Spa` of a standard étale presentation — the statement of
[Huber96] §1.7, Prop. 1.7.1, p. 80, for affinoid perfectoid spaces of characteristic `p` that [ECD]
§6, proof of Prop. 6.4(iv), p. 28, takes from the theory of pseudocoherent sheaves. *Needs:*
[*standard-etale-presentation*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2); [*affinoid-etale-zariski-closed-embedding*](#p5-13-3) (P5.13.3);
[*conormal-description*](#p5-13-3) (P5.13.3); [*local-equations-near-closed-subspace*](#p5-13-3) (P5.13.3); [*jacobian-criterion-char-p*](#p5-13-2) (P5.13.2);
AdicSpacesPartII layer R0 (Huber 1.7.1).

The proposed Lean form uses
`AdicSpace.IsEtaleLocalDescription (CAff.spaHom φ …)` and the algebra structure induced by
`φ`; it comes with the characteristic-`p` converse and with perfectoidness of a presented
target.
The explicit-inverse presentation is `withInverse`. README only (no Lean declaration).

**Checks.**

- *test_noetherian_is_huber* — over a Tate ring with a noetherian ring of definition this is
  Huber's 1.7.1 (AdicSpacesPartII layer R0).
- *test_rational_presentation* — `Y = X(s/t)`: the presentation of
  [*standard-presentation-basic-pieces*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2) (a).
<a id="p5-13-4"></a>

### P5.13.4 finite-stage approximation and Scholze's Proposition 6.4(iv)

The tilting input is layers P1, P3 and P5: For a cofiltered system
`X_i = Spa(R_i, R_i⁺)` of affinoid perfectoid spaces with limit `X = Spa(R, R⁺)`, the tilts form a
cofiltered system of affinoid perfectoid spaces of characteristic `p` with limit `X^♭`, with
`R^{♭+}/ϖ^♭ = R⁺/ϖ = colim R_i⁺/ϖ` (layer P1,
[*completed-colimit-of-perfectoid-tate-rings*](#p1-29) (P1.29) (c), and layer P5, [*completed-colimit-is-perfectoid*](#p5-2) (P5.2)),
and the tilting equivalences `(X_i)_{ét,aff} ≃ (X_i^♭)_{ét,aff}`, `X_{ét,aff} ≃ X^♭_{ét,aff}` of
layer P3 ([*etale-site-tilting-equivalence*](#p3-20) (P3.20), applied levelwise) commute with the
base change functors; hence the comparison functor of [ECD] 6.4(iv) for `(X_i)` is an equivalence
iff the one for `(X_i^♭)` is, which is how [ECD] §6, proof of Prop. 6.4(iv), p. 28, reduces
essential surjectivity to characteristic `p`. *Needs:* layers P1, P3 and P5
([*completed-colimit-of-perfectoid-tate-rings*](#p1-29) (P1.29), [*etale-site-tilting-equivalence*](#p3-20) (P3.20),
[*completed-colimit-is-perfectoid*](#p5-2) (P5.2)).


**Checks.**

- For the identity map take no variables or equations and exponent zero.

**Approximating equations and rational data at a finite stage of a cofiltered limit.**
(*cofiltered-limit-presentation-approximation*) In the situation of layer P5
(`X = Spa(R, R⁺) = lim X_i`, `X_i = Spa(R_i, R_i⁺)` affinoid perfectoid, compatible
pseudouniformiser `ϖ`), write `φ_i : R_i → R`. Prove: (a) `R⁺⟨T⟩/ϖ^M = colim (R_i⁺/ϖ^M)[T]`, so
for every `N ≥ 0`, every finite family of elements of `R⁺⟨T₁, …, T_N⟩` and every `M ≥ 0` there
are `i` and elements of `R_i⁺⟨T⟩` whose images agree with the given ones modulo `ϖ^M R⁺⟨T⟩`
(`exists_approx_restrictedPowerSeries`, with polynomial approximations and the colimit
described by approximation of `R⁺` modulo every `ϖ^M` through monotone directed images); the
same holds for the numerators and denominator of a rational subset of `B^N_X`, and for `M` large
the approximating data define, over `X`, the same rational subset (Tau Ceti
`exists_mem_nhds_forall_rationalSubset_eq_of_sub_mem`); (b) if `(N, F_i, k, π_i)` is a standard
étale presentation of `(C_i, C_i⁺)` over `(R_i, R_i⁺)` and `Y_i := Spa(C_i, C_i⁺)` is affinoid
perfectoid (automatic in characteristic `p`, [*jacobian-criterion-char-p*](#p5-13-2) (P5.13.2)), then the base change
`(N, φ_i(F_i), k, ·)` is a standard étale presentation of
`C_i ⊗̂_{R_i} R = R⟨T⟩/closure(φ_i(F_i))` (`StandardEtalePresentation.baseChange` with its
coordinate formula, for any morphism of complete Tate pairs), and `Y_i ×_{X_i} X = Spa(R⟨T⟩/closure(φ_i(F_i)), integral closure)` as perfectoid spaces, the fibre product with `X` ([ECD] §6, Prop. 6.4
set-up, p. 27: the inverse limit is formed by `ϖ`-adically completing the colimit of the integral
rings; proof of Prop. 6.5, p. 29: uniformity makes `Spa` of the completed colimit the limit, and
the same completeness drives the approximation). The identification `R⁺/ϖ^M = colim R_i⁺/ϖ^M` is
layer P5's [*cofiltered-limits-of-affinoid-perfectoid-spaces*](#p5-3) (P5.3) and the finite-stage
descent of rational data its *descent-of-quasicompact-opens* and
*limit-restriction-to-rational-subsets*; the base change of presentations (b) is the content
here. *Needs:* [*standard-etale-presentation*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2);
layers P2 and P5; AdicSpacesPartII layer R0 (completed tensor products); Tau
Ceti `TauCeti.ValuationSpectrum.exists_mem_nhds_forall_rationalSubset_eq_of_sub_mem`. README only (no Lean declaration).

**Checks.**

- *approximation_pointwise_union* — For the constant directed system `Q₂`, the polynomials selected by `exists_approx_restrictedPowerSeries` for the family `(1,T)` modulo 2 are both nonzero; the first has constant coefficient 1 modulo `2ℤ₂`.
- *approximation_no_common_stage* — The same selected family has nonzero constant coefficient in its first polynomial and nonzero `T` coefficient in its second, and the two polynomials differ. A construction keeping just one member of the family cannot satisfy the simultaneous approximation.
- *approximation_directed* — for a directed, monotone family of image subrings, an upper
  bound of the stages containing `a` and `b` contains both; this is the `hmono` hypothesis of
  `exists_approx_restrictedPowerSeries`.
- *test_linear_base_change* — `F_i = (T − a_i)` with `a_i ∈ R_i⁺`: the base change is
  `T − φ_i(a_i)`, presenting `R` itself; its perturbation to `T − a` with `a ≡ φ_i(a_i) mod ϖ²`
  gives the same space.

**Affinoid étale objects over a cofiltered limit descend to a finite stage, in characteristic
`p`.** (*finite-stage-approximation-char-p*) Let `X_i = Spa(R_i, R_i⁺)`, `i ∈ I`, be a cofiltered
system of affinoid perfectoid spaces of characteristic `p` with limit `X = Spa(R, R⁺)`
(layer P5), and `Y → X` an étale morphism with `Y` affinoid perfectoid ([ECD]
Def. 6.2). Prove that there are `i ∈ I` and an affinoid perfectoid space `Y_i` with an étale
morphism `Y_i → X_i` such that `Y_i ×_{X_i} X ≅ Y` over `X`. Explicitly: let `(N, F, k, π)` be a
standard étale presentation of `O(Y)` over `R` in explicit-inverse form
(*global-standard-etale-presentation*), `F = (F', S·J_{F'} − ϖᵏ)`, and `m₀ = 4k + 2`; choose `i`
and `F'_i ∈ R_i⁺⟨T'⟩^{N−1}` with `φ_i(F'_i) ≡ F'` modulo `ϖ^{m₀}`
(*cofiltered-limit-presentation-approximation*) and put `F_i := (F'_i, S·J_{F'_i} − ϖᵏ)`, again in
explicit-inverse form, so that `φ_i(F_i) ≡ F` modulo `ϖ^{m₀}`; then the Jacobian
`J_{F_i} = J_{F'_i}²` is invertible on `Y_i := V(F_i)` everywhere, with `ϖ^{2k}J_{F_i}^{−1} = S²`
integral — the last equation forces `|J_{F'_i}| ≥ |ϖᵏ|` at every point of `V(F_i)`, also at
points not coming from `X` — `Y_i` is affinoid perfectoid and étale over `X_i`
([*jacobian-criterion-char-p*](#p5-13-2) (P5.13.2) (b)), and `Y_i ×_{X_i} X = V(φ_i(F_i)) ≅ V(F) = Y`
([*cofiltered-limit-presentation-approximation*](#p5-13-4) (P5.13.4) (b), [*standard-presentation-perturbation*](#p5-13-1) (P5.13.1)). The
ring-level steps are `nonempty_ofExplicitInverse` (the system `F_i`
presents its quotient pair with exponent `2k`, whatever `F'_i` is), `StandardEtalePresentation.baseChange`
and the specification of `StandardEtalePresentation.perturb` (the comparison
`V(φ_i(F_i)) ≅ V(F)` over the perfectoid, hence sousperfectoid, `R`). The
same holds for every `j ≥ i` with `F_j` the image of `F_i` ([ECD] §6, proof of Prop. 6.4(iv),
p. 28: perturbed equations define affinoid (sous)perfectoid spaces étale over the base, and
close enough approximations are isomorphic to `Y`). *Needs:* [*standard-etale-presentation*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2);
[*standard-presentation-perturbation*](#p5-13-1) (P5.13.1); [*jacobian-criterion-char-p*](#p5-13-2) (P5.13.2);
[*global-standard-etale-presentation*](#p5-13-3) (P5.13.3); [*cofiltered-limit-presentation-approximation*](#p5-13-4) (P5.13.4);
layer P5.

The proposed Lean form chooses a stage, integral equations
approximating the original equations coefficientwise modulo `ϖ^(4k+2)`, and a perfectoid
étale model with the explicit-inverse equations and Jacobian exponent `2k`. Its pushout
square is the completed base change and fixes the identification with the given target.

**Checks.**

- *test_rational_descends* — `Y = X(s/t)` with `s, t ∈ R⁺`: approximate `s, t` by
  `s_i, t_i ∈ R_i⁺`; the model is `X_i(s_i/t_i)`.
- *test_finite_etale_descends* — `Y` finite étale over `X`: the model is the finite étale
  descent of layer P5 ([ECD] 6.4(i)), recovered from an approximated monic
  presentation.


**Checks.**

- For a constant diagram the original standard presentation is its own finite-stage model.


**Checks.**

- For the identity map take no variables or equations and exponent zero.

**Independence of the finite-stage model from presentations, approximations and indices.**
(*independence-of-presentations*) In the situation of [*finite-stage-approximation-char-p*](#p5-13-4) (P5.13.4), or of
[*affinoid-etale-finite-stage-6-4-iv*](#p5-13-4) (P5.13.4) in any characteristic (`X_i` affinoid perfectoid,
cofiltered; `Y` affinoid étale over `X`), prove: (a) if `(Y_i, α)` and `(Y'_{i'}, α')` are two
affinoid étale models of `Y` — `Y_i` affinoid étale over `X_i` with `α : Y_i ×_{X_i} X ≅ Y`,
obtained from any standard étale presentations, approximations and indices — then there is
`j ≥ i, i'` and an isomorphism `β_j : Y_i ×_{X_i} X_j ≅ Y'_{i'} ×_{X_{i'}} X_j` over `X_j`
whose base change to `X` is `α'^{−1}∘α`. Any two such isomorphisms agree after passage to
a sufficiently large common stage; uniqueness is in the filtered colimit, not asserted at a
fixed stage (the full faithfulness of [ECD] 6.4(ii), layer P5); (b) the rule
`Y ↦ (class of (Y_i, α))` is well defined in the 2-colimit `2-colim_i (X_i)_{ét,aff}` and,
together with the base change functor, gives mutually quasi-inverse equivalences; (c) the model
agrees with the finite-stage descent of `Y` as a qcqs étale object given by PerfectoidSpaces
layer P5 ([ECD] 6.4(ii)): the P5 descent `Z_k` of `Y` is isomorphic over some `X_j` to
`Y_i ×_{X_i} X_j`, and is therefore affinoid for large `j` ([ECD] §6, proof of Prop. 6.4(ii),
p. 28: isomorphism detection at a finite stage; proof of Prop. 6.4(iv), p. 28: full faithfulness
in (iv) follows from (ii)). *Needs:* [*finite-stage-approximation-char-p*](#p5-13-4) (P5.13.4); PerfectoidSpaces
layers P2 and P5 (finite-stage descent of qcqs étale objects).

The proposed Lean forms descend the specified comparison of two
models and detect equality at a later stage.
The statements use cartesian systems of pair algebras: their transition squares and
squares to the completed base are pushouts in `CAff`. Models initially defined at different
indices are compared on the directed tail above a common upper bound. This is the ring-level
form of the isomorphism detection and full faithfulness in [ECD] 6.4(ii), printed p. 28. README only (no Lean declaration).

**Checks.**

- *test_two_rational_presentations* — two presentations of a rational subset
  `Y = X(s/t) = X(s'/t')`: the two models `X_i(s_i/t_i)` and `X_{i'}(s'_{i'}/t'_{i'})` coincide as
  subsets of `X_j` for `j` large, since qc opens descend uniquely ([ECD] 6.4(o)).


**Checks.**

- Using the same presentation twice gives the identity comparison after base change.
- Three models give the same composite comparison after passing to a common stage, by equality detection.

**Affinoid étale objects descend to a finite stage of a cofiltered limit of affinoid perfectoid
spaces.** (*affinoid-etale-finite-stage-6-4-iv*) Let `X_i = Spa(R_i, R_i⁺)`, `i ∈ I`, be a
cofiltered inverse system of affinoid perfectoid spaces of any characteristic indexed by a small
category, and `X = Spa(R, R⁺)` its inverse limit: `R⁺` is the `ϖ`-adic completion of
`colim_i R_i⁺` and `R = R⁺[1/ϖ]` for a compatible choice of pseudouniformisers `ϖ` for large `i`
(layer P5). For each `i` let `(X_i)_{ét,aff}` be the full subcategory of the
étale site ([ECD] Def. 6.2; layer P3) of affinoid perfectoid spaces étale over
`X_i`, and similarly `X_{ét,aff}`; base change is well defined in these categories because fibre
products of affinoid perfectoid spaces are affinoid perfectoid (layer P2), and
perfectoid Tate rings without base field use the [ECD]-generality of layers P1–P3. Prove that the
base change functors `(X_i)_{ét,aff} → X_{ét,aff}`, `Y_i ↦ Y_i ×_{X_i} X`, induce an equivalence
of categories `2-colim_i (X_i)_{ét,aff} → X_{ét,aff}`: (full faithfulness) for
`Y_i, Y'_i ∈ (X_i)_{ét,aff}`,
`colim_{j ≥ i} Hom_{X_j}(Y_i ×_{X_i} X_j, Y'_i ×_{X_i} X_j) → Hom_X(Y_i ×_{X_i} X, Y'_i ×_{X_i} X)`
is bijective; (essential surjectivity) every affinoid perfectoid space `Y` étale over `X` is
`X`-isomorphic to `Y_i ×_{X_i} X` for some `i` and some affinoid perfectoid `Y_i` étale over
`X_i`, the model being unique in the 2-colimit (*independence-of-presentations*). The proof
tilts to characteristic `p` (P3’s [*etale-site-tilting-equivalence*](#p3-20) (P3.20) and P5’s [*completed-colimit-is-perfectoid*](#p5-2) (P5.2)), applies
[*finite-stage-approximation-char-p*](#p5-13-4) (P5.13.4) and uses neither diamonds, nor [ECD] Prop. 11.30, nor [ECD]
Lem. 15.6, nor any six-functor formalism ([ECD] §6, Prop. 6.4(iv), p. 27, and its proof sketch,
p. 28: full faithfulness from (ii), essential surjectivity in characteristic `p` via Huber 1.7.1
and Kedlaya–Liu's pseudocoherent sheaves). *Needs:* P3’s [*etale-site-tilting-equivalence*](#p3-20) (P3.20) and P5’s [*completed-colimit-is-perfectoid*](#p5-2) (P5.2);
[*finite-stage-approximation-char-p*](#p5-13-4) (P5.13.4); [*independence-of-presentations*](#p5-13-4) (P5.13.4); layers
P2, P3 and P5.

The proposed Lean forms give a perfectoid étale stage model and an
isomorphism `CompletedTensor R_i B_i R ≃+* B`, homeomorphic, with its value on each pure
tensor specified and its plus ring mapped onto `B⁺`, and identify the quotient of the disjoint union of stage pair-Hom sets by common extension
with the Hom set over `R`; the equation on representatives pins the comparison. The system
hypotheses express both approximation and eventual detection of congruences modulo every
power of a compatible pseudouniformiser, so they specify the completed colimit, including
its topology. These signatures use directed preorders; a cofinal directed presentation gives
the stated small cofiltered-category form. The source is [ECD] 6.4(iv), printed pp. 27–28.

**Checks.**

- *test_rational_finite_etale_composites* — `Y = X(s/t)` comes from `X_i(s_i/t_i)` for
  approximations `s_i, t_i` (Tau Ceti perturbation of rational subsets); finite étale `Y` comes
  from [ECD] 6.4(i) (layer P5); a composite rational `⊂` finite étale `⊂` rational
  comes from composing these.
- *test_ecd_7_18* — in [ECD] §7, proof of Lem. 7.18, p. 37, an affinoid étale cover of an affinoid
  perfectoid space comes from a finite level of the tower constructed there and splits at the
  next level. More generally, if every finite-stage affinoid étale cover acquires a section
  at some later stage, every affinoid étale cover of the completed limit has a section.
- *test_ecd_7_11* — in [ECD] §7, proof of Lem. 7.11(i), p. 33, composites of affinoid pro-étale
  maps are formed via Prop. 7.10 and 6.4(iv). On complete Huber pairs, two filtered étale
  presentations give a filtered étale presentation whose cocone point has the composite
  structure map over the original base.

**Scholze's Proposition 6.4 assembled.** (*etale-descent-to-finite-stage-6-4*) For a cofiltered
system `X_i = Spa(R_i, R_i⁺)` of affinoid perfectoid spaces with limit `X = Spa(R, R⁺)` (`R⁺`
the `ϖ`-adic completion of `colim R_i⁺`), under the hypotheses of [ECD] Prop. 6.4 with `κ` as in
[ECD] Lem. 4.1: (o) `|X| → lim |X_i|` is a homeomorphism of spectral spaces; (i)
`2-colim (X_i)_{fét} → X_{fét}`; (ii) `2-colim (X_i)_{ét,qcqs} → X_{ét,qcqs}`; (iii)
`2-colim (X_i)_{ét,qc,sep} → X_{ét,qc,sep}`; (iv) `2-colim (X_i)_{ét,aff} → X_{ét,aff}` are
equivalences of categories, and `X` is `κ`-small ([ECD] §6, Prop. 6.4(o)–(iv), p. 27). Parts
(o)–(iii) and the cardinal bound are layer P5 (cofiltered limits of affinoid
perfectoid spaces; finite-stage descent of qcqs étale objects), imported unchanged; part (iv) is
[*affinoid-etale-finite-stage-6-4-iv*](#p5-13-4) (P5.13.4). Prove that the four equivalences are compatible: the
inclusions `X_{fét} ⊆ X_{ét,aff} ⊆ X_{ét,qc,sep} ⊆ X_{ét,qcqs}` (a finite étale space over an
affinoid perfectoid space is affinoid perfectoid, layer P3; affinoid perfectoid
spaces are quasi-compact and separated) commute with the base change functors and with the four
comparison functors from the 2-colimits. This layer does not reprove (o)–(iii), and
layer P5 does not claim (iv); layer P6
(*affinoid-etale-finite-stage-assembly*) cites this compatibility clause and records the division
of labour from its side. *Needs:* [*affinoid-etale-finite-stage-6-4-iv*](#p5-13-4) (P5.13.4);
layers P2, P3 and P5.

The proposed Lean forms identify the same pair pushout with the
geometric pullback, preserving finite étale and étale maps; fix the comparison of two
realizations by both labelled tensor inclusions; and fix its action on
maps by the same equations and their contravariant `Spa` equations. Thus forgetting from
finite étale to affinoid étale to quasi-compact separated étale to qcqs étale retains the
same base-change object, projections and maps. Uniqueness makes these comparisons coherent
under composition and passage to the filtered colimit of Homs. README only (no Lean declaration).

**Checks.**

- *test_constant_system* — for the constant two-sheeted affinoid over `ℚ₂`, completed tensor
  base change along the identity has a unique comparison with the original split pair fixing
  both coprojections. It sends the first characteristic idempotent to the first, not the second.
  For the four descent functors use instead the constant split two-sheeted pair over
  the perfectoid field `K = completion of ℚ₂(μ_{2^∞})` (P1.31). The same tensor comparison
  fixes both idempotents, and all four functors are naturally isomorphic to identities.


**Checks.**

- A constant inverse system gives identity equivalences in all four étale categories.
- Finite étale two-sheeted objects recover the finite étale limit comparison, preserving the two characteristic idempotents.

**The Zariski closed immersions of this layer are strongly Zariski closed.**
(*zariski-closed-immersion-ecd-comparison*) Let `(A, A⁺)` be a perfectoid Tate pair of
characteristic `p`, `X = Spa(A, A⁺)`, `Y = Spa(B, B⁺)` affinoid perfectoid étale over `X` and
`b ∈ (B⁺)^N` as in [*affinoid-etale-zariski-closed-embedding*](#p5-13-3) (P5.13.3). Let
`B^{N,perf}_X := Spa(A⟨T₁^{1/p^∞}, …, T_N^{1/p^∞}⟩, A⁺⟨T^{1/p^∞}⟩)` be the perfectoid relative
ball and `g^perf : Y → B^{N,perf}_X` the map `T_i^{1/pⁿ} ↦ b_i^{1/pⁿ}`. Prove that `g^perf`, the
closed embedding of P5.13.3 composed into the perfectoid ball, is strongly Zariski closed in the
sense of [ECD] §5, Def. 5.7(ii), p. 24, directly: `A⟨T^{1/p^∞}⟩ → B` is surjective, its image
containing the image of `A⟨T⟩`, which is `B`; `B⁺` is the integral closure of the image of
`A⁺⟨T^{1/p^∞}⟩` (it is the integral closure of the smaller image of `A⁺⟨T⟩`, and the roots
`bᵢ^{1/pⁿ}` lie in `B⁺`); and `A⁺⟨T^{1/p^∞}⟩ → B⁺` is almost surjective, i.e. `ϖ^{1/pⁿ}B⁺` lies
in the image for every `n`: the image is open, so it contains `ϖ^c B°` for some `c`, and it is
perfect (the source plus ring is integrally closed in the perfect source ring, hence closed
under the unique p-th roots), so with `b ∈ B⁺` and
`ϖ^c b^{pⁿ} ∈ image` also `ϖ^{c/pⁿ}b ∈ image`. These are the two forms in which strong Zariski
closedness is used ([ECD] §5, Def. 5.7(ii) and Thm. 5.8: surjectivity on the rings with the plus
ring the integral closure of the image, and almost surjectivity on the plus rings); in particular
`g^perf` is a Zariski closed immersion ([ECD] 5.7(i)). The proof
of [*affinoid-etale-finite-stage-6-4-iv*](#p5-13-4) (P5.13.4) does not use [ECD] Thm. 5.8 (Zariski closed implies
strongly Zariski closed), which is not an input of this roadmap: the closed immersions it needs
are into the sousperfectoid ball `B^N_X` and are constructed as quotient mappings
(*affinoid-etale-zariski-closed-embedding*), and the transfer between characteristics is done on
étale categories (P3’s [*etale-site-tilting-equivalence*](#p3-20) (P3.20) and P5’s [*completed-colimit-is-perfectoid*](#p5-2) (P5.2)), not on Zariski closed immersions.
*Needs:* [*affinoid-etale-zariski-closed-embedding*](#p5-13-3) (P5.13.3); P3.17a; PerfectoidSpaces
layer P1.

`Huber.Pair.Hom.quotientMapping_perfect_extension_charP` applies to the extension of the
ball quotient map to the perfectoid ball: any pair map extending that quotient is again
a quotient mapping and is almost surjective on plus rings.
`Huber.Pair.Hom.almost_surjective_of_quotient_charP` states the general ring-level implication:
for each compatible root `ϖ^(1/p^n)` and each integral target element its product lies in
the image of the source plus ring. These are the conditions of [ECD] 5.7(ii), printed p. 24. README only (no Lean declaration).

**Checks.**

- *ambient_perfect_not_plus_perfect* — in the perfection of `𝔽₂(t)`, let
  `B = 𝔽₂[t^(1/2^∞)]` and `S = 𝔽₂ + tB`. The integral closure of `S` is `B` and
  `tB ⊆ S`, but the unique square root `t^(1/2)` is absent from `S`. Thus an ambient
  perfect ring and a conductor bound do not suffice without integral closure of the source plus ring.
- *test_almost_surjective_of_perfect* — if the source plus subring is integrally closed in a
  perfect characteristic-p ring and `ϖ^c S⁺` lies in its image, taking p-power roots inside
  that subring gives the bounded lifts used in the argument. Perfectness of the ambient
  ring alone does not assert perfectness of an arbitrary subring.
- *test_rational_strongly_closed* — `Y = X(s/t)`: `g^perf` identifies `Y` with the Zariski closed
  subset `V(t^{1/pⁿ}T^{1/pⁿ} − s^{1/pⁿ} : n)` of the perfectoid ball, which is strongly Zariski
  closed.


#### Examples for analytic approximation

The worked examples of the layer run along its build. A rational subset `Y = X(s/t)` descends to
`X_i(s_i/t_i)` for approximations `s_i, t_i` of `s, t` (Tau Ceti's perturbation of rational
subsets, `exists_mem_nhds_forall_rationalSubset_eq_of_sub_mem`), and its closed embedding is
`Tᵢ ↦ sᵢ/t` with ideal the closure of `(tTᵢ − sᵢ)` ([*affinoid-etale-zariski-closed-embedding*](#p5-13-3) (P5.13.3),
[*finite-stage-approximation-char-p*](#p5-13-4) (P5.13.4)). For a finite étale `Y`, the model from an approximated
monic presentation agrees with [ECD] 6.4(i) of layer P5. The tube of
`e = T − a` on `B¹_X` is the translation `{|T − a| ≤ |ϖ|} ≅ B¹(|ϖ|)` (*tubular-neighbourhood*),
and the perturbation of `(T − a)` to `(T − a')` with `a ≡ a' mod ϖ²` gives the identity of `A`
(*standard-presentation-perturbation*). `F = T^p − a` in characteristic `p` is rejected by the
Jacobian condition, `J_F = 0`, and a nonempty such locus is not étale ([*standard-etale-presentation*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.2),
[*jacobian-criterion-char-p*](#p5-13-2) (P5.13.2)). Over `ℚ_p⟨T⟩`, strongly noetherian, the pseudocoherent theory
reduces to Kiehl's coherent theory (AdicSpacesPartIII layer R3, Kiehl's theorem on Tate affinoids;
[*pseudocoherent-kiehl-gluing*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A3.1)) and pseudoflatness to flatness
(*rational-localisation-pseudoflat*). The applications in [ECD] are those recorded under
[*affinoid-etale-finite-stage-6-4-iv*](#p5-13-4) (P5.13.4): in the proof of Lemma 7.18, affinoid étale covers descend
to a finite level of a tower; in the proof of Lemma 7.11(i), composites of affinoid pro-étale
maps are formed via Proposition 7.10 and 6.4(iv). Alongside these sit the non-examples that fix
the definitions: the finitely presented but not 2-pseudocoherent `R/xR` over
`k[x, y₁, y₂, …]/(x·yᵢ)`, the strictly but not stably pseudocoherent quotient `A/fA` of [KL16]
Ex. 2.4.2, the non-pseudoflat `ℚ_p⟨T⟩/(T)`, the pushforward `j_*O_U` that is not pseudocoherent,
the dilation `X ↦ ϖX` that is no automorphism of `A⟨X⟩`, and the non-integral equation `(T/p)²`
for which the tube theorem fails.


| Check | Computation or boundary case |
| --- | --- |
| *perturb_zero* | no perturbation recovers the specified quotient identification. |
| *perturb_plus* | For `Fᵢ = Tᵢ² − Tᵢ` and `Gᵢ = (Tᵢ − δ)² − (Tᵢ − δ)`, where `δ = ϖ^(4k+2)`, the perturbation equivalence sends the old idempotent to the integral idempotent `Tᵢ − δ` and excludes its complement over a nonzero target. |
| *perturb_coordinate* | For the same shifted idempotent equation, the coordinate error is `−δ` and differs from `1 − 2Tᵢ + δ`, the error obtained by exchanging the two roots. |
| *test_perturb_linear_identity* | for `T − a` perturbed to `T − a'` the comparison is the identity of `A` with coordinate error `a − a'`. |
| *test_identity_cutoff* | the identity is a quotient mapping with zero kernel; there are no cutoff equations. |
| *approximation_no_common_stage* | The same selected family has nonzero constant coefficient in its first polynomial and nonzero `T` coefficient in its second, and the two polynomials differ. A construction keeping just one member of the family cannot satisfy the simultaneous approximation. |
| *test_transport_direction* | `π ∘ θ` kills the inverse substitute `X − 3` and sends the forward substitute `X + 3` to `6`. |
| *tube_empty_variables* | no equations and unit Jacobian give the whole spectrum. |
| *tube_zero_jacobian* | a positive-dimensional zero equation has no tube. |
| *tube_linear* | the Jacobian cutoff is automatic for `T-a` and `k=0`. |
| *quotient_identity* | the identity is a quotient mapping. |
| *quotient_top* | quotient by the unit ideal has the zero target ring and full plus ring. |
| *quotient_ideal* | the canonical quotient has precisely the integral-image plus ring. |
| *quotient_plus_too_big* | a strictly larger target plus ring is not a quotient mapping. |
| *quotient_coordinate* | quotientEquiv is induced by the actual presentation map. |
| *quotient_scalar* | quotientEquiv preserves the specified A-algebra structure. |
| *quotient_relation* | each defining relation vanishes under quotientEquiv. |
| *presentation_tube_zero* | the tube of a zero-variable presentation is the full spectrum. |
| *presentation_tube_radius* | larger m gives a smaller tube. |
| *presentation_tube_cutoff* | every point satisfies the Jacobian lower bound. |
| *inverse_system_zero_variables* | with no old variables the explicit-inverse system is `S − ϖᵏ`. |
| *jacobian_two_equations* | the explicit-inverse system of `T² − a` has Jacobian `(2T)²`. |
| *base_change_jacobian* | the base-changed Jacobian is the image of `J_F`. |
| *test_translation*, *test_unit_jacobian_product*, *test_linear_regular* | `tubeSet` and `ratLoc` give the scaled coordinate, product comparison preserving plus rings, and regularity with closed ideal on every rational restriction. |
| *test_perturb_rational*, *test_linear_base_change*, *test_two_rational_presentations* | Close rational data give the same rational set and a homeomorphic pair comparison; linear coefficients pass through the base map; comparison of two rational presentations is unique over the base. |
| *test_field_trivial*, *test_field_quadratic* | `StandardEtalePresentation` has the specified equations and exponent zero. The quadratic Check assumes `2` is a unit in `K⁺`, expressing the odd residue-characteristic condition. |
| *test_rational_finite_etale_pieces*, *test_frobenius_locus_not_etale* | `CAff.spaHom` is étale for rational–finite étale composites over a perfectoid base; a nonzero Frobenius quotient in characteristic `p` has zero Jacobian and is not étale. |
| *test_surjective_application*, *test_rational_closed*, *test_rational_embedding*, *test_finite_etale_embedding* | Quotient mappings supply lifts and monic integral equations; rational and monogenic finite étale embeddings retain their coordinates and closed equation ideals. |
| *test_rational_one_piece*, *test_disjoint_union_idempotent* | The unit numerator gives the one-member covering, and the two complementary idempotents give precisely the two disjoint rational components. |
| *test_ratio_is_generator* | The nonzero ratio `ϖ/X` belongs to the rational plus ring and to the subalgebra generated by the prescribed ratios. |
| *test_rational_conormal_basis*, *test_rational_local_equations* | Equation classes give the diagonal basis matrix `t·1`; multiplication by a lift of `t⁻¹` gives the identity derivative matrix. |
| *test_no_tor_independence* | Mathlib's `CategoryTheory.Tor` (`Mathlib/CategoryTheory/Monoidal/Tor.lean`, definition `Tor`) gives `Tor₁^R(R/I,I) ≅ R/I ≠ 0` for the two-variable Tate algebra, although `I/I²` is free of rank two. The example allows any complete algebraically closed nonarchimedean field, including `ℂ_p`. |
| *test_noetherian_is_huber*, *test_rational_presentation* | Étaleness over a noetherian ring of definition agrees with the standard-presentation criterion; the rational example uses the equations `tTᵢ−sᵢ`. |
| *test_rational_descends*, *test_finite_etale_descends*, *test_rational_finite_etale_composites* | The completed-system hypotheses of `affinoid_etale_finite_stage_essSurj` yield models of the same morphism classes, with all three factors of a composite at one stage. |
| *test_constant_system* | The completed tensor of the split two-sheeted `Q₂` pair with `Q₂` has a unique comparison fixing both coprojections and the first characteristic idempotent. Over `K = completion of ℚ₂(μ_{2^∞})`, the constant split pair also tests all four category comparisons using P5's finite étale, qcqs étale, qc separated étale and affinoid étale limit equivalences. |


**Checks.**

- For the identity object use the zero-dimensional ball and zero ideal.

### Characteristic maps and computations

The ring used for P5.1 is P1's `Perfectoid.completedFilteredColimit p D j₀ ϖ`;
`completedColimit` denotes that ring here and does not introduce another completion type.
`Perfectoid.plusDiagram` remembers the **chosen** plus rings. Inside the P1 ring,
`completedColimitPlus` is the topological closure of their images and
`completedColimitPair` equips the ring with that plus ring. The ring's Tate, completeness,
Hausdorff and perfectoid instances are those of P1. The comparison
`completedColimit.plusCompletionEquiv`, characterized by `plusCompletionEquiv_ι`, identifies
this subring with the adic completion of the chosen integral colimit.
`completedColimit.integralPerfectoid` states regularity and the Frobenius quotient
isomorphism, assuming `ϖ^p ∣ p` in the chosen stage plus ring. Its topology is specified
by `completedColimit.hasBasis_nhds_zero`. The construction uses the chosen `R_i⁺`
at every stage. These identifications use [ECD] Proposition 6.5, p. 29.
`compatible_pseudouniformizer_map` carries the plus-ring root system, its divisibility
condition and its topologically nilpotent unit along each transition; on the category
over the chosen stage these images form the compatible system.

On a tail, choose actual compatible elements `t_j ∈ R_j⁺`, with
`φ_a(t_i) = t_j` and `t_j₀ = ϖ`. `completedColimit.quotientDiagram` is the functor
`j ↦ R_j⁺/(t_j^n)`. `plusQuotientEquiv` identifies its ring colimit with `R⁺/(ϖ^n)` for
`n > 0`; `plusQuotientEquiv_ι` sends the class of a stage element to the class of its canonical
image. This is the finite-quotient statement, with the distinguished element fixed before
passing to the colimit ([Bhatt] Lemmas 5.1.1–5.1.2, pp. 33–34; [Stacks] Tag 05GG).
`adicCompletion_kernel_divisible`, `adicCompletion_separatedQuotient` and
`adicCompletion_rootClosed` state the nonseparated completion arguments; the last assumes
`q > 1`, regularity of `t`, and root closure in `A[1/t]`.
`adicCompletion_localized_density` specifies the principal neighbourhood basis after
inverting `t` and proves density of the canonical map from `A[1/t]`.
`NormedAffinoidSystem` specifies complete nonarchimedean norms defining the stage
 topologies, with the chosen plus rings as unit balls and submetric transition maps.
`ringDiagram` retains the structural ring maps. `colimitSeminorm_apply` specifies the
infimum over all representatives, and `NormedColimit` is its Hausdorff completion.
`completedColimit_test_compatibility_sch12` identifies this completion with the adic
completed colimit when the common pseudouniformizer multiplies every norm by the
same number in `(0,1)` ([Sch12] Lemma 6.13, p. 37; [KL15] Definition 2.6.1 and
Remark 2.6.3, p. 53). The comparison agrees with every stage map.

**Checks.** For each of `ringDiagram`, `NormedAffinoidSystem`, `colimitSeminorm`,
and `NormedColimit`, `_zero` gives the zero ring and zero seminorm; `_terminal`
computes the seminorm at a terminal stage and recovers that complete stage;
`_killed` makes an element killed by a transition have zero class and seminorm.

The global declarations use P2's `PerfectoidSpace.Perfd`, `IsAffinoidPerfectoid`,
`globalSections` and `globalPlus`. In `PerfectoidSpace.cofilteredLimit`, `cone` and `isLimit`
specify the limit in that category; `homeomorph_apply` specifies the compatible tuple of
points, and `pairIso_π` identifies the projections with the maps induced by
`completedColimit.ι`. `baseChangeDiagram` uses the actual pullbacks over `Over i`;
`baseChangeHomeomorph_apply` fixes every coordinate of its homeomorphism.
`sectionsDiagram` and `plusSectionsDiagram` use actual pullback homomorphisms.
`globalSections_of_isCardinalFiltered` and `globalPlus_of_isCardinalFiltered`, together with
their `_ι` equations, identify the uncompleted ring colimits when `Iᵒᵖ` is `ℵ₁`-filtered
([ECD] Proposition 6.5, p. 29).
`globalPlusQuotientDiagram` uses compatible global integral sections `t_i`; on every
stage chart their values must be pseudouniformizers. `cofilteredLimit.plusQuotient_ι`
fixes the quotient comparison on every stage section. `cofilteredLimit.globalPlusCompletion`
gives the integral completion comparison, again fixed on every stage section. Applying it
to the rational base-change diagram supplies the completed plus-ring formula in P5.4.
`integral_sections_completion_extension` fixes the image of the adic element;
`integral_sections_localization_extension` gives the unique extension after inverting it;
`integral_sections_chart_continuous` and `integral_sections_localized_continuous` give
continuity on every affinoid chart, with the principal topology explicitly specified.

For a cofiltered diagram `S` of finite profinite sets, `profiniteFunctionDiagram` uses
precomposition on continuous functions. `cofilteredLimit_prodProfinite` identifies its
space limit with the affinoid of continuous functions on `limit S`, and its `_π` equation
identifies each projection with `profiniteProjectionMap`. The underlying topology is fixed
by `continuousFunctionSpaceHomeomorph`; its `_eval` equation sends the point obtained by
evaluation at `s` and a point `x` of the coefficient affinoid to `(s,x)` ([Sch13] end of the
proof of Lemma 4.6, p. 25; [ECD] Proposition 6.5, p. 29).

`tateColimit_spectrum` states the general Tate-pair result without perfectoidness,
completeness or injectivity of the stages. Its input includes compatible open rings of
definition `A_{j,0} ⊆ A_j⁺`, a compatible pseudouniformizer, and the principal-adic topology
on each `A_{j,0}`. `TateColimitModel` carries a Tate pair and a specified ring equivalence to Mathlib's algebraic ring colimit. The theorem fixes
the plus ring, the neighbourhood basis, the canonical ring maps, the homeomorphism and
finite-stage rational descent. Completion of this spectrum uses the existing Tau Ceti
completion homeomorphism ([KL15] Remark 2.6.3, p. 53; [Wedhorn] Proposition 7.48, p. 68).
`cofilteredLimit.rational_descent` states rational descent for the completed perfectoid pair.
`cofilteredLimit.finite_qcOpen_cover` descends a finite cover by compact opens as a cover
at one common stage, including the empty-cover case when the limit is empty.
The independent spectral statements are `spectralLimit_qcOpen`,
`spectralLimit_constructible`, `spectralLimit_eventual_inclusion` and
`spectralLimit_eventual_clopen`, with a specified limit cone and spectral transition maps
([Stacks] Tags 0A30 and 0A2Y).

`Perfectoid.countablePerfectoidSubrings p R ϖ` is precisely the poset `𝒮(R)` of P5.6:
closed subrings containing `ϖ` and its inverse, perfectoid for their subspace topology,
with a countable dense subset. `countableSubring_powerBounded` identifies its power-bounded
subring with the inverse image of `R°`. `countablePerfectoidHull` additionally contains the specified
countable subset and every chosen root. The directedness, covering and cardinal assertions
are `countablePerfectoidSubrings_directed`, `_cover`, `_card` and
`_isCardinalFiltered`. `countableSubringPair` takes the inverse image of the ambient plus
ring; `countableSubringDiagram`, `countableSubringCocone` and `countableSubringIsColimit`
specify the pair presentation. The projection identity of the countable-subring limit isomorphism
fixes the global comparison on all projections. `cofilteredLimit.card_colimit_lt` bounds the uncompleted integral colimit.
`cofilteredLimit.card_lt` retains one common strong-limit
stage bound `κ' < κ`; it does not infer such a bound from separate inequalities `|R_i| < κ`
([ECD] Proposition 6.5, p. 29, for the countable-subring presentation; Proposition 6.4, last clause
and first paragraph of the proof, p. 27, and Lemma 4.1, pp. 19–20, for the cardinal bounds).

`CategoryTheory.FilteredTwoColimit` uses Mathlib's localization of
`Pseudofunctor.Grothendieck F` at arrows whose fibre component is invertible.
`HomRepresentative` carries a stage, two legs and a morphism between the transported
objects. Its relation compares two representatives after a common refinement, including
the pseudofunctor's composition isomorphisms. `homEquiv_symm_mk` identifies the representative
with the composite of the two transport isomorphisms and the stage morphism.
`ι_obj`, `ι_map`, `ιCompIso_id`, `ιCompIso_comp` and `ιCompDescIso_coherence` fix the
inclusions and their coherence. A cocone is Mathlib's `Pseudofunctor.StrongTrans` to the
constant diagram. `descUniqueIso` fixes its comparison isomorphisms before asserting
uniqueness. `equivOfFinal_ι` specifies reindexing, and `fullSubcategory` restricts to a family
of full subcategories stable under transitions ([KL15] Remark 1.2.9, p. 15;
[Berkeley] Remark 7.4.7, p. 54). `fpModuleDiagram` uses
`Module.FinitePresentation` and Mathlib's extension of scalars, with comparison
`fpModuleColimitEquiv_ι` ([Stacks] Tag 05N7).

The geometric pseudofunctors are `finiteStageFiniteEtaleDiagram`, `finiteStageQcqsDiagram`
and `finiteStageSeparatedDiagram`. For every small category `I`,
diagram `X : I ⥤ Perfd p`, stage `i` and object in the indicated slice,
`finiteStageFiniteEtaleDiagram_mapId_hom_app`,
`finiteStageQcqsDiagram_mapId_hom_app` and
`finiteStageSeparatedDiagram_mapId_hom_app` identify the unit component
with `Over.pullbackId`, after transport by `X.map_id`. For arrows
`a : j → i` and `b : k → j`,
`finiteStageFiniteEtaleDiagram_mapComp_hom_app`,
`finiteStageQcqsDiagram_mapComp_hom_app` and
`finiteStageSeparatedDiagram_mapComp_hom_app` identify the composition
component with `Over.pullbackComp (X.map b) (X.map a)`, after transport by
`X.map_comp b a`. Thus the component goes from direct pullback to successive
pullbacks and preserves the projection to the original object. The
2-morphism action is equality transport. The equivalences to the limit
categories additionally require a cofiltered index and affinoid perfectoid
stages ([ECD] Proposition 6.4(i)–(iii), pp. 27–28). Their three equivalences,
ending in `Equiv`, have `_ι`
isomorphisms identifying each stage functor with geometric base change.
`finiteStage_faithfulness`, `_fullness`, `_essentialSurjectivity`, `_separatedness`,
`_surjectivity` and `_isomorphism` state recognition with qcqs hypotheses on the actual
étale slice objects. `finiteStagePullbackIso_fst` fixes the iterated pullback used in the
fullness formula ([ECD] Proposition 6.4(i)–(iii), pp. 27–28).

**Checks for the constructions and maps.** Each name below is a Check specified in this README
only, to become a Lean `example`.
The Checks above also give worked examples of these constructions.

| Construction or map | Check names and discriminating computations |
| --- | --- |
| General Tate colimit carrier | `tateColimitModel_zero`: a zero algebraic colimit forces a zero model; `tateColimitModel_nonzero`: a nonzero colimit cannot have a zero model; `tateColimitModel_terminal`: a terminal index gives a bijective canonical ring map. |
| Contracting disc diagram | `contractingDisc_one_step`: the coordinate maps to `ϖT`; `contractingDisc_two_steps`: it maps to `ϖ²T`; `contractingDisc_constants`: the scalar `ϖ` remains fixed; `completedColimit_test_not_injective`: the nonzero coordinate dies in the completion and the completed pair is `(K, K°)`; `completedColimit_proper_plus`: it is not isomorphic to `(K, K⁺)` when `K⁺ ⊊ K°`. |
| Completed pair, plus ring and inclusions | `completedColimit_test_initial`: a terminal ring index gives an isomorphism; `completedColimit_test_const`: the constant diagram recovers the original pair and each inclusion; `completedColimit_test_zero_terminal`: a zero terminal ring gives the zero plus ring and zero completed ring; `completedColimit_test_profinite`: the first binary coordinate gives a nontrivial idempotent, evaluating to `0` and `1` on the two constant sequences; `uncompleted_cylinders_dense_proper`: the cylinder subring is dense and not closed. |
| Universal map of completed pairs | `completedColimit_lift_identity`: the canonical cocone induces the identity; `completedColimit_lift_terminal`: the lift of a map from a terminal stage restricts to that map; `completedColimit_lift_zero`: every value is zero for a zero target. |
| Quotient diagram | `quotientDiagram_zero`: exponent zero gives the zero ring; `quotientDiagram_unit`: a unit gives the zero ring at every exponent; `quotientDiagram_zero_element`: for positive exponent and zero distinguished element, the quotient map is bijective. |
| Continuous-function pair | `continuousFunctions_empty`: the empty source gives the zero ring; `continuousFunctions_point`: evaluation at the unique point is bijective; `continuousFunctions_two_points`: evaluation gives the product of two copies of the coefficient ring. |
| Binary cylinder diagram | `binaryFunctions_zero_coordinates`: the zeroth stage is one copy of the coefficient ring; `binaryFunctions_one_coordinate`: the next stage is two copies; `binaryFunctions_two_terms`: the transition from one digit to two evaluates the original function on the first digit. |
| Global limit and projections | `cofilteredLimit_test_initial`: the initial stage is the limit; `cofilteredLimit_test_const`: every projection of a constant system is its canonical isomorphism; `cofilteredLimit_test_empty_stage`: a diagram with an empty stage has an empty limit; `cofilteredLimit_test_profinite`: the binary function diagram gives the continuous-function affinoid. |
| Limit lift | `cofilteredLimit_lift_identity`: the limit cone lifts by the identity; `cofilteredLimit_lift_two_terms`: for `1 → 0`, the lift is the component at `1` followed by the inverse stage comparison; `cofilteredLimit_lift_empty`: two cones with specified empty vertex have the same induced map. |
| Base change | Identity: base change of the identity gives the limit; empty: an empty source stays empty; split: a split degree-two cover gives two copies of the limit. |
| Rational restriction | `rationalLimit_empty`: an empty rational domain gives the empty limit; `rationalLimit_whole`: the whole domain gives an isomorphism; `rationalLimit_clopen`: a clopen rational domain remains clopen in the limit. |
| Countable-subring poset | `countableSubrings_separable_top`: a separable ambient ring is its largest member; `countableSubrings_nonseparable_top`: a nonseparable ring is not a member; `countableSubrings_zero`: the zero ring has exactly one member; `countableSubrings_countable_union`: a countable family has a common upper bound; `countableSubrings_nonseparable_no_countable_cofinal`: no countable family is cofinal in the nonseparable case; `countableSubrings_no_countable_tower`: no completed colimit of any countable filtered diagram of separable perfectoid pairs can be a nonseparable pair. |
| Categorical colimit | `filteredTwoColimit_test_const`: a constant discrete Boolean fibre has singleton endomorphism sets and no morphisms between its two objects; `filteredTwoColimit_empty`: empty fibres give an empty colimit; `filteredTwoColimit_bool`: the two objects of a constant discrete Boolean fibre have no cross-morphism; `filteredTwoColimit_test_discrete`: the colimit is equivalent to the discrete category on the set colimit, and a stage element maps to its canonical image; `filteredTwoColimit_test_finitelyPresentedModules`: the comparison is extension of scalars; `filteredTwoColimit_test_notNaive`: an isomorphism first acquired at stage `1` survives. |
| Finite étale diagram coherence | `finiteStageFiniteEtaleDiagram_identity_projection`: the unit is the first pullback projection; `finiteStageFiniteEtaleDiagram_two_steps`: the composition component preserves the projection through both stages in their specified order; `finiteStageFiniteEtaleDiagram_nontrivial_automorphism`: a nonidentity automorphism changes the unit component. |
| Qcqs diagram coherence | `finiteStageQcqsDiagram_identity_projection`: the unit is the first pullback projection; `finiteStageQcqsDiagram_two_steps`: successive pullbacks preserve the projection to the original object; `finiteStageQcqsDiagram_nontrivial_automorphism`: a nonidentity automorphism changes the unit component. |
| Separated diagram coherence | `finiteStageSeparatedDiagram_identity_projection`: the unit is the first pullback projection; `finiteStageSeparatedDiagram_two_steps`: successive pullbacks preserve the projection to the original object; `finiteStageSeparatedDiagram_nontrivial_automorphism`: a nonidentity automorphism changes the unit component. |
| Coherent transport | `transport_empty_unique`: empty fibres give the unique natural comparison; `transport_bool_false`: transport fixes the Boolean object `false`; `transport_automorphisms`: transport is the identity after the constant comparison even in the one-object category with automorphism group `Perm Bool`. |
| Cocone extension | `desc_empty_unique`: the empty-fibre colimit has one functor to a fixed category; `desc_bool_collapse_not_full`: collapsing the two Boolean objects is not full; `desc_bool_identity`: fixing both Boolean objects induces an equivalence. |
| Global quotient diagram | `globalPlusQuotient_zero_exponent`: exponent zero gives zero; `globalPlusQuotient_unit`: a unit gives zero at every exponent; `globalPlusQuotient_zero_section`: the quotient by a positive power of zero retains every section. |
| Profinite function diagram | `profiniteFunctions_empty_stage`: the empty stage gives zero; `profiniteFunctions_singleton_stage`: evaluation gives the coefficient ring; `profiniteFunctions_two_point_stage`: two evaluations give its square. |
| Profinite product | `profiniteProduct_empty`: the product has no points; `profiniteProduct_point`: a singleton recovers the coefficient affinoid; `profiniteProduct_two_points`: two points give its binary coproduct. |
| Full subdiagrams | `fullSubdiagram_empty`: empty full subcategories give an empty colimit; `fullSubdiagram_whole`: all objects of the constant Boolean category recover that category; `fullSubdiagram_singleton`: retaining only `false` gives a category equivalent to the one-point category. |
| Cospan morphisms | `cospan_constant_empty_hom`: no representative connects the two distinct Boolean objects; `cospan_constant_identity`: constant point fibres give a singleton quotient; `cospan_constant_automorphisms`: the quotient recovers the original Hom set, including its automorphisms. |
| Finitely presented module category | `fpModules_zero_ring`: every module over the zero ring is zero; `fpModules_rank_one`: the free rank-one module is finitely presented; `fpModules_infinite_basis`: a countably infinite free module over a nonzero ring is not finitely presented. |
| Étale subcategories | `etaleSubcategories_empty`, `etaleSubcategories_identity`, `etaleSubcategories_split`: the empty arrow, identity and split degree-two arrow belong to both the qcqs and the qc separated subcategories. |
| Detection boundaries | `finiteStage_identity_detection`: in a constant affinoid diagram, pullback to the limit detects whether an arbitrary étale morphism is an isomorphism; `finiteStage_empty_detection`: a map from an empty space to a nonempty space is not surjective. |
| Separatedness and isomorphism detection | `finiteStage_glued_not_separated`: gluing two copies along a nonclosed open gives a nonseparated fold; `finiteStage_glued_becomes_iso`: restriction to the gluing domain makes the fold an isomorphism; `finiteStage_glued_proper`: the original fold is not an isomorphism if the gluing domain is proper. |
| Cantor digit function | `cantorDigits_zero`: the all-zero sequence has value zero; `cantorDigits_singleton`: the sequence supported at `n` has value `p^n` in `ℂ_p`; `cantorDigits_zero_locus`: only the all-zero sequence vanishes, also for `p = 2`; `finiteStage_nonflat_quotient`: its principal quotient of `C(S,ℂ_p)` is not flat; `finiteStage_punctured_cantor`: the complement of the zero sequence is open, noncompact, and not pulled back from any finite coordinate set; `finiteStage_quotient_no_descent`: no algebra over a finite-coordinate product of fields base-changes to this principal quotient; `finiteStage_punctured_cantor_space`: the actual open perfectoid subspace is étale, not quasicompact, and is not the preimage of any subset at a finite stage. |
| Failure of faithfulness without quasicompactness | `finiteStage_nonqc_faithfulness`: over the contracting perfected-disc tower there are two maps with nonquasicompact étale source which agree over the limit and disagree at every finite stage. |

### Examples

The worked examples of this layer are the profinite set `S = lim_n {0,1}^n` over a perfectoid field
`K`, run through every construction: `R⁺ = C^0(S, O_K)` and `R = C^0(S, K)` as the completed direct
limit of the `K^{S_n}`, with `R⁺/ϖ^n` the locally constant functions to `O_K/ϖ^n`, the function
`s ↦ Σ_k s_k ϖ^k` outside every finite stage, and the uncompleted `LC(S, K)` as the non-example; the
limit space `S × Spa(K, O_K)` with underlying space `S × |Spa(K, O_K)|`, its restriction to a clopen
`C × Spa(K, O_K)`, the base change `S × Y_0` of a finite étale `Y_0`, the Cantor-set open `S ∖ {s}`
and the non-quasicompact `X ∖ ({s₀} × Spa(K, O_K))` that do not descend, and finite étale, qcqs
étale and quasicompact separated étale spaces over `S × Spa(K, O_K)` being locally constant on `S`;
the non-injective system `R_n = K⟨T^{1/p^∞}⟩` with `T^{1/p^m} ↦ ϖ^{1/p^m}T^{1/p^m}`, where `ι_0(T) = 0`
and `colim R_n⁺ → R⁺ = O_K` is neither injective nor, for the profinite system, surjective; the
residue-field limit `Spa(K(x), K(x)⁺) = lim_{U ∋ x} U` with `|Spa(K(x), K(x)⁺)| = ⋂ U`; the
non-perfectoid tower `Spa(ℚ_p(p^{1/p^n}), ℤ_p[p^{1/p^n}])`, excluded from the construction;
`C^0({0,1}^{ℵ₁}, ℂ_p)` with its cofinal countably generated subrings `C^0({0,1}^J, ℂ_p)`;
`|C^0({0,1}^ℕ, ℂ_p)| = 2^{ℵ₀}` below every cutoff cardinal; the discrete, constant, finitely
presented module and non-naive 2-colimits; the finitely presented non-flat `C^0(S, K)/(f)` that comes
from no finite stage; and the glued non-separated `Y_1` over `{|T| ≤ |ϖ|}` that becomes separated
over `{|T| ≤ |ϖ|^2}`.

Layer P5 carries no Lean `example` in `Suggested.lean`; its Checks are the computations above.

### Dependencies

Layers P1 (perfectoid Tate rings, completed filtered colimits and compatible roots),
P2 (rational subsets, sheafiness, affinoid perfectoid spaces and fibre products),
P3 (almost purity, étale morphisms, the étale site and sousperfectoid spaces) and
P4 (qcqs and separated morphisms, pro-rational subspaces and residue-field injections).
P5.1–P5.8 use P1 and P2; P5.7 also uses P6.1–P6.2's cardinal bounds, which use P1 and P2.
P5.9–P5.12 use P3 and P4. Other roadmaps: AdicSpacesPartII R0.19.5–R0.19.8 supplies
spectral limits, Tate-pair spectra and finite-stage opens; AdicEtaleGeometry A1.16 supplies
`CAff` and maps into sheafy affinoids; A1.0.3 and A1.0.5 supply algebraic finite étale
colimits and completion; A1.3 supplies local étale descriptions.
P5.13.1–P5.13.4 use the local sousperfectoid theory of P3.17a, the completed limits and qcqs descent
of P5.1–P5.12, AdicEtaleGeometry A1–A2 and A3.1–A3.2's generic pseudocoherent, Newton
and standard-presentation APIs, and AdicSpacesPartII R0 and AdicSpacesPartIII R3's analytic algebra.
Mathlib's strong-limit cardinal arithmetic is used throughout. Mathlib
`CommRingCat.FilteredColimits.colimit`, `CommRingCat.FilteredColimits.colimitCoconeIsColimit`,
`AdicCompletion`, `AdicCompletion.isAdicComplete`, `IsAdicComplete`, `IsIntegrallyClosedIn`,
`CategoryTheory.IsCofiltered`, `CategoryTheory.Over`, `CategoryTheory.Over.initial_forget`,
`CategoryTheory.Functor.Final.colimitIso`, `CategoryTheory.Limits.IsLimit`,
`CategoryTheory.IsCardinalFiltered`, `Cardinal.isRegular_aleph_one`, `Cardinal.IsStrongLimit`,
`Cardinal.mk_set`, `Cardinal.power_le_power_left`, `SpectralSpace`, `IsSpectralMap`,
`WithConstructibleTopology`, `TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system`,
`QuasiSeparatedSpace`, `CategoryTheory.Pseudofunctor`, `CategoryTheory.IsFiltered`,
`CategoryTheory.Functor.IsEquivalence`, `CategoryTheory.Grothendieck`, `CommAlgCat.FiniteEtale`;
Tau Ceti `TauCeti.Huber.Pair`, `TauCeti.Huber.Pair.Hom` with `spaComap`, `continuous_spaComap`,
`spaComap_comp`, `spaComap_preimage_rationalSubset`, `TauCeti.Huber.IsTateRing`,
`TauCeti.Huber.PairOfDefinition`, `TauCeti.Huber.IsPseudoUniformizer` with `hasBasis_nhds_zero`,
`TauCeti.Huber.IsRingOfIntegralElements` with `mem_of_isTopologicallyNilpotent`,
`TauCeti.Huber.powerBoundedSubring`, `TauCeti.ValuationSpectrum` with `comap`, `spa`,
`mem_spa_iff`, `rationalSubset`, `spectralSpace_spa_of_pairOfDefinition`,
`TauCeti.IsProConstructible` with `isCompact`.

## Layer P6: `κ`-small perfectoid spaces and pro-étale morphisms

The layer builds the size and the pro-étale notions on which the pro-étale and v-sites of
DiamondsAndVStacks rest. It defines `κ`-small and uniformly `κ`-small perfectoid spaces for an
uncountable strong limit cardinal `κ`, proves the cover criterion with its uniform bound for
singular `κ`, and proves closure under open subspaces, finite disjoint unions and tilting,
and closure of uniformly `κ`-small spaces under fibre products. It also
records the finite-stage form of ECD Proposition 6.4(iv). It then defines affinoid pro-étale
morphisms as cofiltered limits of affinoid étale ones, pro-étale morphisms as their local form,
and the product `X × S` of an affinoid perfectoid space with a profinite set; proves that
`X_proét^aff` is the pro-category of the affinoid étale site, with its `κ`-bounded refinement; and
proves the stability of (affinoid) pro-étale morphisms under composition, base change,
cancellation, sections and small limits, that Zariski closed immersions and diagonals are
pro-étale, and the comparison with Scholze 2013's affinoid perfectoid objects. Throughout, a
perfectoid space is one of layer P2 (P2.10), locally `Spa(R, R⁺)` for a perfectoid Tate pair in
the sense of ECD Definition 3.1, with no perfectoid base field; étale morphisms are those of layer
P3 (P3.15, ECD Definition 6.2); cofiltered limits of affinoid perfectoid spaces are those of layer
P5 (P5.3).

### P6.1 `κ`-small perfectoid spaces

The first five subsections concern `κ`-small perfectoid spaces: the definition, its uniform variant,
the cover criterion, the closure properties and the finite-stage descent of affinoid étale objects.

**`κ`-small perfectoid spaces.** (*kappa-small-perfectoid-space*) Fix a cardinal `κ` with `ℵ₀ < κ`
that is a strong limit (`2^λ < κ` for every cardinal `λ < κ`); every cutoff cardinal of
[ECD] Lemma 4.1, pp. 19–20, is one, and since Mathlib's
`Cardinal.IsStrongLimit` counts `ℵ₀` as a strong limit, `ℵ₀ < κ` is a separate hypothesis. A
perfectoid space `X` (layer P2 (P2.10): an adic space locally isomorphic to `Spa(R, R⁺)` for a
perfectoid Tate pair `(R, R⁺)` in the sense of ECD Definition 3.1, with no perfectoid base field)
is `κ`-small if (i) its underlying set `|X|` has cardinality `#|X| < κ`, and (ii) for every open
affinoid perfectoid subspace `U = Spa(A, A⁺) ⊆ X` (so `A = O_X(U)`) the ring `A` has cardinality
`#A < κ`. Cardinalities are those of underlying sets; the plus rings carry no condition; clause
(ii) quantifies over all open affinoid subspaces, of which there are at most `2^{#|X|} < κ`, so no
separate count of charts is imposed; neither clause implies the other (a one-point space with a
large field; a disjoint union of `κ` points). The predicate is invariant under isomorphism and
monotone in `κ`; the bound in (ii) is not uniform over the charts, the stronger notion with a
single bound `λ < κ` being P6.2, and for quasicompact `X` the two agree. In `PerfectoidSpace`,
define `IsKappaSmall` as the structure `IsKappaSmall κ X` with the two clauses, with projections
`IsKappaSmall.card_lt` (`#|X| < κ`) and `IsKappaSmall.card_sections_lt` (`#O_X(U) < κ` for every
open affinoid perfectoid `U ⊆ X`); prove transport along `X ≅ Y`,
`IsKappaSmall.mono` (`κ ≤ κ'` carries `κ`-small to `κ'`-small), `IsKappaSmall.restrict` (open subspaces
of `κ`-small spaces are `κ`-small) and `IsKappaSmall.empty`
(for `1 < κ`); define
`KappaSmall`, the full subcategory of perfectoid spaces on the `κ`-small objects, with
the instance `KappaSmall.essentiallySmall`; the affinoid form `IsKappaSmall_affinoid κ (R, R⁺)`
is stated for an arbitrary Huber ring `R` by the three clauses `#Spa(R, R⁺) < κ`, `#R < κ` and
`#R⟨T/s⟩ < κ` for every presentation with `T` generating the unit ideal. For general Huber
rings this is a cardinal predicate, with no assertion that `R` is the ring of global sections.
For perfectoid `R`, such presentations give a rational basis, and for uncountable strong limit `κ`
these clauses are equivalent to the two clauses above and to `#R < κ` alone (P6.3 (d)). The predicate
and its monotonicity are meaningful for all cardinals; uncountability and the strong-limit
condition are hypotheses of the comparison and closure theorems. For `κ`-small `X`,
`X_proét,κ` is the category of `κ`-small perfectoid spaces pro-étale over `X` (ECD Definition 8.1),
on which the sites of
DiamondsAndVStacks layer D2 live ([ECD] §4, Def. 4.2, p. 20; [ECD] §4, after Def. 4.2, p. 20, for
the standing hypothesis on `κ`; [ECD] §8, Def. 8.1, p. 39). *Needs:* layer P2 (P2.10, P2.9); layer P1 (P1.1); Mathlib `Cardinal.IsStrongLimit`, `Cardinal.mk`,
`Cardinal.mk_set`, `CategoryTheory.EssentiallySmall`.

**Checks.**

- Computation: for every cardinal `κ` with `ℵ₀ < κ` and `κ` a strong
  limit, `Spa(C_p, O_{C_p})` is `κ`-small: it has one point and `#C_p = 2^ℵ₀ < κ`.
- Degenerate: the empty perfectoid space has zero points and a
  one-element section ring, so it is `κ`-small exactly when `1 < κ`, including `κ = ℵ₀`.
- At cutoff `2` the empty perfectoid space is small, and every nonempty perfectoid space is
  excluded: an affinoid perfectoid neighborhood of a point has at least two sections.
- Boundary: no affinoid pair
  satisfies the cardinal predicate at `κ = 0` or `κ = 1`; the zero ring satisfies it at `κ = 2`.
- `not_isKappaSmall_Cp_aleph0` (non-example) — `Spa(C_p, O_{C_p})` is not `ℵ₀`-small.
- Strict boundary: for a Huber pair over `R`, `κ`-smallness
  and `κ ≤ κ'` imply `κ'`-smallness, but the pair is never `#R`-small: equality with
  the section-ring cardinal is excluded by the strict bound, including for the zero ring.
- Characterisation: for an affinoid perfectoid space
  `X = Spa(A, A⁺)`, `IsKappaSmall κ X ↔ #A < κ`.
- `not_isKappaSmall_bigField` (non-example) — for `K` the completion, for the Gauss norm, of the
  perfection of `F_p((t^{1/p^∞}))(x_β : β < κ)`, a perfectoid field with `#K ≥ κ`, `Spa(K, K°)` has
  one point but is not `κ`-small: clause (i) does not imply clause (ii).
- `not_isKappaSmall_sigma` (non-example) — the disjoint union of `κ` copies of `Spa(C_p, O_{C_p})`
  is not `κ`-small, although each open affinoid subspace (a finite union of copies) has ring of
  cardinality at most `2^ℵ₀` (equal to `1` for the empty open): clause (ii) does not imply
  clause (i).

The global predicates use `Perfd`, its underlying pre-adic space and its affinoid perfectoid opens
(P2.10), together with `ofPair` from P2–P4.
`IsKappaSmall.card_sections_lt` measures the ring of sections over each open affinoid perfectoid
subspace, so different presentations of an open do not change the condition. The cardinal `κ`
lives in the universe `u` of the spaces and their section rings (`κ : Cardinal.{u}`), so no lifting
of cardinalities is needed. Global versions of the first, second and fourth Checks are stated on
`Perfd`; the proved Lean example states the strictness boundary at `0`, and
`PerfectoidSpace.isKappaSmall_two_iff_isEmpty` states the cutoff-`2` characterisation.

**Checks for the full subcategory.**

- `kappaSmall_inherited_maps` — the Hom type between two `κ`-small objects is exactly the Hom
  type between their underlying perfectoid spaces, including noninvertible maps.
- `kappaSmall_empty_object` — the empty space is an object precisely above the threshold `1`.
- `kappaSmall_excludes_Cp_aleph0` — no object at cutoff `ℵ₀` has underlying space `Spa(C_p,O_Cp)`.

For the large-field Check, construct `GaussPerfectoidField K J` and `gaussPerfectoidField K J`:
a complete perfectoid field with maximal plus ring, an isometric map from `K`, and independent
variables `t_{j,n}` satisfying `t_{j,n+1}^p=t_{j,n}`. Require the Gauss maximum-coefficient norm
on every finite polynomial level and density of the subfield generated by the constants and all
`t_{j,n}`. These equations specify the completion of `K(t_j^{1/p^∞})`; the map on constants
and the variables determine every continuous map out of it. Counting rational expressions and
Cauchy sequences gives `#J ≤ #L ≤ (#K+#J+ℵ₀)^ℵ₀` (the completion estimate of [ECD] Remark 4.3,
p. 20, applied to this dense field).

**Checks for the Gauss field.**

- `gaussField_empty_variables` — for `J=∅`, the map `K → L` is bijective.
- `gaussField_two_terms` — for one variable, `‖a+b t₀‖=max(‖a‖,‖b‖)`.
- `gaussField_distinct_variables` — with two variables, `t_{false,0} ≠ t_{true,0}`.
  Thus the lower cardinal bound counts distinct field elements, even in characteristic two.

Lean: `PerfectoidSpace.IsKappaSmall` (fields `card_lt` and `card_sections_lt`, over the affinoid
perfectoid opens), `IsKappaSmall.mono`, `IsKappaSmall.restrict`, and `isKappaSmall_two_iff_isEmpty`.

### P6.2 uniformly `κ`-small perfectoid spaces

**Uniformly `κ`-small perfectoid spaces.** (*uniformly-kappa-small-perfectoid-space*) Let `κ` be an
uncountable strong limit cardinal. A perfectoid space `X` is uniformly `κ`-small if there is a
cardinal `λ < κ` such that `#|X| ≤ λ` and `#O_X(U) ≤ λ` for every open affinoid perfectoid
subspace `U ⊆ X`; equivalently, `X` is `κ'`-small (see *kappa-small-perfectoid-space*) for some cardinal
`κ' < κ` (take `κ' = λ⁺`, which is `< κ` because `κ` is a limit cardinal). Properties: (a)
uniformly `κ`-small implies `κ`-small; (b) a quasicompact `κ`-small perfectoid space is uniformly
`κ`-small, so the two notions have the same affinoid and the same qcqs objects; (c) if `κ` is regular
(a strongly inaccessible cardinal) the two notions agree; (d) they differ for every singular `κ`
(take `cf κ` summands), in particular for every `κ` of cofinality `ω₁` such as the cutoff
cardinals `κ(µ)` with `cf µ = ω₁` constructed in the proof of ECD Lemma 4.1:
`⊔_{α<ω₁} Spa(K_α, K_α°)`, with perfectoid fields `K_α` whose cardinalities are `< κ` and
cofinal in `κ` (for infinite `λ < κ` the completed perfection of `𝔽_p((t^{1/p^∞}))(x_β : β < λ)` has
cardinality between `λ` and `2^λ < κ`), is `κ`-small (`ℵ₁ ≤ 2^{ℵ₀} < κ` points, finite products of
the `K_α` as affinoid rings) and not uniformly `κ`-small. The uniformly `κ`-small spaces
satisfy the cover criterion and the stability under fibre products that ECD Proposition 4.4 states
for `κ`-small spaces (P6.3, P6.4); ECD's own set-theoretic statements (Proposition 6.4, Lemma 11.22)
are formulated with such a uniform bound `κ' < κ`, and the class is intended for the cutoff
cardinals of [ECD] Lemma 4.1, pp. 19–20, which are in general singular. In
`PerfectoidSpace`, define `IsUniformlyKappaSmall` as the predicate
`∃ λ < κ, #|X| ≤ λ ∧ ∀ U, #O_X(U) ≤ λ`, `U` ranging over the affinoid perfectoid opens; prove the
reformulation `IsUniformlyKappaSmall κ X ↔ ∃ κ' < κ, IsKappaSmall κ' X`,
`IsUniformlyKappaSmall.isKappaSmall` (a), and the statements for (b) (quasicompact `κ`-small spaces
are uniformly `κ`-small) and (c) (the two notions agree for regular `κ`), the affinoid criterion
`IsUniformlyKappaSmall κ (Spa(A, A⁺)) ↔ #A < κ`, `IsUniformlyKappaSmall.isOpen` (open subspaces,
with the same `λ`) and `IsUniformlyKappaSmall.fibreProduct` (P6.4) ([ECD] §6, Prop. 6.4, final
sentence, p. 27; [ECD] §4, proof of Lem. 4.1, p. 20, for `cf κ(ω₁) = ω₁ < κ(ω₁)`, the singular
cutoff of the separating example). *Needs:* *kappa-small-perfectoid-space*;
*complete-tate-ring-cardinality*; layer P2 (P2.8, P2.3); layer P1; Mathlib `Cardinal.iSup_lt_of_lt_cof_ord`, `Cardinal.IsRegular`, `Cardinal.mk_iUnion_le`.

**Checks.**

- Computation: `Spa(C_p⟨T^{1/p^∞}⟩, O_{C_p}⟨T^{1/p^∞}⟩)` is
  uniformly `κ`-small with `λ = 2^ℵ₀` for every uncountable strong limit `κ`.
- Degenerate: the empty perfectoid space is uniformly `κ`-small with
  `λ = 1`: its point set is empty and its ring of sections on the empty affinoid open is the zero
  ring, whose underlying set has one element. For arbitrary cutoffs it is uniformly `κ`-small
  exactly when `1 < κ`; `PerfectoidSpace.isUniformlyKappaSmall_two_iff_isEmpty` states the
  cutoff-`2` characterisation.
- Every nonempty perfectoid space fails uniform smallness at cutoff `2`, since uniform
  smallness implies smallness; the same named characterisation excludes this case.
- Characterisation: for a quasicompact
  perfectoid space `X`, `IsUniformlyKappaSmall κ X ↔ IsKappaSmall κ X`.
- `not_isUniformlyKappaSmall_of_cof_omegaOne` (non-example) — for `κ` with `cf κ = ω₁` and
  perfectoid fields `K_α` (`α < ω₁`) with `#K_α < κ` cofinal in `κ`, `⊔_{α<ω₁} Spa(K_α, K_α°)` is
  `κ`-small and not uniformly `κ`-small.

**Completion cardinality.** (*first-countable-completion-cardinality*)
P6.2 owns `completion_cardinality_bound`: for `G : Type u` with `AddGroup G`,
`UniformSpace G`, `IsUniformAddGroup G`, `T2Space G` and `FirstCountableTopology G`,
a dense subset `A ⊆ G` with `#A ≤ λ` and `ℵ₀ ≤ λ` gives
`#(UniformSpace.Completion G) ≤ λ^ℵ₀`, all cardinals in universe `u`.
Commutativity and completeness of `G` are not required. This is the sequence argument
of [ECD] Remark 4.3, p. 20. DiamondsAndVStacks D0.24 consumes this theorem and
uses cardinal arithmetic to conclude `λ^ℵ₀ ≤ 2^λ < κ` for `λ < κ` and strong-limit `κ`.

The cardinal bounds behind the uniform notion are ring-theoretic
(*complete-tate-ring-cardinality*). Let `A` be a complete Hausdorff Tate ring (Tau Ceti
`IsTateRing`, with its topology complete and separated). (a) If `D ⊆ A` is a dense subset
with `#D ≤ λ` for an infinite cardinal `λ`, then `#A ≤ λ^ℵ₀ ≤ 2^λ`: apply
`completion_cardinality_bound` below to the additive group of `A` (Hausdorffness is needed, else limits of sequences are not
unique; completeness is not needed for (a), only for the setting of (b)). Prove: (b) with `λ_A := #A + ℵ₀`: (b1) for a rational subset `U = R(T/s)` of `Spa(A, A⁺)` the
rational localisation `A⟨T/s⟩` (Tau Ceti `completionLocObj`) has cardinality `≤ λ_A^ℵ₀`, the image
of `A[1/s]` being dense; (b2) a finite étale `A`-algebra `B` with its canonical topology has
`#B ≤ λ_A`, being a quotient of some `Aⁿ`; (b3) for morphisms of complete Tate pairs `A → B` and
`A → C`, the completed tensor product `B ⊗̂_A C` (AdicSpacesPartII R0.3
(*completed-tensor-product*)) and its
uniformisation (AdicSpacesPartII R0.4 (*uniform-completed-tensor-product*)) have cardinality `≤ (#B·#C·ℵ₀)^ℵ₀`, the image of `B ⊗_A C` being dense;
(c) if `κ` is an uncountable strong limit, the bound in (a) is `< κ` when `λ < κ`;
the bounds in (b1)–(b2) are `< κ` when `#A < κ`, and those in (b3) when both `#B < κ`
and `#C < κ`. The
localisation, finite étale and completed tensor clauses (b) are this target's; the first
countability needed here holds because the `ϖⁿA₀` form a countable basis of neighbourhoods
of `0` (Tau Ceti `IsPseudoUniformizer.hasBasis_nhds_zero`) ([ECD] §4, Rem. 4.3, p. 20, for (a);
[ECD] §4, proof of Prop. 4.4, pp. 20–21, for (b1)). *Needs:* *first-countable-completion-cardinality*; Tau Ceti
`TauCeti.Huber.IsTateRing`, `TauCeti.Huber.IsPseudoUniformizer.hasBasis_nhds_zero`,
`TauCeti.Huber.PairOfDefinition.completionLocObj`; AdicSpacesPartII R0.3, R0.4
(*completed-tensor-product*, *uniform-completed-tensor-product*);
Mathlib `FirstCountableTopology`, `UniformSpace.Completion`, `Cardinal.power_mul`,
`Cardinal.power_le_power_right`.

The proposed Lean ring bounds are `rationalLocalization_cardinality`, `finiteAlgebra_cardinality`
and `completedTensor_cardinality`. The tensor statement uses the supplied adic scalar maps;
continuous maps from a Tate base are adic. The finite-algebra bound only requires a finite module.

Lean: `PerfectoidSpace.IsUniformlyKappaSmall`, `IsUniformlyKappaSmall.isKappaSmall`,
`completion_cardinality_bound` (*first-countable-completion-cardinality*),
`IsUniformlyKappaSmall.of_isKappaSmall_of_compactSpace`, and
`isUniformlyKappaSmall_two_iff_isEmpty`.

### P6.3 the cover criterion for `κ`-smallness

**Cover criterion for `κ`-smallness.** (*kappa-small-cover-criterion*) Let `κ` be an uncountable
strong limit cardinal and `X` a perfectoid space. Prove: (a) if `X` is `κ`-small, then `X` is
covered by fewer than `κ` open affinoid subspaces `Spa(A, A⁺)` with `#A < κ` (namely by all of
them: there are at most `2^{#|X|} < κ`); (b) if `X` has a cover `{U_j = Spa(A_j, A_j⁺)}_{j∈J}` by
open affinoid subspaces with `#A_j < κ` for all `j`, then every open affinoid subspace
`U = Spa(A, A⁺) ⊆ X` has `#A < κ` (the sheaf condition embeds `A` into a finite product
of rings of rational subspaces of the cover charts), so such an `X` is `κ`-small as soon as `#|X| < κ`; (c) if moreover
`#J < κ` and `sup_j #A_j < κ` — automatic when `J` is finite, or when `#J < cf κ` — then
`#|X| < κ`, and `X` is `κ`-small, indeed uniformly `κ`-small (see *uniformly-kappa-small-perfectoid-space*);
(d) in particular an affinoid perfectoid space `Spa(A, A⁺)` is `κ`-small iff `#A < κ`, and a
quasicompact perfectoid space is `κ`-small iff it has a finite cover by open affinoid subspaces whose
rings have cardinality `< κ`; (e) the converse of (a) as printed in ECD Proposition 4.4 — a cover by
fewer than `κ` open affinoids with rings of cardinality `< κ`, without a uniform bound, implies
`κ`-small — holds for regular `κ` and can fail as soon as `cf κ ≤ #J < κ`: for `κ` of cofinality `ω₁` (such
as the cutoff cardinal `κ(ω₁)` of the proof of ECD Lemma 4.1) and perfectoid fields `K_α`,
`α < ω₁`, with `#K_α < κ` cofinal in `κ`, the disjoint union of the perfectoid discs
`Spa(K_α⟨T^{1/p^∞}⟩, K_α°⟨T^{1/p^∞}⟩)` is covered by `ω₁ < κ` open affinoids with rings of
cardinality `< κ` and has at least `κ` points. Part (c) needs the uniform bound, and (e) records
exactly where the printed statement needs it. These are the affinoid criterion (in Lean
`isKappaSmall_presentationLimit_iff`: for uncountable strong limit `κ`, `Spa(A, A⁺)` of a perfectoid
pair is `κ`-small iff `#A < κ`) and the cover statement (a cover by open affinoids with `#A_j < κ` and
`#|X| < κ` gives `κ`-small) of P6.1 ([ECD] §4, Prop. 4.4, p. 20, the printed statement, whose correct parts are (a)–(d); [ECD]
§4, proof of Prop. 4.4, p. 20, for (b) and for the step that needs `sup_j #A_j < κ`; [ECD] §4,
proof of Prop. 4.4, p. 21, for the embedding into a finite product in (b)). A countable disjoint
union of copies of `Spa(C_p, O_{C_p})` is `κ`-small for every cutoff `κ` by (c) (`#J = ℵ₀ < cf κ`).
*Needs:*
*kappa-small-perfectoid-space*; *uniformly-kappa-small-perfectoid-space*;
*adic-spectrum-cardinality*; *complete-tate-ring-cardinality*; layer P2 (P2.10, P2.9, P2.8,
P2.3); layer P1; Tau Ceti
`TauCeti.ValuationSpectrum.spectralSpace_spa_of_pairOfDefinition`; Mathlib
`Cardinal.iSup_lt_of_lt_cof_ord`, `Cardinal.mk_iUnion_le`, `Cardinal.mk_set`.

The point count behind (a) and (c) is the following (*adic-spectrum-cardinality*). Let `A` be a
Huber ring and `A⁺ ⊆ A` a subring, with no hypothesis on `A⁺` or on the topology. Prove that
`#Spa(A, A⁺) ≤ 2^{#A·#A}`: a point of `Spv A` is determined by its valuative relation
`{(a, b) ∈ A × A : |a(x)| ≤ |b(x)|}` (Tau Ceti `ValuationSpectrum.ext'`), so `Spa(A, A⁺) ⊆ Spv A`
injects into the power set of `A × A`; for `A` infinite this is `2^{#A}`. Consequently, if `κ` is
an uncountable strong limit and `#A < κ` then `#Spa(A, A⁺) < κ`, and a perfectoid space covered by
open affinoids `Spa(A_j, A_j⁺)`, `j ∈ J`, has at most `#J · sup_j 2^{#A_j·#A_j}` points ([ECD] §4,
proof of Prop. 4.4, p. 20, the injection into the binary relations on `A` and the bound
`2^{#A·#A}`). *Needs:* Tau Ceti `TauCeti.ValuationSpectrum`, `TauCeti.ValuationSpectrum.ext'`,
`TauCeti.ValuationSpectrum.spa`; Mathlib `Cardinal.mk_set`, `Cardinal.mk_prod`,
`Cardinal.mul_eq_self`, `Cardinal.mk_le_of_injective`, `Cardinal.mk_iUnion_le`.

**Check: `not_isKappaSmall_gaussDisc_sigma`.** Let `κ` be a strong limit with
`cf κ = ℵ₁ < κ`, let `L` be a characteristic-`p` perfectoid field with `#L < κ`, and
choose infinite variable sets `J_α` for `α < ω₁` with `#J_α < κ` cofinal in `κ`.
For `K_α = (gaussPerfectoidField p L J_α).field`, the theorem supplies compatible
pseudouniformizer roots and takes the discs on `discPair`. Each disc ring has
cardinality `< κ`; the `ω₁` coproduct inclusions are open immersions and cover the
coproduct. Its point set has cardinality at least `κ`, so it fails `IsKappaSmall`.
Indeed, characteristic `p` supplies compatible roots for every integral field element,
whose evaluations give distinct disc points. The Gauss field cardinal bounds make
the stage sizes cofinal. This is the negative control for the converse printed in
[ECD] Proposition 4.4, pp. 20–21, using the completion estimate of Remark 4.3, p. 20.
This coproduct Check is specified in this README only.

Lean: `isKappaSmall_presentationLimit_iff` and
`PerfectoidSpace.isKappaSmall_of_affinoid_cover` (the cover criterion with a point-set bound).

### P6.4 closure properties of `κ`-small perfectoid spaces

**Closure properties of `κ`-small perfectoid spaces: open subspaces, disjoint unions, fibre
products.** (*kappa-small-closure*) Let `κ` be an uncountable strong limit cardinal. Prove: (a) an
open subspace of a `κ`-small (resp. uniformly `κ`-small) perfectoid space is `κ`-small (resp. uniformly
`κ`-small) (`IsKappaSmall.isOpen`, `IsUniformlyKappaSmall.isOpen`); (b) a disjoint union
`⊔_{j∈J} X_j` is uniformly `κ`-small if `#J < κ` and all `X_j` are `κ'`-small for one `κ' < κ`; finite
disjoint unions of `κ`-small spaces are `κ`-small; (c) (ECD Proposition 4.4, 'in particular') for
morphisms `X → Z ← Y` of perfectoid spaces with `X` and `Y` uniformly `κ`-small (`Z` arbitrary, not
necessarily `κ`-small), the fibre product `X ×_Z Y` (layer P2 (P2.16), over a general perfectoid
Tate base, with affinoid charts given by the uniform completed tensor product) is uniformly
`κ`-small (`IsUniformlyKappaSmall.fibreProduct`): if `#|X|, #|Y| ≤ λ` and all open affinoid rings of
`X` and `Y` have cardinality `≤ λ` (`λ ≥ ℵ₀`), then `X ×_Z Y` is covered by at most `2^λ` open
affinoids `U ×_W V` with rings of cardinality `≤ λ^ℵ₀` and has at most `2^{2^λ}` points; in
particular fibre products of `κ`-small quasicompact (for instance affinoid) perfectoid spaces are
`κ`-small. When `X`, `Y` and `Z` are all affinoid, their fibre product is affinoid. Affinoidness is
not asserted over an arbitrary base: gluing two copies of a perfectoid closed disc along its
non-quasicompact open unit disc gives two affinoid charts whose fibre product is that open disc; (d) for `κ` of
cofinality `ω₁` the unrestricted statement fails: for a perfectoid field `L` of characteristic `p`
with `#L < κ`, and perfectoid fields `K_α ⊇ L` (`α < ω₁`) that are completed perfections of
`L(x_β : β < λ_α)` for the Gauss norm, with infinite `λ_α ≥ #L` cofinal below `κ`, `X := ⊔_α Spa(K_α, K_α°)` is `κ`-small but
`X ×_{Spa(L, L°)} X` is not, since `Spa(K_α ⊗̂_L K_α)` has at least `λ_α` points: the
Gauss-isometric automorphisms obtained by permuting the variables give distinct graph points ([ECD] §4,
Prop. 4.4, p. 20, for (c) with the hypotheses under which it holds; [Berkeley] Lecture 8,
Prop. 8.2.5(3), p. 59, for the fibre products whose size (c) bounds). Part (d) shows that ECD's
'stable under finite products' needs the uniform or quasicompact hypothesis for singular `κ`.
*Needs:*
*uniformly-kappa-small-perfectoid-space*; *kappa-small-cover-criterion*;
*adic-spectrum-cardinality*; *complete-tate-ring-cardinality*; layer P2 (P2.16, P2.13); layer P1;
AdicSpacesPartII R0.4 (*uniform-completed-tensor-product*).

Lean: `PerfectoidSpace.IsUniformlyKappaSmall.fibreProduct` carries the uniform smallness
hypotheses on both factors and permits an arbitrary perfectoid base.

**Checks for small fibre products.**

- For `κ`-small affinoid perfectoid `X → Z` with affinoid perfectoid `Z`,
  `X ×_Z Z ≅ X` is `κ`-small and affinoid.
- For `κ`-small affinoid perfectoid `X = Spa(A,A⁺)` and `Y = Spa(B,B⁺)` over
  affinoid perfectoid `Z = Spa(C,C⁺)`, the completed-tensor presentation of P2.16
  gives an affinoid `κ`-small `X ×_Z Y`.
- Let `K` be a characteristic-p perfectoid field with `#K < κ`. Its perfected
  projective line `X`, obtained by gluing the perfected closed discs with rings
  `K⟨T^{1/p^∞}⟩` and `K⟨T^{-1/p^∞}⟩` along the perfected unit annulus, is
  quasicompact and `κ`-small. Laurent coefficients give `Γ(X,O) = K`, while zero
  and infinity are distinct points, so `X` is not affinoid. With
  `Y = Z = Spa(K,K°)`, the fibre product `X ×_Z Y ≅ X` remains non-affinoid.

**Check: `not_isKappaSmall_gaussField_fibreProduct`.** Under the same strong-limit,
cofinality, characteristic and variable-cardinality hypotheses as the disc Check,
the theorem constructs the scalar maps from `(L,L°)` into the Gauss field pairs and
uses their induced coproduct map `f : X → Spa(L,L°)`. The space
`X = ⊔_{α<ω₁} Spa(K_α,K_α°)` is `κ`-small, whereas `pullback f f` has at least `κ`
points and fails `IsKappaSmall`. Permutations of the infinite variable set `J_α`
give at least `#J_α` distinct graph points in each diagonal component. The scalar
maps are required to equal the Gauss construction's maps on constants. The Check,
specified in this README only, retains both the small source and its non-small self-fibre product.
This tests the finite-product assertion of [ECD] Proposition 4.4, p. 20, in the
slice over `Spa(L,L°)`; it applies also at `p = 2`.

Tilting preserves `κ`-smallness (*kappa-small-tilting*). For a perfectoid Tate ring `A` with tilt
`A♭` (layer P1), prove `#A♭ ≤ (#A)^ℵ₀` and `#A ≤ (#A♭)^ℵ₀·ℵ₀`, the tilt being the inverse limit
along `x ↦ x^p`. Consequently, for an uncountable strong limit `κ` and a perfectoid space `X` with
tilt `X♭` (layer P2 (P2.11)), `X` is `κ`-small iff `X♭` is `κ`-small (tilting invariance of `IsKappaSmall`), and
`X` is uniformly `κ`-small iff `X♭` is; tilting restricts to an equivalence between `κ`-small perfectoid
spaces over a perfectoid space `S` and `κ`-small perfectoid spaces over `S♭`; no base field, any
characteristic ([ECD] §3, after Lem. 3.10, p. 16); so the diamond `Spd` of DiamondsAndVStacks layer D6
keeps `κ`-small objects `κ`-small. *Needs:* layer P1 (P1.27); layer P2 (P2.11, P2.2); *kappa-small-perfectoid-space*;
*uniformly-kappa-small-perfectoid-space*; Mathlib `Cardinal.power_le_power_right`.

For every uncountable strong limit `κ` and every `X : Perfd p`,
`kappaSmallSliceTilt` restricts P2's `sliceTiltFunctor p X` to the full subcategories
whose source spaces are `κ`-small; the base `X` need not be `κ`-small.
The tilting invariance of `IsKappaSmall` identifies the object predicates, and its isomorphism
invariance ensures that the inverse equivalence also preserves them.
`kappaSmallSliceTilt_obj` identifies its underlying object with
`(sliceTiltFunctor p X).obj Y.obj`, and `kappaSmallSliceTilt_map` identifies its
underlying morphism with `(sliceTiltFunctor p X).map f.hom`, under these same
cardinal hypotheses ([ECD] Corollary 3.20, p. 19; Definition 4.2 and the cardinal
estimate in Remark 4.3, p. 20; Lemma 3.10, p. 16).

**Checks.**

- `kappaSmallSliceTilt_object` sends a specified arrow `Y → X` with `κ`-small source
  to the arrow obtained by applying `tiltFunctor`.
- `kappaSmallSliceTilt_morphism` sends the map in a commuting triangle to its
  image under `tiltFunctor`, including triangles between distinct objects.
- `kappaSmallSliceTilt_identity` sends `X → X` to `X♭ → X♭` when `X` is `κ`-small.

Every affinoid perfectoid space is the cofiltered limit of its `κ`-small perfectoid sub-pairs, the
`κ`-bounded form of ECD Proposition 6.5 (*kappa-small-perfectoid-subpairs*). Let `(S, S⁺)` be a
perfectoid Tate pair (ECD Definition 3.1, no base field) with a pseudouniformiser `ϖ` such that
`ϖ^p` divides `p` in `S°`, and let `A ⊆ S` be a subset. Prove that there is a closed subring
`S' ⊆ S` containing `A ∪ {ϖ, ϖ^{-1}}` with `#S' ≤ (#A + ℵ₀)^ℵ₀` such that, with the subspace
topology, `S'` is a perfectoid Tate ring, `S'° = S' ∩ S°`, and `(S', S' ∩ S⁺)` is a perfectoid Tate
pair. Consequently, for an uncountable strong limit `κ` and a morphism
`Y = Spa(S, S⁺) → X = Spa(R, R⁺)` of affinoid perfectoid spaces with `#R < κ`, the set `P_κ(Y/X)`
of closed subrings `S' ⊆ S` of cardinality `< κ` that contain the image of `R` and for which
`(S', S' ∩ S⁺)` is a perfectoid Tate pair, ordered by inclusion, satisfies: (a) it is directed, and
every subfamily of cardinality `≤ λ` whose members have cardinality `≤ λ`, for an infinite
`λ < κ` with `#R ≤ λ`, has an upper bound of cardinality `≤ λ^ℵ₀`. Including `#R` in this
bound also covers the empty subfamily;
in particular it is `ω₁`-directed whenever `cf κ > ω`, as for every cutoff cardinal (ECD
Lemma 4.1 (ii)), but not for `κ = ℶ_ω` when `#S ≥ ℶ_ω`: a countable family of members with
cardinalities below and cofinal in `ℶ_ω` has no upper bound in `P_κ(Y/X)`; (b) `⋃ S' = S` and
`⋃ (S' ∩ S⁺) = S⁺`; (c)
`Y = lim_{S' ∈ P_κ(Y/X)} Spa(S', S' ∩ S⁺)` in perfectoid spaces over `X`, a cofiltered limit of
`κ`-small affinoid perfectoid spaces over `X` whose colimit of plus rings is already complete. For
countable `A` this is P5.6 (every affinoid perfectoid space is an `ω₁`-cofiltered limit of
topologically countably generated ones), of which this is the `κ`-bounded form. For singular `κ` and `#S ≥ κ`, `P_κ(Y/X)` is not `κ`-directed: a family of size `cf κ`
with sizes cofinal in `κ` has no upper bound. If `#S < κ`, it instead has the largest member `S`.
The general directedness statement in (a) uses a uniform bound
([ECD] §6, proof of Prop. 6.5, p. 29, the countable case, whose sub-pair construction is made
explicit and bounded by `κ`; [ECD] §8, proof of Prop. 8.2, p. 39, the `κ`-bounded form used in
DiamondsAndVStacks layer D2 for cutoff-independence, where (a) states the directedness that holds).
*Needs:* layer
P1 (P1.1); layer P5 (P5.3); *kappa-small-perfectoid-space*; *kappa-small-cofiltered-limits*; Tau
Ceti `TauCeti.Huber.IsTateRing`, `TauCeti.Huber.powerBoundedSubring`,
`TauCeti.Huber.isIntegrallyClosedIn_powerBoundedSubring`; Mathlib `Subring.topologicalClosure`.

**Checks for the subpair poset `KappaSubpairs` and for disjoint unions.**

- `kappaSubpairs_top` — if `#S<κ`, the whole subring `S` is the greatest member.
- `kappaSubpairs_empty_family` — for the identity `R → R`, no member can have cardinality
  at most `λ<#R`, even when the family to be bounded is empty.
- `kappaSubpairs_cofinal_sequence` — if a sequence of members has sizes cofinal in `κ`
  (in particular the `ℶ_ω` example), no single member bounds that sequence.
- `kappaSubpairs_two_members` — two members have a common upper bound under the stated
  uncountable strong-limit and base-ring bounds.
- `sigmaPoints_empty`, `sigmaPoints_singleton`, `sigmaPoints_two_components` — geometric
  coproducts have respectively zero points, the original space, and disjoint images of the two
  component inclusions. The homeomorphism `sigmaPoints` sends the image of `x∈X_j` to `(j,x)`.

`KappaSubpairs` is a subtype of the subrings of `S`, ordered by inclusion. Its conditions
are closedness, containment of the image of `R`, the cardinal bound, and existence of a perfectoid
pair mapping homeomorphically onto that closed subring, with plus ring the inverse image of `S⁺`.
The topology and plus ring are induced from `S`. The singular disjoint-union Check is also stated for any small index
set of size `<κ` whose field cardinalities are cofinal in `κ`; this includes the `ω₁` case above.

`exists_small_closed_perfectoid_subpair` states the closed subring construction with the
pseudouniformizer and its inverse included. `KappaSubpairs.diagram` uses the chosen embedded
pairs, with `map_inclusion` fixing each transition. Its cone has vertex `Spa(S,S⁺)`;
`KappaSubpairs.isLimit` gives its universal property. `plusDiagram`, `plusCocone` and
`plusIsColimit` identify the uncompleted integral colimit with `S⁺`.

- `subpair_diagram_identity` — the identity inclusion induces the identity geometric map.
- `subpair_plus_zero` — the integral colimit inclusion sends zero to zero.
- `subpair_top_iso` — a member equal to the whole subring has an invertible inclusion.

For the finite-product Checks, put `D_n = C({0,1}^n,R)` with `R` a perfectoid
Tate pair, use the constant inclusion `D_0 → D_2`, and assume `#D_2 < κ`.
The first-bit subring `B = im(D_1 → D_2)` and the whole subring `C = D_2`
are explicit members; the choices of abstract pairs are compared through their
inclusions into `D_2`. These are finite instances of [ECD] Proposition 6.5,
p. 29, and the profinite-product description after Definition 7.8, p. 32.

- `KappaSubpairs_map_binary_indicator` evaluates `map (B ≤ C)` on the first-bit
  indicator: its value at `(s₀,s₁)` is `s₀`, interpreted as zero or one in `R`.
- `KappaSubpairs_diagram_four_to_two` computes every fibre of the corresponding
  geometric map as two points and asserts surjectivity.
- `KappaSubpairs_diagram_four_to_one` uses the constant subring instead of `B`;
  the geometric map is surjective and each fibre has four points.
- `KappaSubpairs_diagram_strict_direction` has an arrow from `op C` to `op B`,
  no arrow back, and a noninjective map on points. These three geometric Checks
  assume `R` is nontrivial.
- `KappaSubpairs_plusCocone_binary_indicator` follows the first-bit indicator
  through `plusDiagram.map` and `plusCocone`: the all-false and all-true inputs
  give zero and one.
- `KappaSubpairs_plusCocone_complement` follows the complementary indicator;
  the same inputs give one and zero, and the resulting section is nonzero.
  Both indicator Checks assume `R` is nontrivial.
- `KappaSubpairs_plusCocone_parameter` uses the constant subring and a specified
  pseudouniformizer `t ∈ R⁺`; the image has value `t` at all four inputs and is
  invertible in the ambient Tate ring. No nontriviality assumption is needed.

### P6.5 affinoid étale objects descend to a finite stage

**ECD Proposition 6.4(iv) in the form used by the pro-étale calculus: affinoid étale objects and
surjectivity descend to a finite stage.** (*affinoid-etale-finite-stage-assembly*) Let
`X_i = Spa(R_i, R_i⁺)`, `i ∈ I`, be a cofiltered inverse system of affinoid perfectoid spaces of
any characteristic, over no base field, over a small index category, with limit `X = Spa(R, R⁺)`
(layer P5 (P5.3): `R⁺` the `ϖ`-adic completion of `colim R_i⁺`, `R = R⁺[1/ϖ]`). For a perfectoid
space `T` write `T_ét^aff` for the category of affinoid perfectoid spaces étale over `T` (ECD
Definition 6.2, layer P3 (P3.15)). (a) (ECD Proposition 6.4(iv), which is P5.13.4, *affinoid-etale-finite-stage-6-4-iv*) The base change functors induce an equivalence
`2-colim_i (X_i)_ét^aff → X_ét^aff`. (b) (Descent of surjectivity) If `Y_i ∈ (X_i)_ét^aff` and
`Y_i ×_{X_i} X → X` is surjective, there is `j_0 → i` such that `Y_i ×_{X_i} X_j → X_j` is
surjective for every `j → j_0` — the qcqs étale case of P5.12
(*finite-stage-isomorphism-detection* (a), ECD 6.4(ii)), restricted to affinoid étale objects. (c) (Relation to the
quasicompact separated stage of layer P5) If `Y ∈ X_ét^aff` and `Z_i → X_i` is a qcqs étale model
of `Y` (`Z_i ×_{X_i} X ≅ Y`; layer P5 (P5.12), ECD 6.4(ii)), then `Z_i ×_{X_i} X_j` is affinoid
perfectoid and isomorphic over `X_j` to the model given by (a) for all `j` beyond some `j_0`.
Affinoidness at a finite stage is the content of (a) and is not a consequence of ECD 6.4(ii)–(iii)
(P5.13.4 (c)). (d) (Relative form) If `B` is an affinoid perfectoid space and
the `X_i → B` are compatible étale maps (for instance a pro-étale presentation of `X` over `B`),
then every `Y_i ∈ (X_i)_ét^aff` and every `Y_i ×_{X_i} X_j` for `j → i` lies in `B_ét^aff`, with
structure map the projection to `X_j` followed by `X_j → B`, so the finite-stage
models of (a) are affinoid étale over `B`. The compatibility of the equivalences of ECD 6.4 (i)–(iv) with the inclusions
`X_fét ⊆ X_ét^aff ⊆ X_ét,qc,sep ⊆ X_ét,qcqs` is P5.13.4
(*etale-descent-to-finite-stage-6-4*) and is not restated. No step uses v-descent, a diamond atlas
or the comparison of analytic adic spaces with diamonds. The equivalence (a) is imported from
P5.13.4; (c)–(d) are the consequences the proofs of ECD Lemma 7.11(i), Lemma
7.18 and Proposition 8.5 use ([ECD] §6, Prop. 6.4(iv), p. 27,
for (a); [ECD] §6, proof of Prop. 6.4, p. 28, for the division of labour between P5's (ii) and
P5.13.4's essential surjectivity recorded in (c); [ECD] §7, proof of Lem. 7.18, p. 37, for the use of
(a) and (b) in the cover construction of DiamondsAndVStacks layer D1, downstream). *Needs:*
P5.13.4; layer P5 (P5.3, P5.12); layer P3 (P3.15, P3.16); layer P2 (P2.16).

DiamondsAndVStacks consumes this result; its layer D0 is not a prerequisite.

For (a), the assembly target *affinoid-etale-finite-stage-assembly* of this subsection cites
P5.13.4's *affinoid-etale-finite-stage-6-4-iv*, specified in this README: its
essential surjectivity (named `affinoid_etale_finite_stage_essSurj` in its Examples) and its
Hom-colimit bijection are the supplied affinoid-pair statements ([ECD] Proposition 6.4(iv),
pp. 27–28).
P5's `finiteStage_surjectivity` supplies (b); surjectivity remains true after
further base change. `finiteStage_affinoid` retains the eventual affinoidness assertion.
For (c), `finiteStage_affinoid_model_iso` takes a qcqs étale model `Z_i`, an affinoid étale
model `W_l`, and an isomorphism of their pullbacks over `X`. It gives a common stage
`j → i`, `j → l` at which, and after every further `k → j`, the pullback of `Z_i` is affinoid
and isomorphic to the pullback of `W_l` in `smallEtale p X_k`; the isomorphism therefore
commutes with the maps to `X_k` ([ECD] Proposition 6.4(ii), (iv), pp. 27–28).

For (d), `finiteStage_relative` (*affinoid-etale-relative-finite-stage-models*) takes the
affinoid diagram, its natural transformation to the constant diagram at `B`, étaleness of
every component, and an affinoid étale `Y_i`. It places both `Y_i` and each 
`etaleBaseChange` along `X_j → X_i` in `AffinoidEtale p B`. The structure map is the
projection to `X_j` followed by `X_j → B`, and equals the projection to `Y_i` followed by
`Y_i → X_i → B`. This statement also holds for arbitrary perfectoid `B`; the affinoid case
is (d) ([ECD] Proposition 6.4(iv), pp. 27–28; proof of Lemma 7.11(i), p. 33).

**Check.** `finiteStage_relative_nonetale` excludes a map to `B` that is not étale from
`AffinoidEtale p B`, even if its source is affinoid. Identity transitions recover the
original relative model, and empty affinoid models satisfy the same base-change statement.

### P6.6 affinoid pro-étale morphisms

Affinoid pro-étale morphisms are the cofiltered limits of affinoid étale ones.

**Affinoid pro-étale morphisms and pro-étale presentations.** (*affinoid-pro-etale-map*) Let
`f : Y → X` be a morphism of perfectoid spaces (layer P2 (P2.10)). `f` is affinoid pro-étale if
`Y = Spa(S, S⁺)` and `X = Spa(R, R⁺)` are affinoid perfectoid and there are a small cofiltered
category `I`, a functor `i ↦ (Y_i → X)` from `I` to `X_ét^aff` (affinoid perfectoid spaces
`Y_i = Spa(S_i, S_i⁺)` with an étale structure map to `X` in the sense of ECD Definition 6.2, and
`X`-morphisms as transition maps) and an `X`-isomorphism `Y ≅ lim_i Y_i`, the limit taken in
perfectoid spaces (layer P5 (P5.3): `S⁺ ≅` the `ϖ`-adic completion of `colim S_i⁺` for a
pseudouniformiser `ϖ` of `R`, and `S = S⁺[1/ϖ]`). The data `(I, (Y_i), the isomorphism)` is a
pro-étale presentation of `f`. The transition maps are `X`-morphisms between étale `X`-spaces and
hence étale (layer P3, P3.18); they are not required to be surjective or finite étale, which
distinguishes these presentations from those of Scholze 2013 used in AdicEtaleGeometry A1.12;
`I` is small but of arbitrary size; being affinoid pro-étale is the existence of a presentation,
and in a formalisation a presentation is a limit cone in perfectoid spaces over `X` together with
the étaleness of the legs' index objects. `X_proét^aff` denotes the full subcategory of perfectoid
spaces over `X` on the affinoid pro-étale `X`-spaces. In `PerfectoidSpace`, define
`ProEtalePresentation` as the structure (small cofiltered `I`, functor `I ⥤ X_ét^aff`,
`X`-isomorphism of `Y` with its limit), `IsAffinoidProEtale` as `Nonempty (ProEtalePresentation f)`
and `AffinoidProEtale` as `X_proét^aff`; prove `IsAffinoidProEtale.of_presentation`,
`IsAffinoidProEtale.of_isEtale` (an étale morphism of affinoid perfectoid spaces is affinoid
pro-étale, by the constant presentation), `IsAffinoidProEtale.id`,
`ProEtalePresentation.isLimit` (the cone `Y → Y_i` is a limit cone in perfectoid spaces over `X`),
`ProEtalePresentation.plusModEquiv` (`O⁺(Y)/ϖ ≅ colim_i O⁺(Y_i)/ϖ`),
`ProEtalePresentation.homeomorphLimit` (`|Y| ≃ₜ lim_i |Y_i|`, ECD 6.4(o)),
`ProEtalePresentation.range_eq_iInter` (the image of `|Y|` in `|X|` is the intersection of the
quasicompact open images of the `|Y_i|`), `IsAffinoidProEtale.isProConstructible_range` (that image
is pro-constructible and stable under generalisation), `isAffinoidProEtale_iff_canonical`
(see *canonical-pro-etale-presentation*), and the stability lemmas `IsAffinoidProEtale.comp`
(see *affinoid-pro-etale-composition*), `IsAffinoidProEtale.baseChange`
(see *affinoid-pro-etale-base-change*) and `IsAffinoidProEtale.of_comp`
(see *affinoid-pro-etale-cancellation*) ([ECD] §7, Def. 7.8(i), p. 32; [Berkeley] Lecture 8,
Def. 8.2.1, p. 57, the same notion on rings: `(B, B⁺)` is a completed filtered colimit of
perfectoid pairs étale over `(A, A⁺)`). *Needs:* layer P2 (P2.10, P2.9); layer P3 (P3.15, P3.18);
layer P5 (P5.3); AdicSpacesPartII R0.19.5–R0.19.8 and P5.4’s perfectoid limit comparisons; Mathlib `CategoryTheory.IsCofiltered`,
`CategoryTheory.Limits.IsLimit`.

**Checks.**

- `isAffinoidProEtale_rationalSubset` (computation) — for a rational subset
  `U = X(f_1, …, f_n / g)` of an affinoid perfectoid `X`, the inclusion `U ↪ X` is affinoid
  pro-étale with the presentation indexed by a point.
- `isAffinoidProEtale_id` (degenerate) — the identity of an affinoid perfectoid space is affinoid
  pro-étale, as are the map from the empty space `∅ = Spa(0, 0) → X` (`X × ∅`, the constant
  presentation) and every finite étale `Y → X` between affinoid perfectoid spaces (constant
  presentation).
- `isAffinoidProEtale_origin` (computation) — for `K = Q_p(p^{1/p^∞})^∧` and
  `X = Spa(K⟨T^{1/p^∞}⟩, K°⟨T^{1/p^∞}⟩)`, the origin `Spa(K, K°) ↪ X` is affinoid pro-étale with
  the presentation `(X_n = {|T| ≤ |p|ⁿ})_n`, and it is not étale (its image is not open).
- `not_isAffinoidProEtale_openUnitDisc` (non-example) — the open unit disc `{|T| < 1}` of the
  perfectoid closed disc is étale, hence pro-étale, over it but not affinoid pro-étale: it is not
  quasicompact, hence not affinoid.
- `not_isOpenMap_of_isAffinoidProEtale` (non-example) — for a perfectoid field `K` and an infinite
  profinite set `S` with a non-isolated point `x`, `Spa(K, O_K) × {x} → Spa(K, O_K) × S` is
  affinoid pro-étale and not open ([Berkeley] Lecture 8, Rem. 8.2.3): affinoid pro-étale maps are
  not étale-local products.

**Checks for presentations and the affinoid category.**

- `presentation_singleton` — the constant presentation of an affinoid étale map has exactly
  one indexing object.
- `presentation_empty_source_nonempty_index` — the empty open subspace has a one-stage
  presentation whose stage has no points.
- `twoPoint_transition_nonidentity` — the two-sheet presentation in P6.9 has a genuine
  constant-zero transition, which is distinct from the identity over a nonempty base.
- `presentation_rational_constant` — a rational inclusion has a one-object presentation;
  its transition is the identity, without a surjectivity requirement on the inclusion.
- `affinoidProEtale_identity_object`, `affinoidProEtale_empty_object`,
  `affinoidProEtale_excludes_open_disc` — the full category contains the identity and empty
  objects and excludes the non-quasicompact open disc.

### P6.7 pro-étale morphisms

Pro-étale morphisms are the morphisms that are affinoid pro-étale locally on source and target.

**Pro-étale morphisms of perfectoid spaces.** (*pro-etale-map*) A morphism `f : Y → X` of
perfectoid spaces, over no base field, is pro-étale if for every `y ∈ Y` there are an open
neighbourhood `V ⊆ Y` of `y` and an open subset `U ⊆ X` with `f(V) ⊆ U` such that `V` and `U` are
affinoid perfectoid and `f|_V : V → U` is affinoid pro-étale (see *affinoid-pro-etale-map*); ECD's
'open subset `U`' is taken affinoid, as the restriction `f|_V` must be affinoid pro-étale. `V` and
`U` can be chosen rational inside any given neighbourhoods of `y` and `f(y)`
(see *pro-etale-local-shrinking*); `X_proét` denotes the category of perfectoid spaces pro-étale over
`X` with all `X`-morphisms, which are automatically pro-étale (see *pro-etale-stability-and-limits*),
and for `κ`-small `X`, `X_proét,κ` is its full subcategory of `κ`-small objects. Pro-étale morphisms
need not be open nor locally of finite type; being pro-étale is local on the source and, for open
covers, on the target, but not local on the target for the pro-étale topology. This is neither
Mathlib's pro-étale site of a scheme (`AlgebraicGeometry.Scheme.ProEt`, defined by weakly étale
maps) nor Scholze 2013's `X_proét` of a locally noetherian adic space (AdicEtaleGeometry
A1.12). In `PerfectoidSpace`, define `IsProEtale` as the structure above and `ProEtaleOver` as
`X_proét`, with the instance `ProEtaleOver.hasPullbacks` (fibre products, computed in perfectoid
spaces); prove `IsProEtale.of_isAffinoidProEtale`, `IsProEtale.of_isEtale` (étale maps are
pro-étale, *pro-etale-local-shrinking*), `IsProEtale.of_openCover` (if `X = ⋃ U_α` and every
`f⁻¹(U_α) → U_α` is pro-étale then `f` is), `IsProEtale.restrict` (`f|_V : V → U` for open `U ⊆ X`,
`V ⊆ f⁻¹(U)`), `isProEtale_iff_exists_rational` (rational `V ∋ y`, `U ∋ f(y)` with `f(V) ⊆ U` and
`V → U` affinoid pro-étale), `IsProEtale.comp`, `IsProEtale.baseChange`, `IsProEtale.of_comp`
(see *pro-etale-stability-and-limits*) and `IsProEtale.diagonal` (see *diagonals-are-pro-etale*) ([ECD] §7,
Def. 7.8(ii), p. 32; [Berkeley] Lecture 8, Def. 8.2.1, p. 57, the same definition; [Berkeley]
Lecture 8, after Prop. 8.2.5, p. 59, the failure of locality for the pro-étale topology). *Needs:*
*affinoid-pro-etale-map*; layer P2 (P2.10, P2.11); Mathlib `AlgebraicGeometry.Scheme.ProEt`.

**Checks.**

- `isProEtale_openImmersion` (computation) — the inclusion of the open unit disc `{|T| < 1}` into
  the perfectoid closed disc over a perfectoid field is pro-étale.
- `isProEtale_id` (degenerate) — the identity of any perfectoid space is pro-étale.
- `isProEtale_prod_locallyProfinite` (computation) — for a perfectoid space `X` and a locally
  profinite set `S` (e.g. `Q_p`), `X × S := ⋃ X × S_c` over the compact open `S_c ⊆ S` is pro-étale
  over `X` ([Berkeley] Lecture 8, Ex. 8.2.2).
- `not_isOpenMap_of_isProEtale` (non-example) — `Spa(K, O_K) × {x} → Spa(K, O_K) × S` (`x` a
  non-isolated point of a profinite `S`) is pro-étale and not open ([Berkeley] Lecture 8,
  Rem. 8.2.3).

**Checks for the global categories.**

- `proEtaleOver_identity` and `proEtaleOver_open_disc` — the identity and the open disc
  inclusion are objects of the category, although the second is not an affinoid object.
- `proEtaleOver_inherited_maps` — all morphisms over the base are retained.
- `smallProEtale_identity` — the identity is an object whenever its base is `κ`-small.
- `smallProEtale_empty` — for `1<κ`, the empty object lies over every base.
- `smallProEtale_zero_cutoff` — at cutoff zero the category has no objects.

<a id="p6-8"></a>

### P6.8 the product with a profinite set

Products with infinite profinite sets supply affinoid pro-étale maps that are not étale.

**The product `X × S` of an affinoid perfectoid space with a profinite set, and its relative form
`X ×_{S₀} S`.** (*product-with-profinite-set*) Let `X` be an affinoid perfectoid space,
`φ : |X| → S₀` a continuous map to a profinite set and `S → S₀` a continuous map of profinite sets.
Write `S → S₀` as a cofiltered limit of maps of finite discrete sets `S_k → S₀,k` (`k ∈ K`)
compatible with `S → S₀` (Mathlib `Profinite.asLimit`, `DiscreteQuotient`). Each
`φ_k : |X| → S₀,k` is locally constant, so `X = ⊔_{t∈S₀,k} X_{k,t}` with `X_{k,t} := φ_k⁻¹(t)` open
and closed, hence a rational subset cut out by an idempotent of `O(X)` (`O_X` is a sheaf, layer P2
(P2.8)); `X ×_{S₀,k} S_k := ⊔_{s∈S_k} X_{k,image(s)}` is affinoid perfectoid and affinoid étale over
`X`. Define `X ×_{S₀} S := lim_k X ×_{S₀,k} S_k` (limit in perfectoid spaces, layer P5 (P5.3)), and
`X × S := X ×_∗ S = lim_k X × S_k`. Prove: (a) for every perfectoid space `T` over `X`,
`X`-morphisms `T → X ×_{S₀} S` correspond to continuous maps `|T| → S` over `S₀`, so the
construction is independent of the presentation and functorial in `S → S₀`; (b)
`X ×_{S₀} S → X` is affinoid pro-étale with the above pro-étale presentation; (c)
`|X ×_{S₀} S| = |X| ×_{S₀} S`; (d) `O⁺(X × S) = C⁰(S, O⁺(X))` and `O(X × S) = C⁰(S, O(X))`,
continuous maps for the `ϖ`-adic topology on `O⁺(X)` (for `X = Spa(K, O_K)` this is ECD's
`Spa(C⁰(S, K), C⁰(S, O_K))`); (e) for a morphism `T → X`, `T ×_X (X ×_{S₀} S) = T ×_{S₀} S`; (f)
`(X × S)♭ = X♭ × S`; (g) for `S` locally profinite, `X × S` is glued from the `X × S_c` over the
compact open subsets `S_c ⊆ S` and is pro-étale, not affinoid unless `S` is compact, over `X` when `X` is nonempty; for `X = ∅` the product is empty for every `S`.
For nonempty `X` and infinite profinite `S`, `X × S` is not the disjoint union of `#S` copies of `X`. The same construction
`S × X := lim ⊔_{s ∈ S_j} X`, with `|U × S| = |U| × S` and `Hom(V, U × S) = Hom(V, U) × C(|V|, S)`,
is AdicEtaleGeometry A1.12 (*profinite-set-objects*) in Scholze's pro-étale site of a locally
noetherian adic space; here the carrier is the category of perfectoid spaces over an affinoid
perfectoid `X` with ECD's limit, and the relative form `X ×_{S₀} S` is added. In `PerfectoidSpace`,
define `prodProfinite` (`X × S`) and `fibreProdProfinite` (`X ×_{S₀} S`) with the projection
`prodProfinite.fst : X × S → X`; prove `prodProfinite.isAffinoidProEtale` (b),
`prodLocallyProfinite.homEquiv` (`Hom(T, X × S) ≃ Hom(T, X) × C(|T|, S)`),
`prodProfinite.homeomorph` (`|X × S| ≃ₜ |X| × S`),
`fibreProdProfinite.homEquiv` (`Hom_X(T, X ×_{S₀} S) ≃ {continuous |T| → S over S₀}`),
`fibreProdProfinite.homeomorph` (`|X ×_{S₀} S| ≃ₜ |X| ×_{S₀} S`), `prodProfinite.plusEquiv` (d),
`prodProfinite.map` (a continuous `S → S'` induces `X × S → X × S'`, with `map_id`, `map_comp`),
`prodProfinite.baseChange` (the absolute case of (e)), `fibreProdProfinite.baseChange` (e),
`prodProfinite.tilt` (f), `prodLocallyProfinite` with its
projection `prodLocallyProfinite.isProEtale` (g), and `prodProfinite.finiteIso`
(`X × S ≅ ⊔_{s∈S} X` for finite `S`). For finite `S`,
`prodProfinite.isEtale` states that the projection is étale in A1’s local-description sense;
for affinoid `X`, its affinoid source is supplied by `prodProfinite.isAffinoidProEtale`
([ECD] §7, after Def. 7.8, p. 32, the definition
`X × S = lim X × S_i`; [Berkeley] Lecture 8, Ex. 8.2.2, p. 57, the construction for arbitrary
perfectoid `X` and locally profinite `S`; [ECD] §11, Ex. 11.12, p. 58, the relative form
`Z ×_{π0(Z)} (π0(Z) ×_T S)` used for compact Hausdorff spaces).
*Needs:* *affinoid-pro-etale-map*; layer P5 (P5.3); layer P2 (P2.8, P2.3, P2.16, P2.11);
AdicEtaleGeometry A1.12 (*profinite-set-objects*, for the comparison only); Mathlib
`Profinite`, `Profinite.asLimit`, `DiscreteQuotient`, `FintypeCat.toProfinite`, `ContinuousMap`,
`LocallyConstant`.

**Checks.**

- `prodProfinite_finite` (computation) — for a finite discrete `S`, `X × S ≅ ⊔_{s∈S} X` and
  `O(X × S) = O(X)^S`.
- `prodProfinite_punit` (degenerate) — `X × {pt} ≅ X` and `X × ∅ = ∅`.
- `prodProfinite_Zp_sections` (computation) — `O(Spa(C_p, O_{C_p}) × Z_p) = C⁰(Z_p, C_p)` and
  `O⁺ = C⁰(Z_p, O_{C_p})`.
- `not_prodProfinite_eq_sigma` (non-example) — for nonempty `X` and infinite profinite `S`,
  `X × S` is not `⊔_{s∈S} X`: the
  latter is not quasicompact; `C⁰(S, O(X))` is a proper subring of all functions
  `∏_{s∈S} O(X)`. At a non-isolated point `s`, the function equal to `1` at `s` and `0` elsewhere
  is outside the image of the underlying-function map of `prodProfinite.sectionsEquiv`.
  The Check uses a nontrivial affinoid pair and a non-isolated point of `S`.
  For `X = ∅` both products are empty and both section rings are the zero ring.

The projections and coordinate maps are part of the product interface.
`prodLocallyProfinite.homEquiv` sends a morphism to its projection and its continuous
`S`-coordinate. Its `homeomorph` has exactly these two coordinates.
`fibreProdProfinite.homEquiv` is the corresponding equivalence over `X`, with the equation
`a ∘ b = φ ∘ t`; its homeomorphism sends a point to a pair satisfying that equation.
`prodProfinite.sectionsEquiv` and `plusEquiv` use the topologies of the complete pair and its
plus subring. Pullback of a constant section becomes the constant function with that value.
These equations also fix functoriality, base change and tilting.

**Checks for relative and locally profinite products.**

- `fibreProdProfinite_identity` — the projection for `S → S₀ = id` is an isomorphism.
- `fibreProdProfinite_empty` — an empty source `S` gives the empty space, for every `φ`.
- `fibreProdProfinite_point` — when `S₀` is a point, the relative product is `X × S`.
- `prodLocallyProfinite_singleton` — the projection for a singleton is an isomorphism.
- `prodLocallyProfinite_Qp_noncompact` — for nonzero affinoid `X`, `X × Q_p` is not affinoid.
- `prodLocallyProfinite_empty_base` — `∅ × Q_p` is empty.
- `prodProfinite_empty` — `X × ∅` has no points, including for nonempty `X`.
- `pointSection_singleton`, `pointSection_two_points`, `pointSection_empty_base` — the
  section at a singleton is an isomorphism; two different points give different sections
  when `X` is nonempty; over an empty `X` every such section is an isomorphism.
  For a nonisolated point, `not_isOpenMap_of_isAffinoidProEtale` and
  `not_isOpenMap_of_isProEtale` use this same section and its map on points.

### P6.9 the pro-category equivalence

P6.9 owns the perfectoid realization `proEtaleEquivalence` on the carrier
`Pro(C) = (CategoryTheory.Ind Cᵒᵖ)ᵒᵖ`, with its generic Hom and limit API supplied by
AdicSpacesPartII R0.19.9 on Mathlib's carrier. P6.10 owns its bounded
subcategory, and P6.12 owns the cofiltered-limit interface and flattening functor.
DiamondsAndVStacks D0.13 consumes these constructions. The next four subsections
identify `X_proét^aff` with this pro-category and give the `κ`-bounded, Zariski-closed
and limit consequences.

**Affinoid pro-étale spaces over X are the pro-objects of the affinoid étale site.**
(*pro-etale-pro-category-equivalence*) Let `X` be an affinoid perfectoid space, `X_ét^aff` the
category of étale maps `Y → X` from affinoid perfectoid spaces (essentially small and finitely
complete, *affinoid-etale-size-bound*, *canonical-pro-etale-presentation*; `Pro` is taken with small
cofiltered index categories after replacing `X_ét^aff` by a small skeleton) and `X_proét^aff` the
category of affinoid pro-étale maps `Y → X` (see *affinoid-pro-etale-map*). Let `Pro(X_ét^aff)` be the
pro-category (*pro-object-flattening*; the opposite of Mathlib's `Ind` of the opposite
category), with objects formal limits `"lim" Y_i` over small cofiltered categories and
`Hom("lim" Y_i, "lim" Z_j) = lim_j colim_i Hom_X(Y_i, Z_j)`. Prove that the realisation functor
`ρ_X : Pro(X_ét^aff) → X_proét^aff`, `"lim" Y_i ↦ lim Y_i` (limit in perfectoid spaces), is an
equivalence of categories, with quasi-inverse `Y ↦ "lim"_{(T,t)∈E_Y} T`; equivalently, for
affinoid pro-étale `Y = lim Y_i` and `Z = lim Z_j` over `X`,
`Hom_X(Y, Z) = lim_j colim_i Hom_X(Y_i, Z_j)` ([ECD] §7, Prop. 7.10, p. 33; [ECD] §7, proof of
Prop. 7.10, p. 33, the structure of the proof; [Berkeley] Lecture 8, Lem. 8.2.4, p. 57, the Hom
formula). For `X = Spa(C, O_C)` with `C` algebraically closed, `X_ét^aff ≃ FinSet` and the theorem
reads `Pro(FinSet) ≃ Profinite ≃ X_proét^aff`, `S ↦ X × S`
(see *affinoid-pro-etale-over-geometric-point*). *Needs:* *pro-object-flattening*; *affinoid-pro-etale-map*;
*affinoid-etale-size-bound*; *maps-to-etale-factor-through-finite-stage*;
*canonical-pro-etale-presentation*; layer P5 (P5.3); AdicSpacesPartII R0.19.9 (the generic Pro carrier and Hom formula);
Mathlib `CategoryTheory.Ind`, `CategoryTheory.Functor.IsEquivalence`.

The size of the affinoid étale site (*affinoid-etale-size-bound*). Let `X = Spa(R, R⁺)` be an
affinoid perfectoid space and `μ(X) := (#R + ℵ₀)^ℵ₀`. Prove: (a) every `Y ∈ X_ét^aff` (étale in the
sense of ECD Definition 6.2, layer P3 (P3.15)) has `#O_Y(Y) ≤ μ(X)` and `#|Y| ≤ 2^{μ(X)}`, with no
size hypothesis on `X`; (b) `X_ét^aff` is essentially small, with at most `2^{μ(X)}` isomorphism
classes of objects and `#Hom_X(Y, Y') ≤ 2^{μ(X)}`; (c) if `κ` is an uncountable strong limit and
`X` is `κ`-small, then every object of `X_ét^aff` is `κ`-small, `X_ét^aff` has fewer than `κ`
isomorphism classes of objects, and all its Hom-sets have fewer than `κ` elements; in particular
the isomorphism classes of affinoid étale surjections onto `X` form a set of cardinality `< κ`,
the count in ECD's proof of Lemma 7.18 ([ECD] §7, proof of Lem. 7.18, p. 36, for the bound
`μ(X)` and the counting in (b); [ECD] §6, Def. 6.2(ii), p. 26, for the local structure used in (a),
the printed `y ∈ Y` being read `y ∈ V`). *Needs:* layer P3 (P3.15, P3.14); layer P2 (P2.3, P2.8);
*kappa-small-perfectoid-space*; *complete-tate-ring-cardinality*; *adic-spectrum-cardinality*;
Mathlib `Cardinal.power_mul`, `CategoryTheory.EssentiallySmall`.

Maps from a cofiltered limit of affinoid perfectoid spaces to a qcqs étale space factor through a
finite stage (*maps-to-etale-factor-through-finite-stage*). Let `X` be an affinoid perfectoid
space, `(Y_i)_{i∈I}` a cofiltered system of affinoid perfectoid spaces over `X` (compatible
structure maps; no étaleness assumed) with limit `Y = lim_i Y_i` (layer P5 (P5.3)), and `T → X` an
étale morphism from a quasicompact quasiseparated perfectoid space (for instance `T ∈ X_ét^aff`).
Prove that `colim_i Hom_X(Y_i, T) → Hom_X(Y, T)` is bijective: every `X`-morphism `Y → T` factors
through some `Y_i`, and two `X`-morphisms `Y_i → T` that agree on `Y` agree on `Y_j` for some
`j → i` ([ECD] §7, proof of Prop. 7.10, p. 33, the reduction to an étale target; [Berkeley]
Lecture 8, proof of Lem. 8.2.4, p. 58, the route through rational embeddings and finite étale
maps). *Needs:* layer P5 (P5.3, P5.12); layer P3;
layer P2 (P2.16, P2.15).

The canonical presentation (*canonical-pro-etale-presentation*). Let `X` be an affinoid perfectoid
space and `g : Y → X` a morphism from an affinoid perfectoid space. Let `E_Y` be the category of
pairs `(T, t)` with `T ∈ X_ét^aff` and `t : Y → T` an `X`-morphism, with morphisms
`(T, t) → (T', t')` the `X`-morphisms `u : T → T'` with `u ∘ t = t'` — the category of elements of
`Hom_X(Y, −)` on `X_ét^aff`. Prove: (a) `E_Y` is essentially small and cofiltered (finite limits in
`X_ét^aff`); (b) there is a canonical `X`-morphism `c_Y : Y → lim_{E_Y} T` (limit over a small
skeleton, in perfectoid spaces); (c) `g` is affinoid pro-étale iff `c_Y` is an isomorphism
(`isAffinoidProEtale_iff_canonical`); (d) for any pro-étale presentation `Y = lim_{i∈I} Y_i`, the
functor `I → E_Y`, `i ↦ (Y_i, pr_i)`, is initial ([ECD] §7, proof of Lem. 7.11(iv), p. 34, the
finite limits in `X_ét^aff`; [ECD] §7, Lem. 7.13, p. 34, the same canonical index category in
ECD's topological model: the factorisations `X^wl → T → X` through finite disjoint unions of
rational subsets form a cofiltered category with limit `X^wl`). For `Y ∈ X_ét^aff`, `(Y, id)` is
initial in `E_Y` and `c_Y` is the identity. *Needs:* *affinoid-pro-etale-map*;
*affinoid-etale-size-bound*; *maps-to-etale-factor-through-finite-stage*; layer P5 (P5.3); layer
P2 (P2.16); layer P3; Mathlib `CategoryTheory.Functor.Elements`,
`CategoryTheory.IsCofiltered.of_hasFiniteLimits`, `CategoryTheory.Functor.Initial`,
`CategoryTheory.Functor.initial_of_exists_of_isCofiltered`,
`CategoryTheory.Functor.Initial.limitIso`, `CategoryTheory.EssentiallySmall`.

**Checks for the canonical category and the Hom convention.**

- `canonical_etale_initial` — when `Y → X` is affinoid étale, `(Y,id)` is initial in `E_Y`.
- `canonical_empty` — over affinoid `X`, the empty source also gives an initial factorization.
- `canonical_two_arrows` — take `Y=X`, `T=X×{false,true}`, and the section selecting `false`.
  The identity of `T` and the constant-`false` map are distinct for nonempty `X`, but agree
  after that section. This tests the triangle equation `t ≫ u=t'` and its equalizing direction.
- `proHom_precompose_two_points` — on two points, first the constant-zero map and then the
  transposition gives the constant-one map.
- `proHom_postcompose_two_points` — reversing their order gives the constant-zero map.
  `ProEtalePresentation.homStage` precomposes in the source index; `homDiagram` postcomposes
  in the target index. The Hom equivalence identifies a factorized arrow with its class in
  that source colimit at each target stage.

The small indexing model is Mathlib's `SmallModel` of `CanonicalEtaleIndex`, the latter being
`StructuredArrow (Over.mk f)` into the affinoid étale category. `canonicalCone_π` fixes its
components to the factorization maps. The realization `proEtaleEquivalence` uses the opposite
of Mathlib's `Ind` of the opposite small affinoid étale category; its cone is required to be a
limit in perfectoid spaces over `X`.

- `proHom_transition_precompose` and `proHom_transition_postcompose` put these computations
  inside `homStage` and `homDiagram`. The presentation has index `0 → 1`, both stages the
  two-sheet cover, and transition the constant-zero map. Precomposing the transposition gives
  constant one; postcomposing its colimit class gives the class of constant zero.
- `twoPoint_transition_nonidentity` — for nonempty base, that transition differs from the
  identity; an implementation using a constant identity diagram would fail this Check.

`affinoidEtale_objects_cardinality` counts a skeleton, `affinoidEtale_hom_cardinality` bounds
its Hom sets, and `affinoidEtale_isKappaSmall` gives the strong-limit consequence.

### P6.10 `κ`-small affinoid pro-étale spaces

**Bounded pro-category.** P6.10 owns `PerfectoidSpace.HasKappaSmallPresentation`
and `PerfectoidSpace.ProKappa`. For `C : Type u` with `Category.{v} C` and
`κ : Cardinal.{v+1}`, the predicate on `Z : (Ind Cᵒᵖ)ᵒᵖ` means that there are
`I : Type v`, a small cofiltered category structure on `I`, and `D : I ⥤ C` with
`Cardinal.lift #(Σ i j : I, i ⟶ j) < κ` and
`Z ≅ op ((Ind.lim Iᵒᵖ).obj D.op)`. The lift puts the arrow count in the cutoff's
universe; counting all arrows also bounds objects by their identities.
`ProKappa κ C` is Mathlib's full subcategory for this predicate.
`HasKappaSmallPresentation.of_diagram` supplies the predicate from the displayed
strict arrow bound, hence an object of `ProKappa`; no strong-limit or cutoff
hypothesis is required for this construction ([ECD] §7, Proposition 7.10 and
the description of `Pro_κ` on p. 33). DiamondsAndVStacks D0.13 uses these same
carriers; its unlifted cutoff `κ : Cardinal.{v}` corresponds to `Cardinal.lift κ` here.

**`κ`-small affinoid pro-étale spaces are the pro-objects with fewer than `κ` indices.**
(*kappa-bounded-pro-etale-presentations*) Let `κ` be an uncountable strong limit cardinal and `X` a
`κ`-small affinoid perfectoid space. Prove: (a) an affinoid pro-étale `Y → X` is `κ`-small iff it
admits a pro-étale presentation `Y = lim_{i∈I} Y_i` with `#Mor(I) < κ`; all `Y_i ∈ X_ét^aff` are
then `κ`-small (see *affinoid-etale-size-bound*); (b) with `Pro_κ(X_ét^aff)` the full subcategory of
`Pro(X_ét^aff)` of pro-objects isomorphic to one indexed by a cofiltered category with fewer than
`κ` morphisms, the realisation `ρ_X` of *pro-etale-pro-category-equivalence* restricts to an
equivalence `Pro_κ(X_ét^aff) ≃ X_proét,κ^aff` with the full subcategory of `κ`-small affinoid
pro-étale `X`-spaces (ECD states this without proof); (c) every affinoid pro-étale `Y → X` is the
limit, along a directed poset, of `κ`-small affinoid pro-étale `Y_J → X`: for a presentation indexed
by `I`, `J` runs over the cofiltered subcategories (not required to be full) of `I` with fewer than
`κ` morphisms and `Y_J := lim_{i∈J} Y_i` (the pro-étale form of the first step of ECD's proof of
Proposition 8.2; for arbitrary affinoid `Y` see *kappa-small-perfectoid-subpairs*). Fullness cannot
be demanded in (c): the one-object category of an arbitrarily large commutative monoid with an
absorbing zero is cofiltered, but its only nonempty full subcategory contains all endomorphisms,
whereas the non-full subcategory generated by the identity and zero is small and cofiltered
([ECD] §7, Prop. 7.10, p. 33, for (b), where ECD prints 'objets'; [ECD] §7, after Prop. 7.10,
p. 33, the meaning of `Pro_κ`; [ECD] §8, proof of Prop. 8.2, p. 39, for (c)). *Needs:*
*pro-etale-pro-category-equivalence*; *affinoid-etale-size-bound*;
*kappa-small-cofiltered-limits*; *canonical-pro-etale-presentation*;
*kappa-small-perfectoid-space*; *kappa-small-perfectoid-subpairs*; AdicSpacesPartII R0.19.5–R0.19.8 and P5.4’s perfectoid limit comparisons;
[ECD] Lemma 4.1, pp. 19–20 (choice of cardinal); Mathlib `CategoryTheory.IsCofiltered`.

Cofiltered limits of uniformly bounded `κ`-small affinoid perfectoid spaces are `κ`-small, the
set-theoretic clause of ECD Proposition 6.4 (*kappa-small-cofiltered-limits*). Let
`X_i = Spa(R_i, R_i⁺)`, `i ∈ I`, be a cofiltered system of affinoid perfectoid spaces with limit
`X = Spa(R, R⁺)` (layer P5 (P5.3)). Prove: (a) if `#Mor(I) ≤ λ` and `#R_i ≤ λ` for all `i`, with
`λ` infinite, then `#R ≤ λ^ℵ₀` (the ring-level bound `#R ≤ (#Mor(I)·sup_i #R_i)^ℵ₀` of layer P5
(P5.7)), so for an uncountable strong limit `κ > λ`, `X` is `κ`-small and uniformly `κ`-small; (b) if
`κ` is a cutoff cardinal ([ECD] Lemma 4.1, pp. 19–20), `I` is `λ`-small for some `λ < κ`
and all `X_i` are `κ`′-small for one strong limit `κ′ < κ`, then `X` is `κ`-small; if `κ′ = κ_λ` is a
strong limit of cofinality `> λ` (clause (iii) of the cutoff lemma), then `X` is even `κ_λ`-small, so
the bound does not grow with the diagram; (c) the uniform bound cannot be dropped: for `κ` of
cofinality `ω₁` and a tower of perfectoid fields `K_0 ⊆ K_1 ⊆ … ⊆ K_α ⊆ …` (`α < ω₁`) with
`#K_α < κ` cofinal in `κ`, the `X_α = Spa(K_α, K_α°)` are `κ`-small, `I = ω₁` has fewer than `κ`
elements, and the limit (the completed union) is not `κ`-small ([ECD] §6, Prop. 6.4, final sentence,
p. 27, for (b); [ECD] §6, proof of Prop. 6.4, p. 27, including the `κ_λ` refinement). *Needs:* layer P5 (P5.3, P5.7); *kappa-small-perfectoid-space*;
*uniformly-kappa-small-perfectoid-space*; *complete-tate-ring-cardinality*;
*first-countable-completion-cardinality*; [ECD] Lemma 4.1, pp. 19–20 (choice of cardinal);
Mathlib `Cardinal.iSup_lt_of_lt_cof_ord`.

**Check: `not_isKappaSmall_gaussField_cofilteredLimit`.** Let `κ` be a strong limit
with `cf κ = ℵ₁ < κ`, and let `L` be a characteristic-`p` perfectoid field with
`#L < κ`. Choose increasing infinite subsets `S_α` of one variable set, indexed
by `α < ω₁`, whose cardinalities are `< κ` and cofinal in `κ`. Such subsets can
be taken as initial segments of a set of size `κ`. The theorem supplies a functor
of perfectoid pairs identified at stage `α` with
`(gaussPerfectoidField p L S_α).field.toPairCat p`. Its maps preserve constants
and every variable root under `S_α ⊆ S_β`; these are the Gauss-isometric field
inclusions. The opposite space diagram has fewer than `κ` morphisms and every
stage is `κ`-small. Its `cofilteredLimit` has a global section ring of cardinality
at least `κ`, hence is not `κ`-small. This Check, specified in this README only, retains the small
stages and non-small limit. This distinguishes the stagewise bound from the common
bound required by [ECD] Proposition 6.4, final sentence and proof, p. 27; the
completed-union description is that proposition's opening construction.

Affinoid pro-étale spaces over a geometric point are products with profinite sets
(*affinoid-pro-etale-over-geometric-point*). Let `C` be an algebraically closed perfectoid field
and `X = Spa(C, O_C)`. Prove: (a) the functor `F ↦ X × F = ⊔_F X` from finite sets to `X_ét^aff`
is an equivalence; (b) the functor `S ↦ X × S` from profinite sets to `X_proét^aff`
(see *product-with-profinite-set*) is an equivalence: every affinoid pro-étale `Y → X` is
`X`-isomorphic to `X × S` for a profinite set `S`, and `S ≅ |Y|`; (c) the hypothesis `C⁺ = O_C` is
needed: if `C` admits an open bounded valuation subring `C⁺ ⊊ O_C` of rank two, obtained
by composing the rank-one valuation of `C` with a nontrivial rank-one valuation of its residue
field, choose a unit `u ∈ C⁺` of `O_C` with positive residual valuation. The rational locus
`|1| ≤ |u|` is `Spa(C, O_C)`. Its open immersion into `Spa(C, C⁺)` is affinoid étale and not of the form `Spa(C, C⁺) × S` ([ECD] §7, after
Def. 7.8, p. 32, for (b)). *Needs:* *pro-etale-pro-category-equivalence*;
*product-with-profinite-set*; layer P3 (P3.15, P3.7); *pro-object-flattening*;
Mathlib `Profinite.asLimit`, `FintypeCat.toProfinite`, `IsAlgClosed`.

In (a), `geometricPointFiniteEquivalence` is pinned by
`geometricPointFiniteEquivalence_obj`: for an algebraically closed `AffinoidField p` with
`K.plus.toSubring = powerBoundedSubring K`, its value on every `F : FintypeCat` is isomorphic
in `Over X` to `Over.mk (prodProfinite.fst p X (FintypeCat.toProfinite.obj F))`.
Thus the comparison preserves the structure map to `X`; `prodProfinite.finiteIso`
identifies its source with `⊔_F X`. This is the finite-set case of the product construction
in [ECD], the example after Definition 7.8, p. 32.

Checks (with the same algebraic-closedness and plus-ring hypotheses):

- `geometricPointFiniteEquivalence_obj_empty` — `Fin 0` maps to the projection `X × ∅ → X`.
- `geometricPointFiniteEquivalence_obj_singleton` — `Fin 1` maps to the one-copy projection over `X`.
- `geometricPointFiniteEquivalence_obj_pair` — `Fin 2` maps to the split two-copy projection over `X`.

Zariski closed immersions are affinoid pro-étale, by the rational-intersection presentation
(*zariski-closed-immersions-are-pro-etale*): for `X = Spa(R, R⁺)` affinoid perfectoid and `I ⊆ R`
any ideal, the universal perfectoid space `Z_I = lim_F U_F` over `V(I) = ⋂_F U_F` (the limit over
the rational subsets `U_F = X(F/1)`, `F ⊆ I` finite, P4.14 (a)–(b), through which every Zariski
closed immersion with locus `V(I)` factors isomorphically, P4.14) is a pro-rational subspace, hence
affinoid pro-étale over `X` in the sense of *affinoid-pro-etale-map* with pro-étale presentation
`(U_F)_F` by rational open immersions; strong Zariski closedness (P4.15) is not needed ([ECD] §7,
Rem. 7.9, p. 32; [Torsion] §II.2, proof of Lem. II.2.2, p. 14). For
`K = Q_p(p^{1/p^∞})^∧`, `X = Spa(K⟨T^{1/p^∞}⟩, K°⟨T^{1/p^∞}⟩)` and `I = (T)`, `V(I)` is the origin
and `Z_I = Spa(K, K°)`, presented as `lim_n {|T| ≤ |p|ⁿ}`, a Zariski closed immersion that is not
étale. *Needs:* layer P4 (P4.14, P4.12, P4.4); layer P5 (P5.3); layer P2 (P2.3);
*affinoid-pro-etale-map*; Tau Ceti `TauCeti.ValuationSpectrum.rationalSubset`.

**Checks for bounded pro-objects and the rank-two boundary.**

- The constant, zero-and-one and two-stage Checks below apply to `HasKappaSmallPresentation` and
  to `ProKappa` alike.
- The constant pro-object of any `c : C` supplies an object at cutoff `2`.
- `ProKappa 0 C` is empty, including when `C` is empty (a Lean `example`).
- `ProKappa 1 C` is empty even for a nonempty category: every cofiltered
  index has an object and its identity arrow.
- Every constant pro-object admits a presentation with one arrow,
  so it belongs at cutoff `2`.
- No pro-object admits a presentation with fewer than one arrow;
  the bounded subcategories at `0` and `1` are empty.
- A diagram `0 → 1` realizes its initial stage, so its pro-object
  already belongs at cutoff `2`, although that chosen diagram has three arrows.
- `large_zero_monoid_cofiltered` / `zeroMonoidPresentation_cofiltered` — the identity cover
  indexed by a monoid with absorbing zero has a cofiltered index and identity cone legs.
- `large_zero_monoid_arrow_count` / `zeroMonoidPresentation_arrow_count` — the arrow set of
  this presentation has cardinality `#M`, although it presents the identity cover.
- `large_zero_monoid_nonfull` / `zeroMonoidPresentation_nonfull` — for nontrivial `M`, the
  two-element submonoid `{0,1}` embeds faithfully and injectively on objects in the presentation;
  restricting the diagram along this embedding still gives the constant identity cover.
- `large_zero_monoid_proper_embedding` — for `2 < #M`, restricting the identity-cover
  presentation to `{0,1}` gives a cofiltered, faithful, injective-on-objects subdiagram that
  is not full.
- `zeroMonoidPresentation_bounded` — if `κ ≤ #M`, with the universe lift understood,
  the small subdiagram supplied by `bounded_subdiagram_approximation_not_full` is not full.
- `rankTwo_two_points`, `rankTwo_generic_one_point`, `rankTwo_etale_not_product` — a rank-two
  plus ring gives two valuation points; replacing it by the norm unit ball gives one point;
  that rational étale inclusion is not a product with any profinite set.

For the last three Checks, `RankTwoGeometricPoint` carries an algebraically closed perfectoid
field and a proper plus valuation ring whose only valuation overrings inside the norm unit ball
are itself and the unit ball. This is the rank-two case of P4's valuation-overring description.
`rankTwoGeometricPoint` constructs one by taking the completed algebraic closure of a Gauss
field with transcendental residue and composing its norm valuation with a nontrivial rank-one
residue valuation. `generization_rational` identifies the generic-point map with a rational
localization. It does not assert this for an arbitrary proper plus ring.

`bounded_subdiagram_approximation` states the directed approximation with a faithful,
injective-on-objects embedding of each small cofiltered index into the original index.
`zeroMonoidPresentation` constructs the identity-cover presentation for any affinoid perfectoid
`X` and monoid with zero `M`; all its diagram arrows and cone legs are identities. Its three
Checks above test the cone, arrow count and restriction to `{0,1}`.
`bounded_subdiagram_approximation_not_full` applies `bounded_subdiagram_approximation` to
this presentation when `κ` is strong limit, `ℵ₀ < κ`, `X` is `κ`-small and `κ ≤ #M` after
lifting universes. It produces a nonempty cofiltered presentation `Q` and a faithful,
injective-on-objects embedding with fewer than `κ` arrows that cannot be full: fullness
would already require all `M` endomorphisms at any one object. This is a specialization of
[ECD] Proposition 8.2, proof, p. 39.
`cofilteredLimit_uniform_card` retains one common ring-and-arrow bound. The bounded equivalence
is tied to the original realization by `proEtaleKappaEquivalence_forget`.

Lean: `PerfectoidSpace.HasKappaSmallPresentation`, `ProKappa`, `HasKappaSmallPresentation.of_diagram`.

### P6.11 Zariski closed equals strongly Zariski closed

**Zariski closed equals strongly Zariski closed.** (*zariski-closed-equals-strongly-zariski-closed*)
For an affinoid perfectoid space `X = Spa(R, R⁺)` and a morphism `f : Z → X` of perfectoid spaces,
`f` is a Zariski closed immersion if and only if `f` is strongly Zariski closed: the direction
strongly Zariski closed ⇒ Zariski closed, the case of an already given surjective perfectoid
quotient `R → S`, is P4.16 (*strongly-zariski-closed-is-zariski-closed*), and the direction
Zariski closed ⇒ strongly Zariski closed is ECD Theorem 5.8, proved in P4.15 through the
perfectoidisation of semiperfectoid rings ([BS22] Theorem 7.4 and Remark 7.5, via surjectivity of
perfectoidisation). Apply `stronglyZariskiClosed_iff_zariskiClosed`
from P4.15 to obtain the consequences for the pro-étale calculus below. Their completed
character-algebra application is given by `CharacterCompletion.fourier_surjective` and
`fourier_kernel_nontrivial`, with the coefficient conventions below. For every ideal `I ⊆ R`
the map `R → S_I` (layer P4 (P4.14)) is surjective with
kernel `I^perf = {h : |h| = 0 on V(I)}`, `S_I⁺` is the integral closure of the image of `R⁺`, and
`R⁺ → S_I⁺` is almost surjective. For `R = ℂ_p⟨T^{±1/p^∞}⟩` and `I = (T − 1)` this gives
`R ↠ C⁰(ℤ_p, ℂ_p)` (Fresnel–de Mathan), contradicting Scholze's Remark II.2.4 and Bhatt's
Warning 9.4.2. The kernel is strictly larger than `(T − 1)`: writing `f = Σ_a c_a T^a`
(`a ∈ ℤ[1/p]`, `c_a → 0`), `(T − 1)R` is the set of `f` with `Σ_{k∈ℤ} c_{a+k} = 0` for every
`a`, the inverse being `d_a := Σ_{k≥1} c_{a+k}` with `‖d‖ ≤ ‖c‖`, so `(T − 1)` is closed and
`R/(T − 1) = ℂ_p⟨μ_{p^∞}⟩` (the `c_0`-completed group algebra, `T^{k/pⁿ} ↦ [ζ_{pⁿ}^k]`); the
element `Σ_{ζ∈μ_{pⁿ}} [ζ]` of norm `1` maps to the function `pⁿ·1_{pⁿℤ_p}` of norm `|p|ⁿ`, so
the surjection `ℂ_p⟨μ_{p^∞}⟩ → C⁰(ℤ_p, ℂ_p)` is not bounded below and, by the open mapping
theorem, has a nonzero kernel `I^perf/(T − 1)`, consisting of elements of spectral norm `0`.
**Checks for the coefficient and character conventions.**

- `division_by_T_sub_one` — in the rational unit-annulus pair of `discPair`, take
  `T = rationalMap (rootX 0)`. The coefficients `c₀ = −1`, `c₂ = 1`, zero elsewhere,
  give `d₀ = d₁ = 1` by the positive-index tail formula, and their images satisfy
  `(T − 1)(d₀ + d₁T) = c₀ + c₂T²` in this pair's ring.
- `two_character_sum` — `finiteFourier 2 1 (-1)` sends `[0]+[1]` to the values `2,0`.
  The image has norm `1/2` and the group-algebra coefficient norm is `1`.
  These are computations from the coordinate and character formulas in this subsection.

The finite group-algebra restriction of the character map is `finiteFourier`: for a primitive
`pⁿ`-th root `ζ`, it sends `[a]` in `C_p[Z/pⁿ]` to `b ↦ ζ^{ab}` and coefficients to constant
functions. Its coefficient norm is the maximum of the coefficient norms; its target uses the
supremum norm on functions on the finite group. This is the finite-level part of the completed
map above; `fourier_surjective` states surjectivity of the completed map.

**Checks for the Fourier map.**

- `fourier_two_character_norm` — for `p=2`, `n=1`, `ζ=-1`, the group-algebra element
  `[0]+[1]` has coefficient norm `1`; its image takes values `2,0` and has norm `1/2`.
- `fourier_constant` — `c[0]` maps to the constant function `c` at every level.
- `fourier_level_zero` — at level `0`, the group is trivial and the map returns the sole
  coefficient. These are direct computations from the displayed character formula.

The completed carrier `CharacterCompletion` is a complete normed `C_p`-algebra with a dense
algebra map from `C_p[μ_{p^∞}]` whose norm is exactly the coefficient maximum. Here
`pPowerRoots` uses the union of Mathlib's `rootsOfUnity` subgroups. The continuous character
`pPowerCharacter` is fixed by `ζ ↦ (x ↦ ζ^(x mod pⁿ))` whenever `ζ^(pⁿ)=1`.
`CharacterCompletion.fourier_single` fixes the completed transform on the dense group algebra;
`fourier_surjective` and `fourier_kernel_nontrivial` state the completed comparison above.

**Checks for the completion and its characters.**

- `completedFourier_zero` and `completedFourier_constant` — zero maps to zero and `c[1]`
  maps to the constant function `c`.
- `completedFourier_two_characters` — in the completed algebra at `p=2`, `[1]+[-1]`
  has norm `1`, while its Fourier transform takes values `2,0` at `0,1` and has norm `1/2`.
- `completion_distinct_coefficients` — `[ζ]-[η]` has coefficient norm `1` for distinct roots.
- `pPowerRoots_one`, `pPowerRoots_minus_one_two`, `pPowerRoots_two_not_root` — the identity
  is a root, `-1` is a two-power root, and `2` is never a root of unity in `C_p`.
- `pPowerCharacter_at_zero`, `pPowerCharacter_one`, `pPowerCharacter_minus_one` — every
  character is `1` at zero, the identity root gives the constant character, and at `p=2`
  the character of `-1` takes values `1,-1` at `0,1`.

Nothing else in layers P0–P7 uses the stronger direction (the diagonal and Zariski closed
pro-étale statements of this layer need only the elementary one); layer [P8](../PerfectoidTowers/README.md#layer-p8-finite-quotients-and-closed-perfectoid-loci-in-towers) and AdicEtaleGeometry
layer A3 use it downstream ([ECD] §5, Thm 5.8 with proof, p. 25; [ECD] §5, after Def. 5.7, p. 24, the
correction of Scholze's Remark II.2.4; [AWS] Lecture 2, Thm 2.9.12, an independent route; [AWS]
Lecture 2, Ex. 2.4.10, the example `I = (T − 1)`). *Needs:* layer P4 (P4.15, P4.16, P4.12, P4.13,
P4.14).

Sections of affinoid pro-étale maps are affinoid pro-étale (*affinoid-pro-etale-sections*). Let
`f : Y → X` be affinoid pro-étale with pro-étale presentation `Y = lim_{i∈I} Y_i`, and `s : X → Y`
a section of `f` (`f ∘ s = id_X`). Prove that `s` is affinoid pro-étale, with pro-étale
presentation `X ≅ lim_i (X ×_{Y_i} Y) → Y`, where `X → Y_i` is `s_i := pr_i ∘ s` and `Y → Y_i` is
`pr_i` ([ECD] §7, proof of Lem. 7.11, p. 33, the statement and the presentation). *Needs:* *affinoid-pro-etale-map*; layer P3; layer P2 (P2.16);
layer P5 (P5.3).

### P6.12 limits in the affinoid pro-étale category

**`X_proét^aff` has all small limits, computed in perfectoid spaces.** (*affinoid-pro-etale-limits*)
For an affinoid perfectoid space `X`, prove that the category `X_proét^aff` has all small limits,
and that they are computed in perfectoid spaces over `X`: the final object is `id_X`; fibre
products `Y ×_Z Y'` of objects of `X_proét^aff` are the fibre products of perfectoid spaces;
cofiltered limits are the limits of layer P5; every small limit is a cofiltered limit of finite
limits of affinoid perfectoid spaces, and the inclusion `X_proét^aff → (perfectoid spaces over X)`
preserves all small limits. Under `ρ_X` (see *pro-etale-pro-category-equivalence*) the categorical statement — `Pro(C)` has
all small limits for `C = X_ét^aff` finitely complete — uses the opposite of Mathlib
Ind categories and the cofiltered-limit instance below, with
`C` essentially small and finitely complete; what is proved here is that `ρ_X` carries those limits
to the limits of perfectoid spaces ([ECD] §7, Lem. 7.11(iv), p. 33; [ECD] §7, proof of
Lem. 7.11(iv), p. 34, the finite limits in `X_ét^aff`). *Needs:*
*pro-etale-pro-category-equivalence*; *affinoid-pro-etale-base-change*;
*affinoid-pro-etale-cofiltered-limits*; *affinoid-pro-etale-composition*;
*affinoid-pro-etale-cancellation*; *canonical-pro-etale-presentation*; layer P2 (P2.16);
*pro-category-cofiltered-limits*; Mathlib `CategoryTheory.Ind`,
`CategoryTheory.Limits.hasLimits_of_hasColimits_op`, `CategoryTheory.Limits.HasLimits`.

**Cofiltered limits of pro-objects.** (*pro-category-cofiltered-limits*)
Mathlib's `HasFilteredColimits (Ind Cᵒᵖ)` instance and
`CategoryTheory.Limits.has_cofiltered_limits_op_of_has_filtered_colimits`
supply `HasLimitsOfShape I (Ind Cᵒᵖ)ᵒᵖ` directly, for `C : Type u`
with `Category.{v} C` and `I : Type v` a small cofiltered category.
AdicSpacesPartII R0.19.9 supplies the generic cofiltered-limit API; the downstream
consumer DiamondsAndVStacks D0.13 uses the same Mathlib instances. Finite limits in the
affinoid étale category then supply all small limits as used above ([ECD] proof of
Lemma 7.11(iv), pp. 33–34).

**Flattening pro-objects.** (*pro-object-flattening*) P6.12 owns `proFlatten`,
`proFlatten_adjunction` and `proFlatten_obj`; DiamondsAndVStacks D0.13 consumes them.
`proFlatten` sends a pro-object
of pro-objects to the limit of its cofiltered presentation. For any category `C`,
`proFlatten_adjunction` makes this functor right adjoint to the constant pro-object
functor `δ : Pro(C) → Pro(Pro(C))`, with `Pro(C) = (Ind Cᵒᵖ)ᵒᵖ`.
For the adjunction chosen from this theorem, write `η` for its unit and `ε` for its
counit. For every `X : Pro(C)` and `Y : Pro(Pro(C))`, `proFlatten_obj` gives
`δ(η_X) ≫ ε_{δ(X)} = id_{δ(X)}` and
`η_{proFlatten(Y)} ≫ proFlatten(ε_Y) = id_{proFlatten(Y)}`.
These are applications of Mathlib's triangle identities to the flattening
construction used in [ECD] proof of Lemma 7.11(i), p. 33.
The Checks `proFlatten_constant`, `proFlatten_empty`, and
`proFlatten_two` below fix its constant-object behavior.

Base change of affinoid pro-étale maps (*affinoid-pro-etale-base-change*). Let `f : Y → X` be
affinoid pro-étale with pro-étale presentation `Y = lim_{i∈I} Y_i`, and `g : X' → X` a morphism
from an affinoid perfectoid space. Prove that `Y' := X' ×_X Y` is affinoid perfectoid and
`f' : Y' → X'` is affinoid pro-étale with pro-étale presentation `Y' = lim_i X' ×_X Y_i`
(`IsAffinoidProEtale.baseChange`), and that `|Y'| → |X'| ×_{|X|} |Y|` is surjective ([ECD] §7,
Lem. 7.11(ii), p. 33; [ECD] §7, proof of Lem. 7.11, p. 33). *Needs:* *affinoid-pro-etale-map*;
layer P2 (P2.16); layer P3 (P3.16); layer P5 (P5.3); AdicSpacesPartII R0.19.5–R0.19.8 and P5.4’s perfectoid limit comparisons.

Cofiltered limits of affinoid pro-étale `X`-spaces are affinoid pro-étale, the geometric form of
`Pro(Pro(C)) → Pro(C)` (*affinoid-pro-etale-cofiltered-limits*). Let `X` be an affinoid perfectoid
space and `(Y_k)_{k∈K}` a small cofiltered diagram in `X_proét^aff`. Prove that its limit
`Y = lim_k Y_k` in perfectoid spaces (layer P5 (P5.3)) is affinoid pro-étale over `X` and is the
limit of the diagram in `X_proét^aff`; under `ρ_X`, `Y` corresponds to the limit in
`Pro(X_ét^aff)` of the corresponding pro-objects (the flattening `Pro(Pro(C)) → Pro(C)` is
the functor `proFlatten` on opposites of Mathlib Ind categories), so `ρ_X` preserves cofiltered limits ([ECD] §7,
proof of Lem. 7.11, p. 33, the categorical input made geometric; [Berkeley] Lecture 8,
Prop. 8.2.5(3), p. 59, existence of the limits). Prove preservation of the limit by its Hom
universal property and finite-stage factorisation. *Needs:* *pro-etale-pro-category-equivalence*;
*maps-to-etale-factor-through-finite-stage*;
*canonical-pro-etale-presentation*; layer P5 (P5.3); *pro-object-flattening*.

Composites of affinoid pro-étale maps are affinoid pro-étale (*affinoid-pro-etale-composition*).
If `g : Z → Y` and `f : Y → X` are affinoid pro-étale, prove that `f ∘ g : Z → X` is affinoid
pro-étale (`IsAffinoidProEtale.comp`): for pro-étale presentations `Y = lim_{i∈I} Y_i` over `X`
and `Z = lim_{j∈J} Z_j` over `Y`, each `Z_j` is `Y`-isomorphic to `W_j ×_{Y_{i(j)}} Y` for some
`i(j) ∈ I` and `W_j ∈ (Y_{i(j)})_ét^aff` (see *affinoid-etale-finite-stage-assembly*), and the transition maps between the `Z_j` descend after refinement by
*maps-to-etale-factor-through-finite-stage*. Use the category of all such finite-stage
models and compatible refinements, retaining each index `i(j)`: prove it is
cofiltered and that its affinoid étale `X`-diagram has limit `Z`
([ECD] §7, Lem. 7.11(i), p. 33; [ECD] §7, proof of Lem. 7.11, p. 33, the reduction used by
*pro-etale-stability-and-limits*). *Needs:* *affinoid-etale-finite-stage-assembly*;
*maps-to-etale-factor-through-finite-stage*; *affinoid-pro-etale-map*; layer P2 (P2.16); layer
P3.

`CompositionModel` records a pair of stages, an affinoid étale model over the first stage,
and its base-change isomorphism to the second. Its arrows retain both indexing refinements,
the map of models and the compatibility with that isomorphism. `diagram`, `cone`,
`cofiltered`, `essentiallySmall` and `isLimit` state the simultaneous descent construction.

**Checks for simultaneous models.**

- `composition_model_constant` — two one-stage étale presentations admit the original
  composite source as a model.
- `composition_model_empty_stage` — an empty model has an empty base-changed stage.
- `composition_model_two_sheets` — over a nonzero affine base, the two-sheet constant
  presentation has a two-sheet model whose structure map is not an isomorphism.
- `composition_model_identity` additionally checks that the model diagram sends identity
  refinements to identity arrows, in agreement with its functor structure.

Morphisms between affinoid pro-étale `X`-spaces are affinoid pro-étale
(*affinoid-pro-etale-cancellation*). Let `Y → X` and `Y' → X` be affinoid pro-étale. Prove that
every `X`-morphism `g : Y → Y'` is affinoid pro-étale (`IsAffinoidProEtale.of_comp`); consequently,
for `Y' ∈ X_proét^aff`, an affinoid perfectoid space `Z → Y'` is affinoid pro-étale over `Y'` iff
`Z → X` is affinoid pro-étale, so `(Y')_proét^aff ≃ X_proét^aff/Y'` (slice category) ([ECD] §7,
Lem. 7.11(iii), p. 33; [ECD] §7, proof of Lem. 7.11, p. 33). *Needs:*
*affinoid-pro-etale-base-change*; *affinoid-pro-etale-composition*;
*affinoid-pro-etale-sections*; layer P2 (P2.16).

### P6.13 stability of pro-étale morphisms

The last two subsections globalise the affinoid stability results to pro-étale morphisms of
arbitrary perfectoid spaces and compare with Scholze 2013.

**Stability of pro-étale morphisms: composition, base change, cancellation and limits.**
(*pro-etale-stability-and-limits*) Composition (`IsProEtale.comp`), base change along arbitrary
morphisms of perfectoid spaces (`IsProEtale.baseChange`) and cancellation (`IsProEtale.of_comp`)
for pro-étale morphisms are the affinoid statements *affinoid-pro-etale-composition*,
*affinoid-pro-etale-base-change* and *affinoid-pro-etale-cancellation* of P6.12 passed to the
general case by *pro-etale-local-shrinking*, and the limits of `X_proét^aff` for affinoid `X` are
*affinoid-pro-etale-limits* (P6.12); none is restated here. Consequences: `X_proét` (perfectoid spaces pro-étale over `X`, with
all `X`-morphisms) contains `X_ét`, is closed under fibre products, which are computed in
perfectoid spaces (`ProEtaleOver.hasPullbacks`), and for `Y ∈ X_proét`, `Y_proét ≃ X_proét/Y`.
These are the stability assertions the big and small pro-étale sites of DiamondsAndVStacks layer
D2 rest on; pro-étaleness is not claimed to be local on the target for the pro-étale topology
([ECD] §7, Lem. 7.11, p. 33; [Berkeley] Lecture 8, Prop. 8.2.5, p. 58, Berkeley's version of the
same lemma). *Needs:*
*affinoid-pro-etale-base-change*; *affinoid-pro-etale-composition*;
*affinoid-pro-etale-cancellation*; *pro-etale-local-shrinking*; *affinoid-pro-etale-limits*;
*pro-etale-map*; layer P2 (P2.16).

Étale maps are pro-étale, and affinoid pro-étale maps restrict to affinoid opens
(*pro-etale-local-shrinking*). Prove: (a) every étale morphism of perfectoid spaces is pro-étale
(`IsProEtale.of_isEtale`), and every étale morphism between affinoid perfectoid spaces is affinoid
pro-étale (`IsAffinoidProEtale.of_isEtale`); (b) if `f : Y → X` is affinoid pro-étale, `U ⊆ X` is
an open affinoid perfectoid subspace and `V ⊆ f⁻¹(U)` is an open affinoid perfectoid subspace,
then `f|_V : V → U` is affinoid pro-étale; (c) consequently `f : Y → X` is pro-étale iff every
`y ∈ Y` has rational neighbourhoods `V ∋ y` and `U ∋ f(y)`, inside any given neighbourhoods, with
`f(V) ⊆ U` and `f|_V` affinoid pro-étale (`isProEtale_iff_exists_rational`); pro-étaleness is local
on the source (`IsProEtale.of_sourceOpenCover`: if `Y = ⋃ V_α` and all `V_α → X` are
pro-étale, so is `f`), local on the target for open covers (`IsProEtale.of_openCover`), and stable under
restriction to open subspaces of the source and over open subspaces of the target
(`IsProEtale.restrict`) ([ECD] §7, proof of Lem. 7.11, p. 33, the reduction made precise;
[Berkeley] Lecture 8, Def. 8.2.1, p. 57, locality on source and target). *Needs:* *affinoid-pro-etale-map*; *pro-etale-map*;
*affinoid-pro-etale-base-change*; *affinoid-pro-etale-composition*; layer P3 (P3.15); layer P2
(P2.3).

Diagonals and graphs of morphisms of perfectoid spaces are pro-étale (*diagonals-are-pro-etale*).
Let `f : Y → X` be any morphism of perfectoid spaces, with no separatedness or finiteness
hypothesis. Prove: (a) the diagonal `Δ_f : Y → Y ×_X Y` (layer P2 (P2.16)) is pro-étale
(`IsProEtale.diagonal`); if `X = Spa(R, R⁺)` and `Y = Spa(S, S⁺)` are affinoid, `Δ_f` is a Zariski
closed immersion, cut out by the kernel of the multiplication map `S ⊗̂_R S → S`, and affinoid
pro-étale (see *zariski-closed-immersions-are-pro-etale*); (b) for morphisms `Y → X` and `Y' → X` and
an `X`-morphism `g : Y → Y'`, the graph `Γ_g : Y → Y ×_X Y'` is pro-étale, affinoid pro-étale when
`X`, `Y`, `Y'` are affinoid ([ECD] §7, Rem. 7.9, p. 32, for (a); [ECD] §5, proof of Prop. 5.9,
p. 25, the affinoid diagonal is strongly Zariski closed; [ECD] §11, proof of Prop. 11.4, p. 55,
the consumer in DiamondsAndVStacks layer D4).
*Needs:* *zariski-closed-immersions-are-pro-etale*; layer P4 (P4.18, P4.13); layer P2 (P2.16);
*pro-etale-map*; *pro-etale-local-shrinking*; *pro-etale-stability-and-limits*.

`proFlatten` takes the limit of the cofiltered presentation of a pro-object of
pro-objects. The unit and counit equations of `proFlatten_obj` identify the two
composites with identities on constant and flattened objects. The Checks
`proFlatten_constant`, `proFlatten_empty`, and `proFlatten_two` recover a constant
pro-object, the unique endomorphism of the empty finite set, and the four endomorphisms
of the two-point set.

### P6.14 comparison with Scholze 2013's affinoid perfectoid objects

**Scholze 2013's affinoid perfectoid objects of `X_proét` give affinoid pro-étale perfectoid spaces
in the sense of ECD Definition 7.8.** (*scholze-2013-affinoid-perfectoid-comparison*) Let `K` be a
perfectoid field of characteristic `0` (Scholze 2013's setting), `X` a locally noetherian adic
space over `Spa(K, K⁺)`, and `X_proét` Scholze's pro-étale site (AdicEtaleGeometry A1.12, on the
category `pro-X_ét` of AdicEtaleGeometry A1.12). Let `U = lim U_i ∈ X_proét` be affinoid
perfectoid (Scholze 2013, Definition 4.3: a pro-étale presentation by affinoid
`U_i = Spa(R_i, R_i⁺)` whose completed colimit `(R, R⁺)` is a perfectoid affinoid
`(K, K⁺)`-algebra) and `Û := Spa(R, R⁺)`; the comparison is with the base-field-free notion of
this layer restricted to spaces over `Spa(K, K⁺)`. Prove: (a) if `V_i → U_i` is étale and a
composite of rational subsets and finite étale maps, then `V := V_i ×_{U_i} U` is affinoid
perfectoid and `V̂ = V_i ×_{U_i} Û` (fibre product of adic spaces over `K`) is an object of
`Û_ét^aff`, étale in the sense of ECD Definition 6.2 (Scholze 2013, Lemma 4.5(i)); (b) if
`V = lim_k V^{(k)} → U` is a cofiltered limit in `X_proét` of objects `V^{(k)} → U` as in (a), then
`V` is affinoid perfectoid and `V̂ = lim_k V̂^{(k)}` (limit in perfectoid spaces, layer P5) is
affinoid pro-étale over `Û` with pro-étale presentation `(V̂^{(k)})_k` (*affinoid-pro-etale-map*,
`isAffinoidProEtale_scholze2013`); `V ↦ V̂` is functorial, being induced on pro-objects by the
functor of (a) followed by the realisation of *pro-etale-pro-category-equivalence*; (c) boundary:
Scholze 2013's presentations require eventually finite étale surjective transition maps over a
locally noetherian base, ECD's do not; the erratum restricts only the covering families of
`X_proét`, not its objects, so (a)–(b) are unaffected; no converse is asserted (an affinoid
pro-étale `Û`-space need not come from `X_proét`) ([Sch13] §4, Def. 4.3(i), p. 23, the objects
compared; [Sch13] §4, Lem. 4.5(i), p. 23, for (a); [Sch13] §4, proof of Lem. 4.6, p. 25, for (b);
[Sch13E] item (1), p. 1, for (c)).
*Needs:* AdicEtaleGeometry A1.12 (Scholze 2013's pro-étale site, for the comparison only); layer P2
(P2.3); layer P3 (P3.16, P3.15); layer P5 (P5.3);
*affinoid-pro-etale-map*; *pro-etale-pro-category-equivalence*.

The ring presentation in Definition 4.3(i) has the carrier `ScholzeAffinoidTower`:
strongly noetherian Tate stages over a characteristic-zero perfectoid field, finite étale
surjective transitions, and a perfectoid colimit in complete Huber pairs. `map_stage` fixes
the induced completion map on every stage. `rational_baseChange` and `finiteEtale_baseChange`
state the two elementary factors of (a) as pushouts of complete Huber pairs. The global
comparison in (b) additionally uses the classical `AdicSpace.ProEtaleSite` carrier and the
identification of its affinoid pro-objects with these ring presentations.

`ScholzeAffinoidTower.isAffinoidProEtale_scholze2013` states (b) on these completed
ring presentations. Fix a small cofiltered category `J`, towers `B j`, their completed-pair
diagram `D : Jᵒᵖ ⥤ PairCat p`, and an affinoid étale diagram `E` over `A.realization`,
expressed in `affinoidEtaleSmall`. Require `D.obj (op j) = (B j).completedPair` and a
natural isomorphism from the spectra of `D` to the underlying spaces of `E`; this identifies
both the stages and their transition maps. Choose a stage and compatible pseudouniformizer
roots for P5's `completedColimitPair`. For the map `a` from `A.completedPair` to that
completion, require that every spectral projection followed by its map to `A.realization`
equals `spaFunctor.map a.op`. The theorem says this spectral map is affinoid pro-étale and
supplies a `ProEtalePresentation` with index `J` and diagram `E` transported from the small
model. It also gives an isomorphism, **over `A.realization`**, to `proEtaleEquivalence.cone E`,
carrying each projection to the corresponding completed-pair inclusion. This is the
presentation `(V̂^{(k)})`; `proEtaleEquivalence.cone_pt` identifies its vertex with
`ρ_X("lim" E)`. Thus the completion, its structure map, and its specified presentation are
all related in the statement. Source: [Sch13] Lemma 4.6 and its proof, pp. 24–25, together
with [ECD] Definition 7.8(i), p. 32, and Proposition 7.10, p. 33.

`ScholzeAffinoidTower.isAffinoidProEtale_scholze2013_naturality` specifies the map of
realizations for a natural transformation `α : D → E` of small cofiltered affinoid étale
diagrams over `A.realization`. The map obtained by applying `proEtaleEquivalence` to the
induced pro-object map, followed by the `j`th projection, equals the source projection
followed by `α.app j`. Combined with the preceding isomorphisms, this identifies the maps
of completed towers with `ρ_X` on morphisms, including identity and composition, rather than
only identifying their objects ([Sch13] paragraph after Definition 4.3, p. 23;
[ECD] Proposition 7.10, p. 33). These are limits of perfectoid realizations; comparison with
the original noetherian stages uses AdicSpacesPartII R0.20.1–R0.20.2's `IsTildeLimit` and
`IsTildeLimit.baseChange`, with the completed-direct-limit criterion of
*tilde-limits-affinoid-criterion*.

**Checks for the cofiltered realization.**

- `isAffinoidProEtale_scholze2013_constant` — a constant pro-object realizes to its original
  affinoid étale object over the base.
- `isAffinoidProEtale_scholze2013_empty_stage` — an empty stage forces an empty realization;
  the index category itself remains nonempty by cofilteredness.
- `isAffinoidProEtale_scholze2013_two_stages` — for a diagram `0 → 1`, projection to stage
  `0` is an isomorphism and projection to stage `1` is that isomorphism followed by the given
  transition. The transition is retained, with no identity or surjectivity assumption.
- `isAffinoidProEtale_scholze2013_constant_zero_transition` — in the two-stage diagram
  `X × {0,1} → X × {0,1}` with transition `(x,s) ↦ (x,0)`, the realization is the first
  two-sheet cover: the projections are the identity and the constant-zero map. For nonempty
  `X`, the two sheets of the limit survive although the transition is not surjective.

**Checks for the completed tower.**

- `scholze_completion_identity` — the identity map of a tower realizes to an isomorphism.
- `scholze_completion_empty` — a zero-ring stage forces the completed spectrum to be empty.
- `scholze_completion_two_points` — a tower with one object and only its identity arrow,
  whose stage has two spectral points, retains those two points after completion.

<a id="p6-15"></a>

### P6.15 continuous function rings on completed torsor towers

**Continuous function rings on the completed tower are perfectoid.**
(*completed-tower-continuous-functions-perfectoid*) In the situation of [*completed-tower-pair*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.5),
let `p` be prime and topologically nilpotent in `A`. For every `k ≥ 0`, prove that
`C^0(G^k,Â_∞)` is perfectoid, with ring of integral elements `C^0(G^k,Â_∞⁺)`.
For `k = 0` this is [*perfectoid-uniform-completion*](#p1-32-split-tate-rings) (P1.32). In general it is the ring of the
product of `Spa(Â_∞,Â_∞⁺)` with the profinite set `G^k` (P6,
[*product-with-profinite-set*](#p6-8) (P6.8)). Combined with [*completed-tower-self-product*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.5), it makes the
uniform tensor square a perfectoid self-product; the tensor-square identification itself is
built in AdicEtaleGeometry A4.5. This supplies `relation_isPerfectoid` for `ofTower T` in
[*perfectoid-cover-presentation*](#p6-16) (P6.16) ([ECD] §15, Prop. 15.4 and proof of Lem. 15.6, pp. 90–91).
*Needs:* [*perfectoid-uniform-completion*](#p1-32-split-tate-rings) (P1.32); [*completed-tower-self-product*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.5);
P6 (*product-with-profinite-set*); Mathlib `ContinuousMap`, `ProfiniteGrp`.
README only (no Lean declaration).

**Checks.**

- *tower_functions_zero_factors* — `G^0` is a singleton and its function ring is `Â_∞`.
- *tower_functions_one_factor* — the relation ring `C^0(G,Â_∞)` is perfectoid,
  as in *tower_continuousMap_perfectoid*.
- *tower_functions_trivial* — for a trivial-group tower over a perfectoid split base `A`,
  every `C^0(G^k,Â_∞)` is `A`.
<a id="p6-16"></a>

### P6.16 the perfectoid torsor presentation

**Perfectoid torsor presentation.** (*perfectoid-cover-presentation*) Let `p` be a prime and
`(A, A⁺)` a Huber pair with `A` a complete Hausdorff Tate ring, `A⁺` any ring of integral
elements, and no sheafiness of `(A, A⁺)`; existence of a presentation (`exists`) assumes that `p`
is topologically nilpotent in `A`. A *perfectoid torsor presentation* of
`(A, A⁺)` consists of a profinite group `G` (Mathlib `ProfiniteGrp`), a complete Huber pair
`(Ã, Ã⁺)` with `Ã` a perfectoid Tate ring (layer P1), a continuous action of `G`
on `(Ã, Ã⁺)` by automorphisms of Huber pairs, and a `G`-invariant morphism of Huber pairs
`q : (A, A⁺) → (Ã, Ã⁺)`, such that

`(a) Ã ⊗̂ᵘ_A Ã ≅ (C^0(G, Ã), C^0(G, Ã⁺)),  b ⊗ b′ ↦ (g ↦ b·g(b′))   (uniform completed tensor     product, AdicSpacesPartII layer R0); (b) Spa(q) : Spa(Ã, Ã⁺) → Spa(A, A⁺) is surjective and open with fibres exactly the     G-orbits, so |Spa(Ã, Ã⁺)|/G ≅ |Spa(A, A⁺)|.`

`Spa(q)` is moreover spectral, in particular quasi-compact, for every morphism `q` out of the Tate
pair `(A, A⁺)` (Tau Ceti `Huber.Pair.Hom.isSpectralMap_spaComap_of_isTateRing` in
`TauCeti/AlgebraicGeometry/AdicSpace/Spa/MorphismSpectral.lean`), so the quasi-compactness that
[ECD] Prop. 15.4 records is automatic and is not an axiom of the record.

The *perfectoid cover* is `X̃ = Spa(Ã, Ã⁺)`; the *induced perfectoid equivalence relation* is
`R̃ = Spa(C^0(G, Ã), C^0(G, Ã⁺)) ≅ G × X̃`, with `s, t : R̃ ⇉ X̃` induced by `b ↦ (g ↦ b)` and
`b ↦ (g ↦ g·b)`, identity `X̃ → R̃` (evaluation at `1`) and composition induced by
`C^0(G, Ã) → C^0(G × G, Ã)`, `f ↦ ((g,h) ↦ f(gh))` in edge coordinates — the action groupoid of `G` on `X̃`, with all objects affinoid
perfectoid. This is the generalized-affinoid presentation of [ECD] §15: `(A, A⁺)` need not be
sheafy, while `X̃` and `R̃` are affinoid perfectoid. `R̃` is an equivalence relation only in the
category of v-sheaves (DiamondsAndVStacks layer D6): on underlying sets
`|R̃| → |X̃| ×_{|Spa A|} |X̃|` is surjective but not injective. Define
the structure `Huber.PerfectoidTorsorPresentation p P` for `P = (A, A⁺)` (`p.Prime`, `G : ProfiniteGrp`, a
complete perfectoid pair `(Ã, Ã⁺)` with continuous `G`-action, a `G`-invariant `q`, and axioms
(a), (b)); define `ofTower` (for a finite étale torsor tower `T` over `A`, take `G = lim G_i` and
`(Ã, Ã⁺) = (Â_∞, Â_∞⁺)`, assembled field by field from [*perfectoid-uniform-completion*](#p1-32-split-tate-rings) (P1.32),
`completionAction_spec` (*completed-tower-pair*), the self-product identification of
[*completed-tower-self-product*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.5) (axiom (a)), the perfectoid relation from
[*completed-tower-continuous-functions-perfectoid*](#p6-15) (P6.15) and the quotient homeomorphism of
[*spa-of-completed-tower*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.7) (axiom (b))), `exists` (every
complete Tate `ℤ_p`-pair admits a presentation: `ofTower` of `FiniteEtaleTorsorTower.canonical A`),
`relation` (the pair
`(C^0(G, Ã), C^0(G, Ã⁺))` with `relation.source`, `relation.target`,
`relation.unit` and `relation.composition`; `relation.mem_plus` fixes the plus ring
pointwise, `relation.source_apply` and `relation.target_apply` fix the two projections,
and `spaQuotientHomeomorph_mk` fixes the quotient map on classes; `unit_apply`, `composition_apply`, `unit_comp` and
`composition_laws` specify evaluation, the unit laws and associativity, while
`composition_continuous` and `composition_mem_plus` give continuity and preservation of integral elements) and `ofPerfectoid`
(the trivial presentation of a perfectoid pair, `G = 1`); prove `relation_isPerfectoid`
(`C^0(G^k, Ã)` is perfectoid for every `k ≥ 0`; layer P6,
[*product-with-profinite-set*](#p6-8) (P6.8), and for `k = 1` through (a) also P2's
[*uniform-completed-tensor-of-perfectoid-over-any-base*](#p2-14) (P2.14)), `uniformCompletedTensorEquiv` (axiom (a),
with `inlᵘ ↦ s` and `inrᵘ ↦ t`), `spaQuotientHomeomorph` (`|Spa(Ã, Ã⁺)|/G ≃ₜ |Spa(A, A⁺)|` induced
by `Spa(q)`), `spa_relation_homeomorph` (`|Spa(C^0(G, Ã), C^0(G, Ã⁺))| ≃ₜ G × |Spa(Ã, Ã⁺)|`, under
which `s` and `t` become `(g, x) ↦ x` and `(g, x) ↦ g⁻¹·x`; layer P6,
[*product-with-profinite-set*](#p6-8) (P6.8)), with Tau Ceti’s `TauCeti.Huber.Pair.Hom.spaComap_preimage_rationalSubset` (the
preimage under `Spa(q)` of `R(T/s)` is `R(q(T)/q(s))`) cited. For `ofTower T hp`, apply the
effective descent along the tower: every finite étale `Ã`-algebra
with a continuous semilinear `G`-action is equivariantly isomorphic to `Ã ⊗_A B` for a
finite étale `A`-algebra `B` (*finite-etale-effective-descent-along-tower*) ([ECD] §15, Prop. 15.4, p. 90 (the
presentation of `Spd(A, A⁺)` by an affinoid perfectoid space with a `G`-torsor, of which this is
the analytic side); [ECD] §15, proof of Prop. 15.4, p. 91 (the topological quotient (b));
[Berkeley] Lecture 10, Lem. 10.1.7, p. 75; [ECD] §15, after Def. 15.5, p. 91 (generalized, not
necessarily sheafy, adic spaces)). *Needs:* [*completed-tower-pair*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.5);
[*perfectoid-uniform-completion*](#p1-32-split-tate-rings) (P1.32); [*completed-tower-self-product*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.5);
[*completed-tower-continuous-functions-perfectoid*](#p6-15) (P6.15); [*spa-of-completed-tower*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.7);
[*finite-etale-effective-descent-along-tower*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.5); [*finite-etale-torsor-tower*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.3); AdicSpacesPartII layer
R0 (uniform completed tensor product); layers
P1, P2 and P6; AdicEtaleGeometry layer A1
(*generalized-adic-presentation*); Tau Ceti `Huber.Pair`, `Huber.Pair.Hom`,
`ValuationSpectrum.spa`, `ValuationSpectrum.spaComap`; Mathlib `ProfiniteGrp`, `ContinuousMap`. README only (no Lean declaration).

**Checks.**

- *nonprime_indices* — `PerfectoidTorsorPresentation` over `(Q₂,ℤ₂)` is empty at parameters 0 and 1 and inhabited at parameter 2.
- *tower_carriers* — `ofTower T` has exactly `T.group` and `Uniformization T.colim`;
  a spurious extra factor in the group cannot satisfy these identities.
- *tower_integral_elements* — its plus pair is `T.completion P`, with the completed integral
  elements of the tower.
- *tower_structure_map* — `q(a) = T.completionStructureHom P(a)`; together with
  `completionStructureHom_apply`, this is the map through every finite level. Since the carrier,
  plus ring and structure map of `ofTower T` are those of [*completed-tower-pair*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.5), its computed
  value and degenerate case are *completion_test_Qp* and *completion_test_trivial*.
- *perfectoid_identity_map* (degenerate) — `ofPerfectoid` has `q(a) = a`.
- *perfectoid_identity_plus* — its plus pair is the original `P`, even when the plus ring
  is properly contained in the power-bounded subring.
- *perfectoid_trivial_action* — its one-element group acts by `a ↦ a`.
- *relation_evaluation* — the point corresponding to `(g,x)` is `f ↦ x(f(g))`, as fixed by
  `spa_relation_homeomorph_symm`.
- *relation_source* — source applied to that point is `x` (`spa_relation_source`).
- *relation_identity* (degenerate) — at `g = 1`, target also gives `x`; in general it is
  `b ↦ x(g·b)` (`spa_relation_target`). The two-term `C₃` computation in
  *galoisTensorMap_inverse_support* distinguishes this from the other action convention.
- *quotient_projection* — `spaQuotientHomeomorph([x]) = Spa(q)(x)`.
- *quotient_fibre* — equality of these values is exactly the existence of an element sending
  one point to the other; injectivity before taking orbits is not asserted.
- *quotient_trivial* (degenerate) — the quotient homeomorphism of `ofPerfectoid` sends `[x]`
  to `x` in the original adic spectrum.
- *test_Qp* (computation) — `(ℚ_p, ℤ_p)` has a presentation with ring topologically
  isomorphic to `ℂ_p`.
- *test_perfectoid* (degenerate) — if `(A, A⁺)` is perfectoid, `G = 1` and `(Ã, Ã⁺) = (A, A⁺)` is a
  presentation (`C^0(1, A) = A = A ⊗̂ᵘ_A A`).
- *test_relation_not_injective* (non-example) — for `(ℚ_p, ℤ_p)` the map
  `|R̃| = Gal(ℚ̄_p/ℚ_p) × |X̃| → |X̃| ×_{|Spa ℚ_p|} |X̃| = point` is surjective but not injective:
  `R̃ ⇉ X̃` is not an equivalence relation on underlying sets.
- *test_finite_level* (compatibility) — for a presentation `ofTower T` and a level `i`,
  two points of `Spa(Ã, Ã⁺)` have the same image in `Spa(A_i, A_i⁺)` exactly when
  they lie in an orbit of `ker(G → G_i)`.
- *test_nonuniform* (compatibility) — `(A, A⁺)` admits a presentation if and only if
  `(A^u, A^{u+})` does.
- *perfectoid_not_Qp* (non-example) — `ℚ_p` is not perfectoid (no `ϖ ∈ ℤ_p` has `ϖ^p ∣ p`;
  [ECD] §3, Rem. 3.3, p. 14), so `ofPerfectoid` does not apply to `(ℚ_p, ℤ_p)`, whose
  presentations come from the tower.
<a id="p6-17"></a>

### P6.17 comparison interfaces with diamonds

**`Spd` of a Tate Huber pair and the perfectoid torsor.**
(*spd-of-tate-huber-pair-and-perfectoid-torsor*) The analytic half of [ECD] §15 — Lem. 15.3,
p. 90, and the identification of underlying spaces in the proof of Prop. 15.4, pp. 90–91 —
supplied by this layer is the declaration `Huber.PerfectoidTorsorPresentation.exists`: for a prime
`p`, every Huber pair `(A, A⁺)` with `A` a complete Hausdorff Tate ring in which `p` is
topologically nilpotent has a perfectoid torsor presentation ([*perfectoid-cover-presentation*](#p6-16) (P6.16),
assembled from [*finite-etale-torsor-tower*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.3), [*completed-tower-pair*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.5), [*perfectoid-uniform-completion*](#p1-32-split-tate-rings) (P1.32),
[*completed-tower-self-product*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.5), [*completed-tower-continuous-functions-perfectoid*](#p6-15) (P6.15) and
[*spa-of-completed-tower*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.7), with no sheafiness assumption). The
v-sheaf statements of [ECD] Lems. 15.1–15.2 and Prop. 15.4, pp. 89–91 — `Spd ℤ_p` and
`Spd(A, A⁺)` are v-sheaves, `Spd(A_i, A_i⁺) → Spd(A, A⁺)` and `Spd(Â_∞, Â_∞⁺) → Spd(A, A⁺)` are
torsors, `Spd(A, A⁺)` is a spatial diamond with `|Spd(A, A⁺)| = |Spa(A, A⁺)|` — are
DiamondsAndVStacks layer D6's targets *spd-of-a-tate-pair* (D6.1) and *spd-is-a-spatial-diamond*
(D6.3), which consume this record (for a Tate `ℤ_p`-algebra that is not complete, through its
completion, as in the conventions of this layer) and justify nothing here. *Needs:*
[*perfectoid-cover-presentation*](#p6-16) (P6.16). README only (no Lean declaration).


**Checks.**

- The analytic presentation of `ℚ_p` has completed pair `(ℂ_p,O_ℂ_p)`.
- For a perfectoid pair the analytic input uses the original plus ring and trivial group.
- On the underlying space, a nontrivial stabilizer gives distinct groupoid arrows with the same endpoints; set-level injectivity cannot replace the diamond interpretation.

**The diamond of an analytic adic space and its étale site.**
(*diamond-of-analytic-adic-space-and-etale-site-15-6*) [ECD] §15, Def. 15.5 and Lem. 15.6, p. 91
(`Y^♢` for an analytic adic space `Y` over `ℤ_p`, `|Y^♢| = |Y|`, `Y^♢_ét ≅ Y_ét`,
`Y^♢_fét ≅ Y_fét`) are DiamondsAndVStacks layer D6's targets *gluing-and-the-diamond-functor*
(D6.4) and *etale-site-comparison* (D6.5); the analytic input this layer supplies to the
affinoid case consists of the following separately named interfaces for the presentation of
`(A, A⁺)` from a finite étale torsor tower: the quotient
`|Spa(A, A⁺)| = |Spa(Â_∞, Â_∞⁺)|/G` with rational subsets corresponding (*spa-of-completed-tower*),
the equivalences `(Â_∞)_fét ≅ 2-colim (A_i)_fét` and their variants for `C^0(G, Â_∞)` and
`C^0(G × G, Â_∞)` (*finite-etale-invariance-along-tower*), and the descent of `FÉt(A)` along the
tower (*finite-etale-effective-descent-along-tower*), the sites `Y_fét`, `Y_ét` of a possibly
non-sheafy `Spa(A, A⁺)` being AdicEtaleGeometry layer A1's ([*finite-etale-site*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.6), [*etale-site*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.6),
[*etale-site-generalized*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.9) on [*generalized-adic-presentation*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.16); [KL15] §8.2, Lem. 8.2.17(a), p. 162)
([ECD] §15, proof of Lem. 15.6, p. 91). *Needs:* [*perfectoid-cover-presentation*](#p6-16) (P6.16);
[*spa-of-completed-tower*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.7); [*finite-etale-invariance-along-tower*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.5);
[*finite-etale-effective-descent-along-tower*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A4.5); AdicEtaleGeometry layer A1 ([*finite-etale-site*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.6),
[*etale-site*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.6), [*etale-site-generalized*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.9), [*generalized-adic-presentation*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.16)). README only (no Lean declaration).

#### Presentation examples

For `(ℚ_p, ℤ_p)`, the tower of finite Galois extensions has uniform completion `ℂ_p`.
Its profinite limit group `G` gives `ℂ_p ⊗̂ᵘ_{ℚ_p} ℂ_p ≅ C^0(G, ℂ_p)` and
`|Spa ℂ_p|/G = |Spa ℚ_p|`. For `(ℚ_p⟨T⟩, ℤ_p⟨T⟩)`, the completed tower is
perfectoid although the original ring is not: approximate p-power roots of `T`
modulo `ϖ^p` exist in its power-bounded subring. The incomplete ring `ℚ_p[T]`
illustrates the completeness hypothesis. A perfectoid pair has the trivial
presentation with `G = 1`.

The following Checks specify the presentation's carrier, action, plus ring,
quotient and groupoid maps; their statements are README targets.

| Definition | Checks |
| --- | --- |
| `Huber.PerfectoidTorsorPresentation` | *presentation_prime*, *test_Qp*, *test_perfectoid*, *test_relation_not_injective* |
| `Huber.PerfectoidTorsorPresentation.q` | *tower_structure_map*, *perfectoid_identity_map*, *presentation_base_invariant* |
| `Huber.PerfectoidTorsorPresentation.ofTower` | *tower_carriers*, *tower_structure_map*, *test_finite_level*, *completion_test_Qp*, *completion_test_trivial* |
| `Huber.PerfectoidTorsorPresentation.relation` | *relation_plus_constants*, *relation_plus_pointwise*, *relation_plus_target* |
| `Huber.PerfectoidTorsorPresentation.relation.source` | *relation_source_values*, *relation_unit_source*, *relation_source* |
| `Huber.PerfectoidTorsorPresentation.relation.target` | *relation_target_values*, *relation_unit_target*, *relation_identity* |
| `Huber.PerfectoidTorsorPresentation.relation.unit` | *relation_unit_value*, *relation_unit_source*, *relation_unit_target* |
| `Huber.PerfectoidTorsorPresentation.relation.composition` | *relation_composition_value*, *relation_composition_units*, *relation_composition_associative*, *relation_composition_noncommutative* |
| `Huber.PerfectoidTorsorPresentation.uniformCompletedTensorEquiv` | *presentation_tensor_left*, *presentation_tensor_right*, *presentation_tensor_plus* |
| `Huber.PerfectoidTorsorPresentation.spaQuotientHomeomorph` | *quotient_projection*, *quotient_fibre*, *quotient_trivial* |
| `Huber.PerfectoidTorsorPresentation.spa_relation_homeomorph` | *relation_evaluation*, *relation_source*, *relation_identity* |
| `Huber.PerfectoidTorsorPresentation.ofPerfectoid` | *perfectoid_identity_map*, *quotient_trivial*, *test_perfectoid*, *perfectoid_not_Qp* |

- *presentation_prime* — the perfectoid parameter cannot be 0 or 1.
- *presentation_base_invariant* — q is fixed pointwise by the action.
- *relation_plus_constants* — a constant function is integral exactly at its value.
- *relation_plus_pointwise* — every value, not just the identity value, must be integral.
- *relation_plus_target* — an orbit function of an integral element is integral.
- *relation_source_values* — source has the same value at two group elements.
- *relation_target_values* — target is the orbit function and starts at a.
- *relation_unit_value* — the unit selects identity evaluation.
- *relation_unit_source* — source restricted along the unit is the identity.
- *relation_unit_target* — target restricted along the unit is the identity.
- *relation_composition_value* — two edges multiply in traversal order.
- *relation_composition_units* — either identity edge can be deleted.
- *relation_composition_associative* — rebracketing three edges does not change the value.
- *presentation_tensor_left* — the left tensor insertion is constant.
- *presentation_tensor_right* — the right insertion evaluates the action.
- *presentation_tensor_plus* — the isomorphism preserves the entire plus ring.
- *test_spa_target_convention* — evaluation at `(g, x)` followed by the target map gives the
  pullback of `x` along the ring action of `g`, which is `g⁻¹ • x`: the ring action is pulled back
  contravariantly (it follows from `spa_relation_target`).
- *tower_continuousMap_perfectoid* — for `ofTower T`, every `C^0(G^k, Â_∞)` is perfectoid,
  `k = 0` included (it follows from `relation_isPerfectoid`).
- *action_inverse_three* — On the regular `ℤ/3` torsor, `galoisTensorMap` of `1 ⊗ δ₁` has value 1 at group coordinate 1 and the identity point, but value 0 at group coordinate 2.
- *relation_composition_noncommutative* — For a presentation with group identified with permutations of three letters, `relation.composition` of the function detecting `g(0) = 1` gives 1 on `((01),(12))` and 0 in the opposite order.

### Examples

The worked examples of this layer are the checks above: `Spa(C_p, O_{C_p})`, the perfectoid disc
and the empty space as `κ`-small and uniformly `κ`-small spaces, with the one-point space over a field
of cardinality `≥ κ`, the disjoint union of `κ` points and the `ω₁`-indexed disjoint unions over
cofinal fields `K_α` as the non-examples that separate the notions for singular `κ`; rational
subsets, the identity and the origin of the perfectoid disc as affinoid pro-étale maps, the open
unit disc as a pro-étale map that is not affinoid pro-étale, and `Spa(K) × {x} ↪ Spa(K) × S` as a
pro-étale map that is not open; `X × S` for finite `S`, for `S = {pt}`, `∅` and `Z_p`, against
the disjoint union of `#S` copies; and, in the subsections, `Pro(FinSet) ≃ Profinite ≃ Spa(C, O_C)_proét^aff`, the origin `Spa(K, K°) = lim_n {|T| ≤ |p|ⁿ}` as a Zariski closed immersion
that is affinoid pro-étale and not étale, and `R ↠ C⁰(ℤ_p, ℂ_p)` for `I = (T − 1)`.

`doubled_disc_overlap` uses the geometric pushout of two closed discs along the open unit disc.
The fibre product of the two chart inclusions is not affinoid, testing the affine-base hypothesis
in the fibre-product statement.

The Lean `example`s of `Suggested.lean` for this layer check that no perfectoid space is `0`-small
(smallness is a strict bound) and that `ProKappa 0 C` is empty.

### Dependencies

Layers P1 (P1.1, P1.27), P2 (P2.2, P2.3, P2.8–P2.11, P2.13, P2.15, P2.16), P3 (P3.7, P3.14–P3.16,
P3.18), P4 (P4.3, P4.7, P4.10, P4.12–P4.15, P4.18) and P5 (P5.3, P5.7, P5.12) of this roadmap;
Mathlib Ind categories, AdicSpacesPartII R0.19.5–R0.19.8 and P5.4’s perfectoid limit comparisons and P6.2’s completion bound. Other roadmaps,
by subsection: P6.2 uses AdicSpacesPartII R0.3 and R0.4 (*completed-tensor-product*,
*uniform-completed-tensor-product*), and P6.4 uses R0.4; P6.5 uses P5.13.4 (affinoid
étale objects at a finite stage), which uses P2.10, P3.17a.4, P3.20, P5.2, P5.7 and
P5.9–P5.12, together with AdicEtaleGeometry A1.0.5's completion comparison. P5.7 uses only
P6.1 and P6.2. P6.8 and P6.14 use AdicEtaleGeometry A1.10–A1.12 (the pro-étale site and
its profinite-set objects); P6.14 uses AdicSpacesPartII R0.20.1 for the comparison with
noetherian stages; P6.9–P6.12 use AdicSpacesPartII R0.19.9's generic Pro API.
P6.15–P6.17 use P1.32, P2, P6.8 and AdicEtaleGeometry A4.2–A4.5 and A4.7's
generic torsor-tower, completion, descent and orbit-spectrum constructions.
Mathlib `Cardinal`
(`IsStrongLimit`, `IsRegular`, `iSup_lt_of_lt_cof_ord`, `mk_set`, `mk_iUnion_le`, `power_mul`,
`power_le_power_right`), `CategoryTheory.Ind`, `CategoryTheory.IsCofiltered`,
`CategoryTheory.Functor.Initial`, `CategoryTheory.Functor.Elements`,
`CategoryTheory.EssentiallySmall`, `CategoryTheory.Limits.IsLimit`, `Profinite.asLimit`,
`DiscreteQuotient`, `FintypeCat.toProfinite`, `ContinuousMap`, `LocallyConstant`,
`UniformSpace.Completion`, `Subring.topologicalClosure`, `IsAlgClosed`,
`AlgebraicGeometry.Scheme.ProEt`; Tau Ceti `TauCeti.Huber.IsTateRing`,
`TauCeti.Huber.IsPseudoUniformizer`, `TauCeti.Huber.PairOfDefinition.completionLocObj`,
`TauCeti.Huber.powerBoundedSubring`, `TauCeti.ValuationSpectrum.spa`,
`TauCeti.ValuationSpectrum.ext'`, `TauCeti.ValuationSpectrum.rationalSubset`.

## Layer P7: tilde-limits and Frobenius-controlled towers

The layer builds tilde-limits `X ~ lim_i X_i` of analytic adic spaces and their perfectoid form,
the residue-field variant of Scholze's perfectoid paper, and the good affinoid perfectoid subsets
through which a tilde-limit is computed at a finite level; Frobenius-controlled integral towers
with their completion, the uniformity bound and the perfectoid Frobenius criterion that makes their
completed generic fibre a perfectoid tilde-limit; preperfectoid spaces and the strong completion;
cofinality witnesses for towers indexed by compact open subgroups; the uniqueness and
representability of perfectoid tilde-limits, their comparison with inverse limits of represented
functors (the comparison with diamonds belongs to DiamondsAndVStacks), their restriction to opens
and rational subsets, étale and finite étale base change and gluing; the limit of a tower of perfectoid spaces with affinoid transition maps; the
comparison of étale topoi along a tilde-limit of noetherian adic spaces; and the regular finite flat
towers of Česnavičius whose completion is perfectoid. AdicSpacesPartII R0.20.1 owns the intrinsic
tilde-limit relation; this layer adds the perfectoid specialisations.

### P7.1 tilde-limits of analytic adic spaces

**Tilde-limits of analytic adic spaces.** (*tilde-limit-of-analytic-adic-spaces*) Let `I` be a
small cofiltered category, `(X_i)_{i ∈ I}` a diagram of analytic adic spaces (the Foundations of
adic spaces roadmap (AdicSpaces), layer 5) whose transition maps are quasicompact and
quasiseparated, and `X` an analytic adic space with compatible morphisms `φ_i : X → X_i`. Two
separate conditions: (T) the continuous map `|X| → lim_i |X_i|` is a homeomorphism; (D) every
`x ∈ X` has an open affinoid neighbourhood `U = Spa(A, A⁺)` such that, writing the morphisms
`U → X_i` through open affinoid subspaces `Spa(A_i, A_i⁺) ⊆ X_i`, the union over `i` of the images
of `A_i → A` is dense in `A`. `X ~ lim_i X_i` (`X` is a tilde-limit of the `X_i`) means (T) and (D).
Neither `X` nor the `X_i` is required to be quasicompact or locally noetherian; the perfectoid
tilde-limits of P7.2 are the tilde-limits whose `X` is perfectoid. AdicSpacesPartII R0.20.1 owns the
intrinsic relation `AdicSpace.IsTildeLimit` with its affinoid criterion
(*tilde-limits-affinoid-criterion*: `AdicSpace.IsTildeLimit.affinoid` and the completed-direct-limit
criterion, the latter specified in AdicSpacesPartII R0.20.1's README only); P7.1 uses only this sub-target. The base change
`AdicSpace.IsTildeLimit.baseChange` (*tilde-limit-base-change*, AdicSpacesPartII R0.20.2) is used in P7.12.
P7 specializes the intrinsic relation to analytic Yoneda carriers.
Here `AdicSpace.IsTildeLimit` expresses its analytic Yoneda form (the
predicate (T) and (D) on `X`, the system and the cone); prove `IsTildeLimit.homeomorph` (the
homeomorphism `|X| ≃ lim_i |X_i|`) and `IsTildeLimit.dense_image` (the density clause (D) at every
point) ([SW13] §2.4, Definition 2.4.1, pp. 19–20, which does not require the spaces to be analytic).
The comparison with the arbitrary-open density condition of [Huber96] §2.4,
Definition 2.4.2, p. 125 is `PerfectoidSpace.IsTildeLimit.toHuber` in P7.2. *Needs:* AdicSpaces
layer 5; Mathlib `CategoryTheory.IsCofiltered`, `TopCat.limitCone`, `Dense`.

**Comparison with the intrinsic supplier.** `AdicSpace.IsTildeLimit.ofAdicSpace_iff`
states, for a small cofiltered diagram `D : I ⥤ TauCeti.AdicSpace` and cone `c`,

`AdicSpace.IsTildeLimit (D ⋙ ofAdicSpace) (ofAdicSpace.mapCone c)   ↔ AdicSpacesPartII.AdicSpace.IsTildeLimit D c.`

Its hypotheses are analyticity of `ofAdicSpace.obj c.pt` and of every
`ofAdicSpace.obj (D.obj i)`, and `AdicSpace.Qcqs (ofAdicSpace.map (D.map a))`
for every transition `a`. Honesty follows from the `TauCeti.AdicSpace` carriers.
Neither compactness of the levels nor a perfectoid apex is assumed. These
extra analytic and transition hypotheses belong to P7's Yoneda form; the
PartIV relation itself imposes only topology and density. Transport identifies
the point maps with the same cone projections and identifies sheafy affinoid
charts and their ring maps ([SW13] Definition 2.4.1, pp. 19–20; AdicSpacesPartII R0.20.1).

**Checks for the comparison.**

- `IsTildeLimit_ofAdicSpace_constant` — an analytic identity cone satisfies the
  Yoneda relation, including the nonperfectoid case `Spa(ℚ_p, ℤ_p)`.
- `IsTildeLimit_ofAdicSpace_empty` — a diagram of empty honest spaces with empty
  apex satisfies both relations.
- `IsTildeLimit_ofAdicSpace_nonanalytic` — the intrinsic relation holds for the
  identity cone on a nonanalytic honest space, whereas the Yoneda analytic form
  fails. Thus the analytic hypothesis in the equivalence is essential.

**Density on Yoneda charts.** (*yoneda-tilde-limit-dense-charts*)
`AdicSpace.HasDenseCharts` is the chart-density clause in the analytic Yoneda
comparison, using `AdicSpace.chartImage` and open affinoid factorizations
([SW13] Definition 2.4.1, pp. 19–20). The general adic-space predicate and
completed-affinoid criterion remain AdicSpacesPartII R0.20.1’s supplier contracts.

**Checks.**

- `tildeLimit_const` (degenerate) — `X ~ lim X` for the constant diagram.
- `tildeLimit_perfectoid_torus` (computation) — `Spa ℂ_p⟨T^{±1/p^∞}⟩ ~ lim_n Spa ℂ_p⟨T^{±1/p^n}⟩`,
  with (D) witnessed by the Laurent polynomials in `T^{1/p^n}` ([Sch13] Example 4.4, p. 23;
  [SW13] Proposition 2.4.2, p. 20). Downstream,
  [`PerfectoidTowers.Perfectoid.TorsorDescent.rootTorusTower_exists`](../PerfectoidTowers/README.md#layer-p9-continuous-torsor-descent-with-coefficients) contains this example as its
  specialization to `ℂ_p`, with a chosen compatible system of primitive roots of unity, its
  `ProetaleGaloisTower.tilde` field giving the global predicate and its coordinate equations the
  finite-level maps; PerfectoidTowers is not an input here.
- `tildeLimit_not_uncompleted` (non-example) — the uncompleted colimit `colim ℂ_p⟨T^{±1/p^n}⟩` is
  not complete, so it is not the ring `O(X)` of the tilde-limit (its `Spa`, as an adic space, is
  that of its completion, Tau Ceti `TauCeti.ValuationSpectrum.spaCompletionHomeomorph`); the
  completion is required for the ring, not for the space. The example states that the
  union of the finite-level function images in `perfectedTateAlgebra.laurent` is dense
  and is not a complete subset: the convergent series `∑_{n≥0} p^{n+1}T^{1/p^{n+1}}`
  lies outside it. The inherited topology is the common `p`-adic Tate topology
  ([Sch13] Example 4.4, p. 23; [SW13] Proposition 2.4.2, p. 20).
- `isTildeLimit_affinoid` (compatibility) — the affinoid predicate `IsTildeLimit_affinoid` on
  a filtered system of Huber pairs `(A_j, A_j⁺)` with a cone to `(R, R⁺)` (no perfectoid
  hypothesis on `R`) is (T) for `Spa(R, R⁺) → lim Spa(A_j, A_j⁺)` (an embedding with image the
  compatible families) together with (D) for the single affinoid `U = Spa(R, R⁺)` and the levels
  `V = Spa(A_j, A_j⁺)`: `⋃_j im(A_j → R)` dense in `R`; when the source and target pairs are complete and sheafy with Tate coordinate rings, it
  implies (T) and (D) above and is the
  form in which the checks of P7.2 are stated. The matching example applies
  `PerfectoidSpace.IsTildeLimit_affinoid.isTildeLimit` to this same cone.
- `isTildeLimit_affinoid_Qp` (computation) — the identity cone on `Spa(ℚ_p, ℤ_p)`
  is an analytic tilde-limit, without a perfectoid hypothesis.

- `tildeLimit_empty` (degenerate) — the identity cone on the zero affinoid has empty point space
  and is a global tilde-limit.
- `tildeLimit_density_needed_global` (non-example) — the global cone from `Spa(C_p)` to the
  constant `Spa(Q_p)` system fails section density, despite having the correct point space.

- `tildeLimit_test_malformedDiagram` (non-example) — on the one-object system of the
  dual-number pair over `ℂ_p`, setting the identity transition to `a + bε ↦ a` violates
  the diagram identity: `ε ↦ 0 ≠ ε`. The affinoid predicate rejects it.
- `tildeLimit_test_dualNumberTwoTerm` (convention) — the reduction homomorphism from the
  dual-number pair to `ℂ_p` satisfies `IsTildeLimit_affinoid` for the one-object system.
  In its source `(a + bε)² = a² + 2abε`; the same homomorphism sends this square to `a²`
  and kills the nonzero element `ε`, including when `p = 2`. This uses the affinoid comparison
  with AdicSpacesPartII R0.20.1, with the nonreduced behaviour of [SW13] Proposition 2.4.4, p. 20.

For a small filtered category `J`, a functor `B : J ⥤ CAff` and a cocone `b`,
`PerfectoidSpace.IsTildeLimit_affinoid.isTildeLimit` assumes that all `B_j` and `b.pt`
are sheafy with Tate coordinate rings (completeness and Hausdorffness are part of `CAff`).
The affinoid predicate is evaluated on the ring maps `B.map` and `b.ι.app`.
Its conclusion is `AdicSpace.IsTildeLimit (B.op ⋙ S) (S.mapCone b.op)`, where `S`
is the lift of `spaYFunctor` to `YonedaAdicSpace`: the apex is `ofSpaY b.pt`, and the
projection to `ofSpaY (B.obj j)` is induced by `b.ι.app j`. Thus (T) uses these precise
maps and (D) is witnessed by the whole affinoid chart. Maps between these affinoids
are qcqs; no separate transition hypothesis is needed. This is the analytic Yoneda
comparison with AdicSpacesPartII R0.20.1's `IsTildeLimit.affinoid`, not another intrinsic
criterion ([SW13] Definition 2.4.1, pp. 19–20).

The global carrier of this analytic tilde-limit is the Yoneda-adic category supplied by
AdicEtaleGeometry A1.16, into which perfectoid spaces embed through `YonedaAdicSpace.ofAdicSpace`.
`YonedaAdicSpace.IsAnalytic` means a cover by Tate affinoids ([Berkeley] Definition 4.3.2,
printed p. 33). Affinoid sheafiness is AdicEtaleGeometry's `R.Sheafy` for an object `R` of `CAff`, namely Tau
Ceti's `IsSheafyForEveryPresentation` on the plus ring. `AdicSpace.IsHonest` means that every point
lies in an open affinoid chart satisfying this predicate; the same predicate supplies the
finite-level hypothesis of `Perfectoid.FrobeniusTower.isTildeLimit` and the sheafiness
hypotheses of `GoodAffinoid.isUniformCompletion` and `IsPreperfectoid.closedSubspace`.
**Checks.** `IsHonest_empty` holds for an empty point space; `IsHonest_sheafy_affinoid`
holds for `ofSpaY R` given `R.Sheafy`; `IsHonest_perfectoid` holds for every
`PerfectoidSpace.Perfd p`, using its sheafy perfectoid charts (P2.6).
Both the source and the levels of `IsTildeLimit` are analytic and honest. `Qcqs` tests inverse
images of affinoid open charts for quasicompactness and quasiseparatedness. `chartImage` includes
all finite-level affinoid factorizations, with their ring maps. `HasDenseCharts` imposes
density on an open cover. The homeomorphism is fixed by
`IsTildeLimit.homeomorph_apply`: its coordinates are the cone maps. For initial reindexing,
analyticity, honesty and qcqs transitions of the original diagram remain hypotheses ([SW13] Definition 2.4.1, pp. 19–20).

### P7.2 perfectoid tilde-limits

The next three subsections fix the perfectoid form of the tilde-limit, the residue-field variant of
Scholze's perfectoid paper, and the good affinoid perfectoid subsets through which a tilde-limit is
computed at a finite level.

**Perfectoid tilde-limits `X ~ lim X_i` of analytic adic spaces.** (*perfectoid-tilde-limit*) Fix a
prime `p`. Let `I` be a small cofiltered category, `i ↦ X_i` a functor from `I` to analytic adic
spaces (AdicSpaces layer 5: every point analytic, sheafy) whose transition morphisms
`f_a : X_j → X_i` (`a : j → i`) are quasicompact and quasiseparated, `X` a perfectoid space in the
sense of P2.10 ([ECD] Definition 3.19: locally `Spa(R, R⁺)` with `R` a perfectoid Tate ring in the
sense of [ECD] Definition 3.1, of characteristic `0` or `p`, no base field; sheafy by P2.6), and
`φ = (φ_i : X → X_i)_i` a compatible family of morphisms of adic spaces (`f_a ∘ φ_j = φ_i`). The
pair `(X, φ)` is a perfectoid tilde-limit of `(X_i)`, written `X ~ lim_i X_i`, if it is a
tilde-limit in the sense of *tilde-limit-of-analytic-adic-spaces*, i.e. (a) the continuous map
`|X| → lim_i |X_i|` into the limit of the underlying topological spaces (Mathlib `TopCat.limitCone`)
is a homeomorphism, and (b) every `x ∈ X` has an open affinoid neighbourhood `U = Spa(A, A⁺) ⊆ X`
such that the union, over `i ∈ I` and over open subsets `V ⊆ X_i` with `φ_i(U) ⊆ V`, of the images
of the restriction maps `O_{X_i}(V) → A` is dense in `A`. Conventions: (i) the index category is
cofiltered and only the transition maps, not the `X_i`, are required to be quasicompact and
quasiseparated; (ii) no perfectoid base field is fixed: the `X_i` may be of characteristic `0` or
`p`, and `X` may lie over a different base than the `X_i` ([CHJ] Theorem 2.8: the `X_i` over `Q_p`,
`X` perfectoid; Scholze's universal cover of an affinoid over `Q_p` with limit over `C_p`); (iii)
the two conditions (a) and (b) are independent and both are required (the checks
`isTildeLimit_test_density_needed` and `isTildeLimit_test_topology_needed`); (iv) a tilde-limit is
not a categorical limit in adic spaces: `X ~ X_0` may hold with `X ≇ X_0` ([SW13] §2.4, after
Proposition 2.4.4, p. 20). Equivalent forms of (b), used throughout the layer: (b′) there is an
open cover of `X` by affinoids `U = Spa(A, A⁺)` for which `colim_{(i, V)} O_{X_i}(V) → A` has dense
image, `V` running over open affinoids of `X_i` through which `U → X_i` factors ([SW13] §2.4,
Definition 2.4.1, p. 19); (b″) the same with `U` affinoid perfectoid and `V` a quasicompact open at
one level (the good affinoid perfectoids of P7.4; [Survey] §2, Definition 2.20 and Remark 2.21,
p. 11). The equivalences (b) ⇔ (b′) ⇔ (b″) given (a) are *good-affinoid-perfectoid-basis*. Only the
density clause is formulated with sections; the weaker residue-field variant of [Sch12] Definition
7.14 is P7.3. In `PerfectoidSpace`, abbreviate `IsTildeLimit` by `AdicSpace.IsTildeLimit` on the underlying
adic spaces, `IsTildeLimit.homeomorph` (`|X| ≃ lim_i |X_i|` induced by `φ`) and
`IsTildeLimit.dense` (every `x ∈ X` has an affinoid open neighbourhood `U` with
`⋃_{i, V ⊇ φ_i(U)} im(O_{X_i}(V) → O_X(U))` dense in `O_X(U)`, with `U` sheafy);
prove `IsTildeLimit.toHuber`: for a perfectoid apex `X`, honest analytic levels `X_i`,
qcqs transition maps, and qcqs point spaces of `X` and every `X_i`, the perfectoid
predicate is equivalent to Huber's two conditions. The first is a homeomorphism
induced by the cone projections; the second uses sections on arbitrary opens, as
specified by `PerfectoidSpace.openChartImage` below. Huber's setting is qcqs
pseudo-adic spaces and adic morphisms ([Huber96] §2.4.1 and Definition 2.4.2, p. 125);
this comparison concerns honest analytic adic spaces, regarded as pseudo-adic
spaces with their full point spaces. Morphisms between these analytic spaces are
adic (AdicSpacesPartII R0.1, `Huber.IsAdicHom.of_isTateRing`). Prove also `IsTildeLimit.iff_goodCover` (`X ~ lim X_i`
iff (a) holds and `X` is covered by good affinoid perfectoids; *good-affinoid-perfectoid-basis*),
`IsTildeLimit.lift` (for a perfectoid space `Y`, compatible maps `ψ_i : Y → X_i` factor uniquely as
`φ_i ∘ g`; *tilde-limit-perfectoid-uniqueness*), `IsTildeLimit.uniqueIso` (two perfectoid
tilde-limits of the same system are uniquely isomorphic compatibly with the cones),
`IsTildeLimit.ofIso` (transport along an isomorphism `X ≅ X'` commuting with the cones),
`IsTildeLimit.open_baseChange` (restriction to the preimage of an open subset at a finite level;
*tilde-limit-rational-restriction*), `IsTildeLimit.reindex` (invariance under an initial change of
index category; *tilde-limit-cofinal-change*), `IsTildeLimit.const` (the constant system on a
perfectoid space with the identity cone is a tilde-limit) and `IsTildeLimit.spectralSpace` (if all
`X_i` are qcqs, `|X|` is a spectral space and each `φ_i` is a spectral map; [Sch12] §7, Remark 7.15,
p. 44; AdicSpacesPartII R0.19.5–R0.19.8 and P5.4’s perfectoid limit comparisons) ([SW13] §2.4, Definition 2.4.1, pp. 19–20). *Needs:*
*tilde-limit-of-analytic-adic-spaces*; P2.10; P2.6; Mathlib
`CategoryTheory.IsCofiltered`, `TopCat.limitCone`.

**Checks.**

- `isTildeLimit_test_perfectoidDisc` (computation) — for an integral perfectoid ring `S`,
  a regular parameter `π`, `π`-adic completeness and `π^p ∣ p`, the generic cone of
  `FrobeniusTower.disc S π hπ hp` satisfies the global `PerfectoidSpace.IsTildeLimit`.
  Its stages are the completed disc algebras with transition `T ↦ T^p`.
- `isTildeLimit_test_perfectoidTorus` (computation) — for a complete nonarchimedean
  perfectoid normed field `K` of characteristic zero, topologically nilpotent `p`,
  `PseudoUniformizerRoots p K`, and compatible primitive `p^n`-th roots of unity,
  the torus tilde-limit is `K⟨T^{±1/p^∞}⟩` with transitions `T ↦ T^p`.
  The example uses `FrobeniusTower`: each integral stage is the completion of
  `K°[T, T⁻¹]` along `ϖ`, the transitions fix coefficients and raise `T` to its
  `p`-th power, and the completed generic cone satisfies `PerfectoidSpace.IsTildeLimit`.
  The projection pulls the level-`n` coordinate back to `T^{1/p^n}`.
  This existence statement holds without choosing roots of unity; with compatible
  primitive roots, [`PerfectoidTowers.Perfectoid.TorsorDescent.rootTorusTower_exists`](../PerfectoidTowers/README.md#layer-p9-continuous-torsor-descent-with-coefficients)
  equips it with the Galois action downstream (not an input here) ([Sch13] Example 4.4, p. 23,
  and Lemma 4.10(iii–iv), pp. 26–27).
- `isTildeLimit_test_empty` (degenerate) — the identity cone on a zero perfectoid
  pair is a perfectoid tilde-limit. Together with `isTildeLimit_test_const` and the global
  `isTildeLimit_test_density_needed`, this tests the global predicate itself.
- `isTildeLimit_test_const` (degenerate) — for a perfectoid space `X` and the constant system
  `X_i = X` (`I` the one-object category) with `φ = id`, `X ~ lim X_i`; more generally any
  isomorphism `X ≅ X_0` of adic spaces with `X` perfectoid exhibits `X ~ X_0`.
- `isTildeLimit_test_not_categorical` (non-example) — set `K = C_p`. The quotient
  `φ : K[ε]/(ε²) → K` satisfies the single-affinoid predicate `IsTildeLimit_affinoid`
  for the one-object diagram, while `φ` is not injective. Thus this affinoid
  tilde-limit condition does not force the cone map to be an isomorphism.
  The pair on `DualNumber C_p` has plus ring `O_{C_p} + C_p ε`, and `φ(a + bε) = a`.
  Its product topology has ring of definition `O_{C_p} ⊕ O_{C_p} ε`.
  Every valuation kills `ε`, so the map on spectra is a homeomorphism of one-point
  spaces; the coefficient map is surjective and kills the nonzero element `ε`.
  This concretely illustrates the distinction following [SW13] Proposition 2.4.4, p. 20.
- `isTildeLimit_test_density_needed` (non-example) — the point `Spa(C_p, O_{C_p})` with its map to
  the constant system `Spa(Q_p, Z_p)` satisfies (a) (both spaces are one point) but not (b) (`Q_p`
  is not dense in `C_p`); (a) alone does not define tilde-limits.
- `isTildeLimit_test_topology_needed` (non-example) — for a nonzero perfectoid Tate ring `R` and
  `X = Spa(R⟨T^{1/p^∞}⟩)`, mapping `X` into the first summand of `X_n := D_n ⊔ D_n`
  (`D_n = Spa(R⟨T^{1/p^n}⟩)`) satisfies (b) but not (a) (`|X| → lim |X_n|` misses the second
  summand); (b) alone does not define tilde-limits. The example uses `FrobeniusTower.disc`
  for an integral perfectoid ring `S`, a regular nonunit `ϖ`, `ϖ`-adic completeness and
  `ϖ^p ∣ p`. Natural first and second inclusions, with a binary coproduct at every level,
  specify the doubled diagram. The first-summand cone has `AdicSpace.HasDenseCharts` and
  an embedding on points, but misses a compatible family in the second summands; it
  therefore fails the global `PerfectoidSpace.IsTildeLimit` predicate. The nonunit
  hypothesis excludes the empty generic fibre ([SW13] Definition 2.4.1, pp. 19–20).
- `isTildeLimit_test_huber_compat` (compatibility) — for a perfectoid `X`, honest analytic
  `X_i`, qcqs transitions, and qcqs `X` and `X_i`, `IsTildeLimit.toHuber` identifies the
  perfectoid predicate with the projection-compatible homeomorphism and arbitrary-open
  section density of [Huber96] Definition 2.4.2, p. 125.

`PerfectoidSpace.openChartImage D c R u` is the subset of `R.A` obtained from
sections on all finite-level opens: choose `i`, an open immersion `v : V → D_i`,
and a factorization `f : Spa^Y(R) → V` with `f ≫ v = u ≫ φ_i`. A coefficient `r`
belongs to the image exactly when some `s : globalFunctions V` pulls back to
`toGlobalFunctions R r`. For sheafy `R`, this is the image in `O_X(U) = R.A`;
`V` need not be affinoid or quasicompact. These are the sections in [Huber96]
Definition 2.4.2(b), p. 125, using AdicEtaleGeometry A1.16's `globalFunctions`,
`globalFunctions.map`, `toGlobalFunctions`, and `IsOpenImmersion`.
`IsTildeLimit.dense` gives this density without requiring qcqs total spaces.

**Checks for `openChartImage`.**

- `openChartImage_constant` — the identity cone on `Spa^Y(R)` has image all of `R.A`,
  since the whole finite-level open and its identity restriction supply every coefficient.
- `openChartImage_zero` — for a zero coordinate ring and any chart map into a cone
  over a cofiltered category, the image is `{0}`; the index category is nonempty.
- `openChartImage_nondense` — for the constant `Spa(Q_p, Z_p)` system with apex
  `Spa(C_p, O_{C_p})`, even sections on arbitrary opens have nondense image in `C_p`.
  The only nonempty open of the finite-level point is the whole point.

The predicate `PerfectoidSpace.IsTildeLimit p D c` records that `c.pt` is perfectoid and
that `AdicSpace.IsTildeLimit D c` holds. Thus its topology and density are exactly those of P7.1.
`IsTildeLimit.lift` quantifies over every perfectoid source and every compatible cone, and
`uniqueIso` requires compatibility with every projection ([SW13] Proposition 2.4.5, pp. 20–21).

### P7.3 residue-field tilde-limits

**Residue-field tilde-limits `X ~′ lim X_i` over locally noetherian adic spaces.**
(*residue-field-tilde-limit*) Let `I` be a small cofiltered category, `(X_i)` an `I`-indexed system
of quasicompact quasiseparated locally noetherian analytic adic spaces (AdicSpacesPartII R0.6:
covered by `Spa(A, A⁺)` with `A` a complete strongly noetherian Tate ring; the transition maps are
then automatically qcqs), `X` a perfectoid space and `φ = (φ_i : X → X_i)` a compatible family.
Then `X ~′ lim X_i` (Scholze's notation `X ~ lim X_i` of Definition 7.14, written `~′` in [SW13]
after Theorem 2.4.7) means: (a) `|X| → lim_i |X_i|` is a homeomorphism, and (c) for every `x ∈ X`
with images `x_i = φ_i(x)`, the map of completed residue fields `colim_i k(x_i)^ → k(x)^` has dense
image, where `k(y)^` is the completion of the residue field of `O_{Y,y}` for the valuation of `y`
(equivalently, `colim_i k(x_i) → k(x)` has dense image for the valuation topologies). No perfectoid
base field is fixed ([Sch12] works over a perfectoid field `K`; the definition and the results of
P7.15 use only that `X` is perfectoid and the `X_i` are qcqs locally noetherian); by Remark 7.15,
(a) forces `|X|` to be spectral, in particular `X` quasicompact and quasiseparated. The density
condition is pointwise on residue fields; the sectionwise condition of P7.2 implies it
(see *tilde-limit-implies-residue-field-tilde-limit*). In `PerfectoidSpace`, define
`IsResidueFieldTildeLimit` (the structure with fields `homeomorph` and `residueField_dense`),
`IsResidueFieldTildeLimit.homeomorph` (`|X| ≃ lim |X_i|`) and
`IsResidueFieldTildeLimit.residueField_dense` (for every `x`, `colim_i k(x_i)^ → k(x)^` has dense
image); prove `IsResidueFieldTildeLimit.spectralSpace` (`|X|` is spectral and `X` is qcqs; Remark
7.15), `IsResidueFieldTildeLimit.of_isTildeLimit` (every tilde-limit over qcqs locally noetherian
`X_i` is a residue-field tilde-limit), `IsResidueFieldTildeLimit.of_bijective` (for `X`
quasicompact and quasiseparated, bijectivity of `|X| → lim |X_i|` together with (c) already gives
(a); *residue-field-tilde-limit-etale-base-change*). The map into the compatible families
is defined by the cone legs, with compatibility given by cone naturality. Also prove
`IsResidueFieldTildeLimit.etale_baseChange` (for `Y → X_i` étale,
`Y ×_{X_i} X ~′ lim_j Y ×_{X_i} X_j`; Proposition 7.16) and `IsResidueFieldTildeLimit.reindex`
(invariance under an initial change of index category) ([Sch12] §7, Definition 7.14, pp. 43–44;
[Sch12] §7, Remark 7.15, p. 44; [SW13] §2.4, after Theorem 2.4.7, p. 21). *Needs:* AdicSpacesPartII
R0.6; P2.10; P2.5; AdicSpacesPartII R0.19.5–R0.19.8 and P5.4’s perfectoid limit comparisons; Mathlib `SpectralSpace`, `IsSpectralMap`.

**Checks.**

- `isResidueFieldTildeLimit_test_disc` (computation) — for a perfectoid field `K`,
  `Spa(K⟨T^{1/p^∞}⟩, K°⟨T^{1/p^∞}⟩) ~′ lim_n Spa(K⟨T^{1/p^n}⟩, K°⟨T^{1/p^n}⟩)`: the underlying
  spaces agree by the tilde-limit of the disc tower, and at every point the colimit of the residue
  fields is dense.
- `isResidueFieldTildeLimit_test_point` (degenerate) — for a perfectoid field `K`, the constant
  system `Spa(K, K°)` (noetherian, as `K` is a field) with the identity satisfies
  `Spa(K, K°) ~′ lim Spa(K, K°)`.
- `isResidueFieldTildeLimit_test_not_dense` (non-example) — `Spa(C_p, O_{C_p}) → Spa(Q_p, Z_p)`
  (constant system) satisfies (a) but not (c): `Q_p` is not dense in `C_p`.
- `isResidueFieldTildeLimit_test_of_tildeLimit` (compatibility) — if `X ~ lim X_i` in the sense of
  P7.2 and all `X_i` are qcqs locally noetherian, then `X ~′ lim X_i`.

- `isResidueFieldTildeLimit_test_empty` (degenerate): the zero affinoid's constant
  system has no residue-field conditions. The point, empty, closed-field-extension non-example
  and section-density implication are examples of `IsResidueFieldTildeLimit`.

An affinoid description of the pointwise condition uses
`AdicSpace.CompletedResidue R x = Completion (WithVal (residueFieldValuation x))`.
`residueMap` is the canonical evaluation from `R`; `residueImage` consists of fractions of
functions from a common finite-level affinoid chart, with nonzero denominator at the point and a
commuting square into the cone. The source chart may shrink around the point for each fraction.
`residueRestriction` is the continuous residue-field isomorphism for that open immersion,
fixed on evaluated functions by `residueRestriction_map`. It transports the fraction into the
fixed completed residue field. `affinoidPoints_naturality` requires the affinoid point
identifications to commute with the pinned maps of valuation spectra. Around each point of the source, `IsResidueFieldTildeLimit`
requires such fractions to be dense. The chart point is identified with the given global point
through the supplied spectrum homeomorphism. This expresses the residue condition through
valuation fields ([Sch12] Definition 7.14, pp. 43–44).

### P7.4 good affinoid perfectoid subsets

**Good affinoid perfectoid subsets of a tilde-limit.** (*good-affinoid-perfectoid*) Let
`(X_i)_{i∈I}` be as in P7.2, `X` a perfectoid space with a compatible cone `φ`. A good affinoid
perfectoid of `(X, φ)` is a triple `(W, i_0, V)` where `W = Spa(R, R⁺) ⊆ X` is an affinoid
perfectoid open subset (P2.9), `i_0 ∈ I` and `V ⊆ X_{i_0}` is a quasicompact open subset such that
(i) `W = φ_{i_0}^{-1}(V)`, and (ii) writing `V_a := f_a^{-1}(V) ⊆ X_j` for `a : j → i_0` (the slice
category `I/i_0`, cofiltered), the ring map `colim_{a ∈ (I/i_0)^op} O_{X_j}(V_a) → R` has dense
image, for the topology of `R` (no topology is placed on the colimit). When moreover the `V_a` are
affinoid, `V_a = Spa(A_a, A_a⁺)`, the pair `(R, R⁺)` is determined by the system
(see *good-affinoid-uniform-completion*). This is the form of Scholze's survey (the paragraph before
Proposition 2.22: dense image and homeomorphism with the limit of the finite-level affinoids), of
[Torsion] Theorem III.1.2(iii) and Definition III.3.5(i) (`V` the preimage of an affinoid `V_m` at
finite level, `lim H⁰(V_m, O) → H⁰(V, O)` with dense image), and of [CHJ] Theorem 2.8 (standard
affinoids pulled back from level `K(p^N)`). Under `X ~ lim X_i` the homeomorphism
`|W| ≅ lim_a |V_a|` is automatic, since `W` is the preimage of `V`. In `PerfectoidSpace`, define
`GoodAffinoid` (the data `(W, i_0, V)` with `W = φ_{i_0}^{-1}(V)` and dense image of
`colim_a O(V_a)` in `O(W)`) and `GoodAffinoid.level` (the level `i_0` and the finite-level
quasicompact open `V ⊆ X_{i_0}`); prove `GoodAffinoid.preimage_eq` (`W = φ_{i_0}^{-1}(V)`),
`GoodAffinoid.denseRange` (`colim_a O_{X_j}(V_a) → O_X(W)` has dense range),
`GoodAffinoid.homeomorph` (under `X ~ lim X_i`, `|W| ≃ lim_a |V_a|`), `GoodAffinoid.rationalSubset`
(a rational subset of `W` is good at a possibly larger level; *good-affinoid-perfectoid-basis*),
`GoodAffinoid.raise` (`(W, i_0, V)` is good at every level `j → i_0` with the preimage `V_a`),
`GoodAffinoid.isUniformCompletion` (if all `V_a` are affinoid, `O_X(W)` is the uniform completion
of `colim_a O(V_a)`; *good-affinoid-uniform-completion*) and `GoodAffinoid.finiteEtaleEquiv` (if
all `V_a` are affinoid, `FÉt(O_X(W)) ≃ 2-colim_a FÉt(O(V_a))`) ([Survey] §2, after Remark 2.21,
p. 11; [Survey] §2, Definition 2.20, p. 11; [Torsion] §III.1, Theorem III.1.2 (i)–(iii), p. 28;
[CHJ] §2, Theorem 2.8, p. 11). *Needs:* P2.9; Mathlib `CategoryTheory.IsCofiltered`.

**Checks.**

- `goodAffinoid_test_disc` (computation) — in the perfectoid disc tower over a perfectoid Tate
  ring `R`, `W = {|T| ≤ |ϖ|} ⊆ Spa(R⟨T^{1/p^∞}⟩)` is good at level `0` with `V = {|T| ≤ |ϖ|}`: its
  coordinate ring is the rational localisation `R⟨T^{1/p^∞}⟩⟨T/ϖ⟩`, and the image of
  `colim_n R⟨T^{1/p^n}⟩⟨T/ϖ⟩` contains the image of `R[T^{1/p^∞}, T/ϖ]`, which is dense.
- `goodAffinoid_test_empty` (degenerate) — the empty subset, with `V = ∅` at any level, is a good
  affinoid perfectoid (`R = 0`).
- `goodAffinoid_test_not_dense` (non-example) — for `Spa(C_p)` over the constant system
  `Spa(Q_p)`, `W = Spa(C_p)` is the preimage of `V = Spa(Q_p)` but `Q_p` is not dense in `C_p`, so
  `W` is not good.
- `goodAffinoid_test_ecd_limit` (compatibility) — if all `X_i = Spa(R_i, R_i⁺)` are affinoid
  perfectoid, the limit `Spa(R, R⁺)` of [ECD] Proposition 6.5 (`R⁺` the `ϖ`-adic completion of
  `colim R_i⁺`) is good at every level with `V = X_{i_0}`.

- `goodAffinoid_test_const` (computation): the whole constant perfectoid affinoid
  is good at the sole level and its chart has full image. The empty and nondense Checks likewise
  quantify over the full `GoodAffinoid` record, including its finite open.

The affinoid pieces of a tilde-limit are computed by their finite-level coordinate rings
(*good-affinoid-uniform-completion*): let `X ~ lim X_i` be a perfectoid tilde-limit and
`(W, i_0, V)` a good affinoid perfectoid (see *good-affinoid-perfectoid*), `W = Spa(R, R⁺)`, such that
every `V_a` (`a : j → i_0`) is affinoid, `V_a = Spa(A_a, A_a⁺)` (for instance when the transition
maps are affinoid), and let `(C, C⁺)` be the completed direct limit of the `(A_a, A_a⁺)`
(*tilde-limit-from-completed-colimit*, Tate case with a pseudouniformiser `ϖ ∈ A_{i_0}` that is
also one of `R`). Then (i) the canonical map `(C, C⁺) → (R, R⁺)` identifies `(R, R⁺)` with the
uniformization of `(C, C⁺)` ([KL15] §2.8, Definition 2.8.13, p. 63; AdicSpacesPartII R0.4),
`R⁺` being the integral closure of the closure of the image of `C⁺`, `Spa` being unchanged by
uniformization; in particular `R` (perfectoid, hence uniform, P1.1) is determined by the system
`(V_a)`; (ii) `GoodAffinoid.finiteEtaleEquiv` composes algebraic finite-stage descent
(AdicEtaleGeometry A1.0.3), AdicEtaleGeometry A1.0.5's `finiteEtale_completedColimit` for `L = colim_a A_a → C = L̂`, and
AdicEtaleGeometry A4.4's
`Huber.Pair.uniformization.baseChange_finiteEtale_equivalence` for `C → Cᵘ`,
then uses (i) to identify `Cᵘ` with `R`. This gives the scalar-extension equivalence
`2-colim_a FÉt(A_a) → FÉt(R)`, also for every rational localisation of `W`
pulled back from a finite level ([KL15] §2.8, Proposition 2.8.16(a), p. 64;
[Berkeley] Lecture 7, Theorem 7.4.8 and its following corollary, printed p. 54;
[SW13] §2.4, proof of Proposition 2.4.5, p. 21 (the form of (i) inside the uniqueness proof)).
*Needs:* *good-affinoid-perfectoid*; *tilde-limit-from-completed-colimit*; AdicSpacesPartII R0.4;
P1.1; AdicEtaleGeometry A1.0.5; AdicEtaleGeometry A1.16 (the Yoneda carrier), A1.0.3 and A4.4; Tau Ceti
`TauCeti.ValuationSpectrum.mem_of_forall_vle_one`.

`GoodAffinoid` records the perfectoid pair of the chart, its open immersion, the finite level,
the quasicompact open immersion there, the equality of inverse images and density. Sections on
later inverse images are morphisms to the supplied affine line; equality after restriction to
an affinoid chart is equality of these morphisms. Consequently the density field also covers
nonaffinoid finite-level opens. `IsTildeLimit.iff_goodCover` assumes the topology of the limit,
analytic honest levels and qcqs transitions, and characterizes density by coverage with this
data ([Survey] Proposition 2.22, p. 11).

`affinoidFiniteEtaleDiagram` uses the finite étale categories of the specified
finite-level affinoids and algebraic scalar extension on arrows. Its Checks
`affinoidFiniteEtaleDiagram_identity`, `affinoidFiniteEtaleDiagram_zero` and
`affinoidFiniteEtaleDiagram_two` compute the identity transition, the zero algebra
and the split rank-two algebra. `GoodAffinoid.isUniformCompletion` gives the
universal mapping property against complete uniform Tate pairs;
`GoodAffinoid.finiteEtaleEquiv` identifies its inverse functor on every finite-stage
object with this scalar extension.

**Topology of a good chart.** (*good-affinoid-homeomorph*)
`GoodAffinoid.homeomorph` restricts the tilde-limit homeomorphism to the inverse
image of its finite-level open. Its coordinate formula is the given chart followed
by each cone projection ([SW13] Definition 2.4.1, pp. 19–20).

**Raising a good chart.** (*good-affinoid-raise*) `GoodAffinoid.raise` takes a
good chart and a qcqs transition `j → W.level`. `GoodAffinoid.raise_spec` fixes
the new level as `j`, preserves the chart image, and identifies the new finite-level
open with the inverse image of the old one ([Survey] proof of Proposition 2.22, p. 11).

**Finite étale algebras on a good chart.** (*good-affinoid-finite-etale-equivalence*)
`GoodAffinoid.finiteEtaleEquiv` uses a perfectoid tilde-limit, a good chart, and
a compatible sheafy Tate affinoid presentation of the finite-level inverse images.
Its inverse is scalar extension along the specified maps into the good chart’s ring;
`affinoidFiniteEtaleDiagram` supplies the diagram. It composes the
algebraic colimit equivalence of AdicEtaleGeometry A1.0.3, A1.0.5's
`finiteEtale_completedColimit`, the imported
`Huber.Pair.uniformization.baseChange_finiteEtale_equivalence`, and the ring
identification from `GoodAffinoid.isUniformCompletion`. The two completion steps
use [Berkeley] Theorem 7.4.8 and its corollary, printed p. 54, and [KL15]
Proposition 2.8.16(a), p. 64, respectively. The colimit may be nonseparated;
no injectivity or finite étaleness of the transition maps is required.

### P7.5 Frobenius-controlled integral towers

The next two subsections define the integral towers whose transition maps are eventually Frobenius
modulo `ϖ^p` and compute their completed generic fibre; that this generic fibre is a perfectoid
tilde-limit is P7.13.

**Frobenius-controlled integral towers.** (*frobenius-controlled-integral-tower*) Fix a prime `p`.
A Frobenius-controlled integral tower `(A_•⁺, ϖ, m_0, w)` consists of: (1) a sequence of rings
`A_0⁺ → A_1⁺ → A_2⁺ → ⋯` with transition maps `t_m : A_m⁺ → A_{m+1}⁺`; (2) an element `ϖ ∈ A_0⁺`
(its images again written `ϖ`), the compatible pseudouniformiser, such that every `A_m⁺` is
`ϖ`-torsion-free and `ϖ`-adically complete (Mathlib `IsAdicComplete (Ideal.span {ϖ}) A_m⁺`) and
`p ∈ ϖ^p A_0⁺` (the condition `ϖ^p | p` of [ECD] Definition 3.1, placed on the integral model);
then every `A_m⁺/ϖ^p` is an `F_p`-algebra and absolute Frobenius induces a ring homomorphism
`Φ_m : A_m⁺/ϖ → A_m⁺/ϖ^p`, `x ↦ x^p` (well defined because
`(x + ϖy)^p ≡ x^p mod pϖA_m⁺ + ϖ^pA_m⁺ = ϖ^pA_m⁺`); (3) the eventual Frobenius lifting condition
modulo `ϖ^p`: an index `m_0` and, for every `m ≥ m_0`, a ring isomorphism
`w_m : A_{m+1}⁺/ϖ ≅ A_m⁺/ϖ^p` such that (i) `t_m ∘ w_m = Φ_{m+1}` as maps
`A_{m+1}⁺/ϖ → A_{m+1}⁺/ϖ^p` (`t_m` reduced modulo `ϖ^p`), and (ii) the `w_m` are compatible with
the transition maps: `w_{m+1} ∘ t_{m+1} = t_m ∘ w_m` as maps `A_{m+1}⁺/ϖ → A_{m+1}⁺/ϖ^p` (`t`
reduced modulo `ϖ` on the left and modulo `ϖ^p` on the right). The `w_m` are data, exist only from
`m_0` on, and characteristic `p` is allowed (`p = 0 ∈ A_m⁺`, as for relative Frobenius towers over
a perfect base). The generic fibres are the complete Tate rings `A_m := A_m⁺[1/ϖ]` with ring of
definition `A_m⁺` (`ϖ`-adic topology) and the Huber pairs `(A_m, A_m⁺⁺)` with `A_m⁺⁺` the integral
closure of `A_m⁺` in `A_m`; `X_m := Spa(A_m, A_m⁺⁺)`. The finite levels `A_m` are not assumed
perfectoid (in the sources they are strongly noetherian). Source form: if all `A_m⁺` are flat
algebras over an integral perfectoid ring `O` with `ϖ ∈ O` (`O = Z_p^cycl` or `O_K` for a
perfectoid field `K`) and `t_m` modulo `ϖ^p` is identified, `O`-linearly, with the relative
Frobenius of `A_{m+1}⁺/ϖ^p` over `O/ϖ^p`, then `w_m` is the composite of the canonical isomorphism
`A_{m+1}⁺/ϖ ⊗_{O/ϖ, Φ_O} O/ϖ^p ≅ (A_{m+1}⁺/ϖ^p)^{(p)} ≅ A_m⁺/ϖ^p` with
`A_{m+1}⁺/ϖ ≅ A_{m+1}⁺/ϖ ⊗_{O/ϖ, Φ_O} O/ϖ^p` (`ϖ` is required to be a non-zero-divisor, `ϖ^p | p`, and `O` is
`ϖ`-adically complete; then `Φ_O : O/ϖ ≅ O/ϖ^p` is bijective by [BMS] Lemma 3.10(i)); in [Torsion] `ϖ = p^{(1−ε)/p}` and `ϖ^p = p^{1−ε}`.
Every construction below takes a chosen regular parameter as an explicit input. Regularity is not a
consequence of integral perfectoidness (`𝔽_p` is integral perfectoid with `π = 0`), and the
regularity required at every stage is not weakened: P7.6 uses it for the `ϖ`-torsion-freeness of
the completion. In `Perfectoid`, define
`IntegralTower` (the rings `A_m⁺`, the transition maps `t_m` and a parameter `ϖ` marked at every
stage and preserved by the transitions, regular (`ϖ`-torsion-freeness), `ϖ`-adically complete and
with `ϖ^p ∣ p` at every stage, which gives `p ∈ ϖ^pA_0⁺`), with `IntegralTower.quotient` (the stage
`A_m⁺/ϖ^k`), `IntegralTower.quotientMap` (the transition modulo `ϖ^k`) and `IntegralTower.diagram`
(the direct system, built with Mathlib's `Functor.ofSequence`); `FrobeniusTower` (an
`IntegralTower` with the index `m_0` and the isomorphisms `w_m` satisfying the lifting conditions
(i), (ii)), `FrobeniusTower.frobMod`
(`Φ_m : A_m⁺/ϖ →+* A_m⁺/ϖ^p`, built from Mathlib `frobenius` on the `F_p`-algebra `A_m⁺/ϖ^p`),
`FrobeniusTower.genericFibre` (the Huber pairs `(A_m, A_m⁺⁺)` and `X_m`), `FrobeniusTower.Hom`
(compatible ring maps `A_m⁺ → B_m⁺` sending `ϖ` to `ϖ` and intertwining the `w_m`; identity and
composition, with a common eventual starting index), `FrobeniusTower.ofRelativeFrobenius` (the tower from flat `ϖ`-adically complete
algebras over an integral perfectoid `O` with a non-zero-divisor `ϖ`, `ϖ^p | p`,
`O` `ϖ`-adically complete, whose transition maps reduce modulo `ϖ^p` to relative
Frobenius over `O/ϖ^p`), `IntegralTower.const` (the constant tower on a ring `A` with a chosen
regular parameter `π` as explicit input, `A` being `π`-adically complete with `π^p ∣ p`) and
`FrobeniusTower.constPerfectoid` (its Frobenius control with `w = Φ`, which exists exactly when
`A/π → A/π^p` is bijective, for instance when `A` is integral perfectoid for this `π` (P1.24), as
for `R°` with a pseudouniformiser of a perfectoid Tate ring) and `FrobeniusTower.perfection` (the relative
Frobenius tower of a flat `t`-adic algebra over a perfect base of characteristic `p`;
*perfection-tilde-limit*); the lifting conditions are the fields `FrobeniusTower.transition_comp_lift`
(`t_m ∘ w_m = Φ_{m+1}` for `m ≥ m_0`) and `FrobeniusTower.lift_comp_transition`
(`w_{m+1} ∘ t_{m+1} = t_m ∘ w_m` for `m ≥ m_0`); prove `FrobeniusTower.shift` (dropping the first `k` terms, lowering `m_0`, gives a Frobenius-controlled
tower with the same completion) and that
`Φ : colim_m A_m⁺/ϖ → colim_m A_m⁺/ϖ^p` is bijective (*frobenius-tower-uniformity-bound*)
([Torsion] §III.2.3, proof of Corollary III.2.19, p. 42 (transition maps equal to relative
Frobenius modulo `p^{1−ε} = ϖ^p`); [Torsion] §III.2.3, proof of Corollary III.2.19, pp. 42–43 (the
colimit Frobenius isomorphism `R_∞/ϖ ≅ R_∞/ϖ^p`); [Torsion] §III.2.3, proof of Corollary III.2.19,
p. 43 (a tower of flat formal schemes over `Z_p^cycl`); [BHW] §5.2, proof of Theorem 5.11, p. 22 of the arXiv manuscript
(relative Frobenius modulo `p^{1−δ}`)). *Needs:* Mathlib `IsAdicComplete`, `Ideal.span`,
`frobenius`, `integralClosure`; Tau Ceti `TauCeti.Huber.IsTateRing`,
`TauCeti.Huber.IsPseudoUniformizer`; P1.1; P1.3.

**Checks.**

- Computation: for a perfectoid Tate pair `(R, R⁺)` and `ϖ ∈ R⁺` with
  compatible `p`-power roots and `ϖ^p | p` in `R⁺`, `A_m⁺ = R⁺⟨T^{1/p^m}⟩` with inclusions `t_m` and
  `w_m : (R⁺/ϖ)[T^{1/p^{m+1}}] → (R⁺/ϖ^p)[T^{1/p^m}]`, `r T^{a/p^{m+1}} ↦ Φ(r) T^{a/p^m}`, is
  Frobenius-controlled with `m_0 = 0`; likewise the torus `R⁺⟨T^{±1/p^m}⟩`.
- Degenerate: for an integral perfectoid ring `S` that is
  `ϖ`-torsion-free and `ϖ`-adically complete with `ϖ^p | p`, the constant tower `A_m⁺ = S`,
  `t_m = id`, with `w_m = Φ : S/ϖ ≅ S/ϖ^p` (bijective by [BMS] Lemma 3.10) is Frobenius-controlled.
- Non-example: for a perfectoid field `K` with
  pseudouniformiser `ϖ`, the constant tower `A_m⁺ = K°⟨T⟩` with identity transition maps is not
  Frobenius-controlled for any `m_0`: `t_m ∘ w_m = Φ` would make `T` a `p`-th power modulo `ϖ^p` in
  `K°⟨T⟩`, which it is not. The Check places the variable class outside the image of
  `(disc S ϖ).frobMod n` for every stage, with the disc hypotheses below and `ϖ` a nonunit.
  Reducing further modulo `ϖ` identifies the obstruction with the exponent of `T` in
  `(S/ϖ)[T]`.
- Compatibility: the tower `A_m⁺ = K°⟨T⟩` with transition maps
  `T ↦ T^p` is isomorphic, as a direct system, to the compatible-root disc tower `K°⟨T^{1/p^m}⟩`
  (via `T ↦ T^{1/p^m}` at level `m`), hence Frobenius-controlled; the definition is invariant under
  isomorphism of direct systems. In the integral presentation `disc S ϖ`, the example
  asserts that each `toCompleted n` is injective, that successive inclusions commute
  with `transition n`, and that `discCompletionEquiv` sends `aT^k` at stage `n` to
  `aT^{k/p^n}`. Thus each stage is isomorphic to its finite-root image, with
  coefficients fixed and the transition direction specified. It uses the regularity,
  integral-perfectoid and completeness hypotheses of `disc`
  ([SW13] discussion before Definition 2.3.4, p. 17; [Torsion] proof of
  Corollary III.2.19, pp. 42–43).
- Characterisation: for a perfect complete nonarchimedean
  field `k` of characteristic `p`, a pseudouniformiser `t`, and a flat `t`-adically complete
  `k°`-algebra `B`, use the base twists `B_m = B ⊗_{k°, F^{-m}} k°` and the transitions
  `b ⊗ c ↦ b^p ⊗ c`. These preserve `t` from the right factor and form the tower of P7.13.

- Convention: for `p = 2`, `(disc S ϖ).frobMod n`
  sends the class of `a+bT` modulo `ϖ` to `a²+b²T²` modulo `ϖ²`.
  This uses the integral-perfectoid, completeness and parameter hypotheses of `disc` below.
- Convention: over `𝔽₈`, choose
  `a³+a+1=0`, so `a⁸=a` and `a²≠a⁴`. On `perfectionStage` for `𝔽₈[T]`,
  `perfectionTransition` sends `(a+T)⊗a` to `(a²+T²)⊗a`, keeping the right factor fixed;
  `inverseFrobeniusIterate` sends `a` to `a⁴` at stage one. Thus the tensor twist is
  distinguished from forward coefficient Frobenius ([Torsion] Definition III.2.18, p. 42).
- Convention: the composite of two `discPower 2`
  maps sends `a+bT` to `a+bT⁴` in `discIntegral S ϖ`, for any commutative coefficient ring.
  It fixes coefficients; `frobMod` squares them. The associated generic-fibre diagram is
  contravariant, as specified by `genericDiagram_step`.
- Non-example: squaring modulo `4` is not additive:
  `(1 + 1)² = 0` but `1² + 1² = 2`. Correspondingly, no `IntegralTower 2` can have a
  stage identified with `ℤ₂` whose marked parameter is `2`: its `divides` field would
  require `4 ∣ 2`. This tests the hypothesis used by `frobMod` to make its target
  characteristic two ([Torsion], proof of Corollary III.2.19, pp. 42–43). More generally, at
  `p = 2` a stage with parameter `2` is the zero ring, since `4 ∣ 2` with `2` regular makes `2` a unit.
- Non-example (regularity): `𝔽_p` is integral perfectoid with `π = 0` but has no regular parameter
  for which it is adically complete (`0` is a zero-divisor, and a nonzero parameter is a unit, for
  which adic separatedness forces the zero ring), so it has no constant tower; more generally no
  stage of any `IntegralTower` is a field (a Lean `example`).
- Convention: the consecutive maps of `IntegralTower.diagram` are the transition maps.

- Degenerate: all quotients of a tower of zero rings are zero.
- Non-example: a nonzero stage cannot have zero
  marked parameter, by regularity.
- Non-example: no Frobenius tower can be identified
  with the constant restricted-disc system with identity transitions and the same parameter.
- `frobeniusTowerHom_test_identity` (computation) — the identity tower morphism fixes `1+T`
  in the disc's zeroth stage.
- `frobeniusTowerHom_test_zero` (degenerate) — every two morphisms into a tower of zero rings
  agree on every stage element.
- `frobeniusTowerHom_test_parameter_not` (non-example) — a morphism cannot kill the marked
  parameter of a nonzero target stage.
- `genericFibre_test_zero` (degenerate) — a zero integral stage has zero localized coordinate ring.
- `genericFibre_test_parameter_inverse` (computation) — the marked parameter becomes a unit
  in every generic fibre, also in characteristic `p`.
- `genericFibre_test_transition_twoTerms` (compatibility) — the map between generic fibres sends
  `1+ϖ²` to `1+ϖ²` at the next stage.
- `relativeTwist_test_zero` (degenerate) — a zero next stage has zero scalar extension.
- `relativeTwist_test_base` (computation) — the constant base stage gives `O/ϖ^p` after twisting.
- `relativeTwist_test_nonzero` (non-example) — this base twist is nonzero when `ϖ` is not a unit.

- `perfection_test_parameter` (computation) — the marked right scalar `1 ⊗ t` is fixed.
- `perfection_test_zero` (degenerate) — a zero input algebra gives zero stages and zero completion.
- `perfection_test_twoTransitions` (convention) — two transitions send `b ⊗ c` to `b^(p²) ⊗ c`.
  These and the relative-Frobenius Characterisation above test the constructed tower.
- `perfection_lift_scalar` (computation) — at `n = 0`, the right coefficient of `1 ⊗ c`
  becomes `c^p` under the quotient lift.
- `perfection_lift_left` (convention) — for every `n`, the lift sends `[b ⊗ 1]` to `[b ⊗ 1]`.
- `perfection_lift_nilpotent` (non-example) — if `ε^p = 0` and `[ε ⊗ 1]` is nonzero
  modulo the stage-zero parameter's `p`th power, the transition sends `ε ⊗ 1` to zero
  but the lift of `[ε ⊗ 1]` is nonzero. This applies to the square-zero generator above,
  including `p = 2`.

- `frobeniusMod_test_twoTerms` computes the disc quotient map on `a+bT` as
  `a^p+b^p T^p`; `frobeniusMod_test_zero` makes it bijective on a zero stage, while
  `frobeniusMod_test_disc_not` denies surjectivity for the nonzero residue disc.
- `shift_test_zero` fixes `1+ϖ²` after a shift by zero; `shift_test_start` makes the new
  starting index zero once the shift reaches the old starting index;
  `shift_test_empty` preserves zero completion after any shift.

`IntegralTower` records the complete integral models before imposing Frobenius control.
Its parameter is marked at every stage and preserved by the transition maps; regularity and
`ϖ^p | p` are stagewise requirements. Its `diagram` iterates the consecutive ring maps.
`FrobeniusTower` adds the eventual quotient isomorphisms and both commuting squares above.
`Hom` preserves the parameter and transitions at every level, and the quotient lifts from
a recorded common index onward. Composition raises this index past the start of the intermediate
tower as well as those of the two morphisms. `discIntegral S ϖ` is
`AdicCompletion (ϖ) S[T]`; `discPower` is fixed on polynomial images by substitution `T ↦ T^p`.
`disc S ϖ` has these stages, transition maps and coefficient-Frobenius lifts. The hypotheses are
that `S` is integral perfectoid, `ϖ` is regular, `ϖ^p | p`, and `S` is `ϖ`-adically complete.
The constant-disc non-example assumes `ϖ` is not a unit, so that its residue polynomial ring
is nonzero.

`FrobeniusTower.discPowerHom S ϖ k` is the endomorphism of `disc S ϖ` given
at every stage by `T ↦ T^k`, with coefficients fixed and starting index zero.
It has the same integral-perfectoid, regularity, divisibility and completeness
hypotheses as `disc`. The lift calculation raises coefficients to their `p`th
powers while leaving the coordinate exponent unchanged; this is the affine
relative-Frobenius calculation in [Torsion] Definition III.2.18 and the proof
of Corollary III.2.19, pp. 42–43.

- `discPowerHom_constant` evaluates `k = 0` on `a+bT`, obtaining `a+b`.
- `discPowerHom_quadratic` evaluates `k = 2` at every stage, obtaining `a+bT²`.
- `discPowerHom_lift_coefficients` computes both sides of `Hom.lift_comm` for
  the quadratic map on `[a+bT]`: both are `[a^p+b^pT²]` modulo `ϖ^p`.
  These three Checks test `Hom` as well as `discPowerHom`.
- `Hom_disc_lift_forced` tests a family of disc ring maps preserving the
  parameter and transitions, without assuming quotient-lift compatibility:
  compatibility follows because substitution `T ↦ T^p` is injective modulo
  `ϖ^p`. Thus such a family on this disc cannot violate `lift_comm`.

`relativeTwist` is the tensor product
`(A_{m+1}⁺/ϖ) ⊗_{O/ϖ, Φ_O} (O/ϖ^p)`; the scalar map into the next stage sends the base parameter
to its marked parameter. This equation also proves containment of the principal ideals
needed for the quotient scalar map; for the constant base tower it is the identity equation.
`relativeTwistLeft` sends `x` to `x ⊗ 1`. The base Frobenius is
bijective, hence this map is bijective. `ofRelativeFrobenius` takes flat models, scalar-linear
transitions and scalar-linear identifications of these twists with `A_m⁺/ϖ^p`; the characteristic
equations specify relative Frobenius and the right scalar inclusion. Compatibility of the
quotient lifts with transitions is a conclusion. These are the
relative-Frobenius presentations in [Torsion], proof of Corollary III.2.19, pp. 42–43.

The constructor `FrobeniusTower.perfection` uses a perfect characteristic-`p` base `O`, a flat
`O`-algebra `B`, a regular `t ∈ O`, and `t`-adic completeness of `B`. Its stage `n` is
`B ⊗_{O,F^{-n}} O`; `perfectionTensor n b c` is `b ⊗ c`. The parameter is `1 ⊗ t`, and
`perfectionTransition_tensor` sends `b ⊗ c` to `b^p ⊗ c`. This keeps the right scalar fixed
([Torsion] §III.2.3, Definition III.2.18, p. 42). With these hypotheses, the lift specification of
`FrobeniusTower.perfection` gives the chosen base-change isomorphism
`w_n : B_{n+1}/(1 ⊗ t) → B_n/(1 ⊗ t)^p` by
`w_n([b ⊗ c]) = [b ⊗ c^p]`. The tensor balancing follows from
`F^{-(n+1)}(a)^p = F^{-n}(a)`; perfectness of `O` gives the inverse on coefficients.
The Frobenius square alone does not specify this lift: for the flat algebra
`B = O[ε]/(ε²)`, the transition kills `ε`, while the lift retains its class whenever
`O/(t^p)` is nonzero. No reducedness hypothesis is imposed on `B`.

Checks of the relative constructor itself use its scalar-linear presentations:
`ofRelativeFrobenius_test_scalar` computes the transition on `1+ϖ` from the base;
`ofRelativeFrobenius_test_zero` gives zero completed generic fibre for zero models;
`ofRelativeFrobenius_test_parameter_not` excludes a zero marked parameter in a nonzero stage.

The following Checks evaluate the constructor's quotient lift, under exactly its
flatness, completeness, marked-parameter and relative-presentation hypotheses
([Torsion], proof of Corollary III.2.19, pp. 42–43):

- `ofRelativeFrobenius_lift_scalar` sends `[c]` from the scalar ring to `[c^p]`.
- `ofRelativeFrobenius_lift_tensor` writes `j(x)=x⊗1` and computes the lift on
  `j⁻¹(x⊗c)` as `e(x⊗1)·c`, for `c ∈ O/ϖ^p`. The right tensor scalar is
  retained; a scalar entering through the left factor is Frobenius-powered.
- `ofRelativeFrobenius_lift_two_terms` specifies `e(x⊗1)=[a]` and
  `e(y⊗1)=[b]`, then computes the constructed lift of `[cx+dy]` as
  `[c^p a+d^p b]`. This tests the presentation on two independent terms and
  fixes which coefficients are powered.

For the constant constructor, `constPerfectoid_test_zero` gives zero generic completion and
`constPerfectoid_test_lift_twoTerms` sends `a+b` modulo `ϖ` to `a^p+b^p` modulo `ϖ^p`.

**Relative Frobenius constructor.** (*relative-frobenius-tower*)
`FrobeniusTower.ofRelativeFrobenius` is the scalar-linear constructor of P7.5,
with flat integral models, a marked non-zero-divisor, completeness, and the specified
relative Frobenius presentations. Its Checks are the scalar, zero-model and
nonzero-parameter examples above. The characteristic-`p` perfection construction
has the separate label *perfection-tilde-limit* ([Torsion] Definition III.2.18, p. 42).

Lean: `Perfectoid.IntegralTower`, `IntegralTower.const`, `IntegralTower.quotient`,
`IntegralTower.quotientMap`, `IntegralTower.diagram`, `FrobeniusTower`.

### P7.6 completion and inversion of a Frobenius-controlled tower

**Completion and inversion of a Frobenius-controlled tower.** (*frobenius-tower-completion*) For a
Frobenius-controlled integral tower `(A_•⁺, ϖ, m_0, w)` (see *frobenius-controlled-integral-tower*)
define: `A_∞⁺ :=` the `ϖ`-adic completion of the direct limit `colim_m A_m⁺` (Mathlib
`AdicCompletion (Ideal.span {ϖ})` of the colimit in `CommRingCat` of `IntegralTower.diagram`, for
which `Ring.DirectLimit` is an equivalent model); `A_∞ := A_∞⁺[1/ϖ]`, topologised so that
`A_∞⁺` is an open ring of definition with the `ϖ`-adic topology (a complete Tate ring with
pseudouniformiser `ϖ`); `A_∞⁺⁺ :=` the integral closure of `A_∞⁺` in `A_∞` (a ring of integral
elements). This is the completed direct limit of Tate pairs of
AdicSpacesPartII R0.19.7 (*spa-of-filtered-colimit-of-tate-pairs*) applied to `(A_m, A_m⁺⁺)` with the rings of definition
`A_m⁺`, which for perfectoid levels is the construction of P5.1–P5.2; what is added for a
non-perfectoid tower is the completion/inversion formula: (i) `A_∞⁺/ϖ^k A_∞⁺ = colim_m A_m⁺/ϖ^k A_m⁺` for every
`k ≥ 1`; (ii) `A_∞⁺` is `ϖ`-torsion-free (the `ϖ`-torsion-freeness of the `A_m⁺` is what makes the
direct limit and its completion `ϖ`-torsion-free), so `A_∞⁺ ⊆ A_∞`; (iii) `(A_∞, A_∞⁺⁺)` is the
completed direct limit of the Huber pairs `(A_m, A_m⁺⁺)` in the sense of
*tilde-limit-from-completed-colimit* and of AdicSpacesPartII R0.19.7 (compatible rings of definition `A_m⁺` with
ideals of definition `ϖA_m⁺`, so `ϖA_{m'}⁺ = (ϖA_m⁺)A_{m'}⁺`); (iv) the construction depends only on the tail
of the tower (shift invariance) and is functorial for morphisms of towers. No uniform completion is
taken: by *frobenius-tower-uniformity-bound* the `ϖ`-adic topology on `A_∞⁺` is already the
spectral topology of `A_∞`. In `Perfectoid`, define `IntegralTower.colimitParameter` (the image of
`ϖ` in `colim_m A_m⁺`) and `IntegralTower.completedPlus` (`A_∞⁺`, defined for the underlying
integral tower of a Frobenius tower),
`FrobeniusTower.completed` (`A_∞ = A_∞⁺[1/ϖ]` with `A_∞⁺` an open ring of definition),
`FrobeniusTower.completedPair` (the Huber pair `(A_∞, A_∞⁺⁺)`; Tau Ceti `Huber.Pair`) and
`FrobeniusTower.toCompleted` (the compatible maps `A_m⁺ → A_∞⁺` and
`(A_m, A_m⁺⁺) → (A_∞, A_∞⁺⁺)`); prove the quotient formula (simp:
`A_∞⁺/ϖ^k ≅ colim_m A_m⁺/ϖ^k` for all `k ≥ 1`), that `A_∞⁺` is `ϖ`-torsion-free, the instance that
`A_∞` is a complete Tate ring with pseudouniformiser `ϖ` (Tau Ceti `Huber.IsTateRing`,
`Huber.IsPseudoUniformizer`), the universal property (compatible maps `A_m⁺ → B` into a
`ϖ`-adically complete ring sending the common `ϖ` to the element defining the adic topology of `B`
extend uniquely to a continuous map `A_∞⁺ → B`, uniqueness from density of `colim A_m⁺`), shift
invariance (the completion of a shifted tower is canonically the same), functoriality
`completed_map` (a morphism of towers induces a continuous map of completions; identity and
composition) and `FrobeniusTower.isCompletedDirectLimit` (`(A_∞, A_∞⁺⁺)` is the
completed direct limit of the `(A_m, A_m⁺⁺)`) ([Torsion] §III.2.3, proof of Corollary III.2.19,
p. 43 (the completion formula); [Torsion] §III.2.3, Definition III.2.18 (ii), p. 42 (the same
formula for the perfection tower, with `R^{perf} = R^{perf+}[t^{-1}]`); [ECD] §6, Proposition 6.5,
p. 29 (the analogous formula for towers of perfectoid rings, which the construction extends to
non-perfectoid finite levels)). *Needs:* *frobenius-controlled-integral-tower*; AdicSpacesPartII R0.19.7; P5.1;
Mathlib `AdicCompletion`, `Ring.DirectLimit`, `IsAdicComplete`, `integralClosure`; Tau Ceti
`TauCeti.Huber.Pair`, `TauCeti.Huber.IsTateRing`.

**Checks.**

- Computation: for the compatible-root disc tower over
  `(R, R⁺)`, the completion is `R⁺⟨T^{1/p^∞}⟩` (the `ϖ`-adic completion of `R⁺[T^{1/p^∞}]`) and
  `A_∞ = R⟨T^{1/p^∞}⟩`.
- Degenerate: for the constant tower of an integral
  perfectoid `S` (`ϖ`-torsion-free, `ϖ`-complete), the completion is `S` and `A_∞ = S[1/ϖ]`.
- Non-example: for the disc tower over a nonzero `R`, the element
  `Σ_{n≥1} ϖ^n T^{1/p^n}` lies in `A_∞⁺` but in no `A_m⁺`: the direct limit itself is not complete,
  so the formula needs the completion.
- Characterisation: `A_∞⁺/ϖ^p ≅ colim_m A_m⁺/ϖ^p` as
  rings, compatibly with the maps from the `A_m⁺`.
- Compatibility: if every `A_m⁺` is the ring of integral
  elements of a perfectoid Tate ring, the completion agrees with the ring `R⁺` of [ECD] Proposition
  6.5 for the limit of the `Spa(A_m, A_m⁺)`.

- Degenerate: completion and subsequent
  localization of a tower of zero rings remain zero. The constant Check tests bijectivity of the
  canonical stage map; the quotient Check evaluates the marked parameter modulo its square.

- `completedPair_test_zero` gives the whole zero ring as the plus ring.
- `completedPair_test_inverse_not` excludes `ϖ⁻¹` from the plus ring when the generic fibre
  is nonzero.
- `completedPair_test_field` recovers `(C_p,C_p°)` from its constant integral tower, with the
  equivalence fixed on every integral coefficient.
- `completed_map_quadratic_root` applies `completed_map (discPowerHom S ϖ 2)`
  to the first root `x = T^{1/p}`, obtaining `x² ≠ x`. Under
  `discCompletionEquiv` this is the monomial of exponent `2/p` with coefficient
  one. In addition to the disc hypotheses, `ϖ` is a nonunit; reduction modulo
  `ϖ` distinguishes the two monomials, including when `p = 2`
  ([Torsion] Definition III.2.18, p. 42).
- `completed_map_test_identity` fixes `1+ϖ²`; `completed_map_test_zero` sends every element to
  zero in a zero target; `completed_map_test_parameter_not` forbids killing the parameter of
  a nonzero target completion.

The constructors use `IntegralTower.colimitRing`, its canonical stage maps and their common
`colimitParameter`. `completedPlus` is its principal-adic completion; `completed` localizes at
the completed parameter. The topology is characterized by `completed_nhds_zero`: images of
powers of the parameter ideal form a neighbourhood basis at zero. `completedPair` takes the
integral closure of this integral model in its localization. The quotient comparison
fixes every finite-stage element; `completed_map_stage` and
`completed_shift_stage` fix the maps on the same dense images. The completion lift fixes the
parameter in the target complete ring, so its ring map is automatically continuous for the
principal-adic topologies. These formulas specialize the completion in [Torsion], proof of
Corollary III.2.19, p. 43, without assuming perfectoid finite stages.

`discCompletionEquiv` identifies the integral completion with the principal-adic completion
of the monoid algebra on the nonnegative `p`-power rational exponents. Its monomial equation
sends the level-`n` monomial `aT^k` to `aT^(k/p^n)`. The disc Check computes the image of the
first root. The missing-series Check uses `Σ_{n≥0} ϖ^(n+1) T^(1/p^(n+1))` and asserts that its
sum is outside every finite generic fibre.

`IntegralTower.quotientDiagram k` is the direct system of the stage quotients by `ϖ^k`.
`quotientColimit` and `quotientColimit_mk` identify its colimit with the quotient of the ring
colimit, fixing every stage element. The quotient formula above combines this
comparison with completion and fixes its stage maps.
`isCompletedDirectLimit` asserts the full `IsCompletedColimitOfTower` predicate for the
constructed generic fibres, their maps, their integral models and their marked parameter.
`perfectoidDiagram` regards the same generic-fibre diagram as a diagram in `Perfd` when its
levels are perfectoid; the compatibility Check above compares its P5 limit with the
constructed completion by an isomorphism commuting with every projection.

Checks `completedQuotient_test_zeroPower`, `completedQuotient_test_parameter` and
`completedQuotient_test_twoTerms` respectively compute the zero ring for exponent zero, zero
for the marked parameter modulo itself, and `1` for `1+ϖ²` modulo `ϖ²`.

Lean: `IntegralTower.colimitParameter`, `IntegralTower.completedPlus`.

### P7.7 preperfectoid adic spaces

**Preperfectoid adic spaces.** (*preperfectoid-space*) Let `X` be an analytic adic space in the
generality of [SW13] §2.1 (an analytic Yoneda-adic space, AdicEtaleGeometry A1.16; the sheafy
adic spaces of AdicSpaces layer 5 form a full subcategory). For an open affinoid `U = Spa(A, A⁺) ⊆ X`
with a pseudouniformiser `ϖ ∈ A⁺`, the strong completion `(Â, Â⁺)` of `(A, A⁺)` is the completion
of `(A, A⁺)` for the topology on `A` that gives `A⁺` the `ϖ`-adic topology (`Â⁺` the `ϖ`-adic
completion of `A⁺`, `Â = Â⁺[1/ϖ]`; independent of `ϖ`); it is taken for the `ϖ`-adic topology on
the plus ring, not the natural topology of `A`. `X` is preperfectoid if `X` has a cover by open
affinoids `U_α = Spa(A_α, A_α⁺)` whose strong completions `(Â_α, Â_α⁺)` are perfectoid Tate pairs
([ECD] Definition 3.1). Over a perfectoid field `K` this is [SW13] Definition 2.3.4 (they complete
for the `p`-adic topology on `A⁺`); [SW13] Definition 2.3.9 defines `X` over `Spa(Q_p, Z_p)` to be
preperfectoid when every base change to a perfectoid field of characteristic `0` is, and
Proposition 2.3.10 compares the two when `X` lies over a perfectoid `K`. The natural topology of
`A` and the `ϖ`-adic topology of `A⁺` can differ, and then `A` itself need not be perfectoid: the
nonreduced point `Spa(K[ε]/(ε²), O_K + Kε)` is preperfectoid with strong completion `(K, O_K)`.
Locally closed subspaces of preperfectoid spaces are preperfectoid ([SW13] Propositions 2.3.7 and
2.3.11). In `PerfectoidSpace`, define `strongCompletionPair` (for a Huber pair `(A, A⁺)` with
pseudouniformiser `ϖ ∈ A⁺`: `(Â, Â⁺)` with `Â⁺` the `ϖ`-adic completion of `A⁺` and
`Â = Â⁺[1/ϖ]`) and `IsPreperfectoid` (`X` is covered by open affinoids with perfectoid strong
completion); prove `strongCompletionPair_indep` (the strong completion does not depend on the
choice of `ϖ`), `IsPreperfectoid.of_isPerfectoid` (instance: perfectoid spaces are preperfectoid),
`IsPreperfectoid.openSubspace` (open subspaces of preperfectoid spaces are preperfectoid),
`IsPreperfectoid.closedSubspace` (closed, hence locally closed, subspaces are preperfectoid; [SW13]
Proposition 2.3.7, p. 18), `IsPreperfectoid.baseChange` (for `X` over a perfectoid field `K`, preperfectoid
over `Spa(Q_p, Z_p)`, i.e. after all base changes to perfectoid fields, implies preperfectoid over
`K`; [SW13] Proposition 2.3.10) and `strongCompletion` (the strong completion `X̂`
of P7.8) ([SW13] §2.3, Definition 2.3.4, pp. 17–18; [SW13] §2.3, Definition 2.3.4, p. 18 (the
`p`-adic, i.e. `ϖ`-adic, topology on the plus ring); [SW13] §2.3, Remark 2.3.5, p. 18 (the
non-reduced example)). *Needs:* AdicEtaleGeometry A1.16; P1.1; P2.10; Mathlib `AdicCompletion`.

**Checks.**

- `isPreperfectoid_test_nonreducedPoint` (computation) — for a perfectoid field `K`,
  `Spa(K[ε]/(ε²), O_K + Kε)` is preperfectoid: the `ε`-component is infinitely `ϖ`-divisible in
  `O_K + Kε`, so the `ϖ`-adic completion of the plus ring is `O_K` and the strong completion is
  `(K, O_K)`.
- `isPreperfectoid_test_perfectoid` (degenerate) — every perfectoid space is preperfectoid, with
  every affinoid perfectoid its own strong completion.
- Boundary: the zero-variable formal disc has generic fibre `Spa(ℚ_p, ℤ_p)`, whose strong
  completion is itself. It fails `IsPreperfectoid` because `ℚ_p` is not perfectoid
  ([ECD] Remark 3.3, p. 14), although every base change to a characteristic-zero perfectoid
  field `K` is `Spa(K, K°)`, so it satisfies [SW13] Definition 2.3.9.
- `isPreperfectoid_test_disc_not` (non-example) — for a perfectoid field `K`, the closed unit disc
  `D = Spa(K⟨T⟩, K°⟨T⟩)` is not preperfectoid: every nonempty open affinoid of `D` is `Spa(A, A⁺)` with `A` a
  reduced one-dimensional `K`-affinoid algebra and `A⁺` bounded, so the strong completion is
  `(A, A⁺)` itself, and `A` is not perfectoid (for `A = K⟨T⟩`, `T` is not a `p`-th power modulo
  `ϖ^p` in `K°⟨T⟩`; in general the reduction `Ã = A°/A°°` is a reduced finitely generated
  `k`-algebra of dimension one, whose fraction fields are not perfect, so Frobenius on `A°/ϖ` is
  not surjective).
- `isPreperfectoid_test_overField` (compatibility) — for `X` over a perfectoid field `K` of
  characteristic `0`, `X` is preperfectoid in this sense iff it is preperfectoid over
  `Spa(K, O_K)` in the sense of [SW13] Definition 2.3.4: `K° ⊂ A⁺` contains a pseudouniformiser `ϖ`
  with `ϖ^p | p` and `p | ϖ^N`, so the `ϖ`-adic and `p`-adic topologies on `A⁺` agree.
  The example expands the field-relative condition on the same Yoneda-adic open charts:
  their `strongRing` with parameter `p` is perfectoid. It states the equivalence with
  `IsPreperfectoid` and records the two divisibilities and equality of Mathlib's
  `Ideal.adicTopology` on each chart's plus ring
  ([SW13] Definition 2.3.4, pp. 17–18).

- `strongCompletionPair_test_perfectoid` (compatibility) — the canonical map on a perfectoid
  pair is bijective.
- `strongCompletionPair_test_zero` (degenerate) — a zero pair has zero strong integral and
  generic completions.
- `strongCompletionPair_test_nonreducedPoint` (computation) — for the dual-number point over
  `C_p`, an identification of its strong completion with `C_p` sends `a+bε` to `a`.
- `isPreperfectoid_test_empty` (degenerate) — the empty affinoid satisfies the local cover condition.

`strongPlus R ϖ` is the existing principal-adic completion of `R⁺`; `strongRing R ϖ hϖ`, with `hϖ` asserting that `ϖ` is a pseudouniformiser in `R`,
is its localization at the completed parameter. `strong_nhds_zero` fixes the topology by the
images of powers of `(ϖ)`. In `strongCompletionPair` the plus ring is the image of the completed
plus ring. The map from the original pair is fixed on `R⁺` by
`strongCompletionPair.toCompletion_plus`; its extension uses that `ϖ` is a pseudouniformiser
in `R`. `IsPreperfectoid` requires an analytic source and an open cover by pairs with perfectoid
strong completion ([SW13] Definition 2.3.4 and Remark 2.3.5, pp. 17–18).

**Closed subspaces.** (*preperfectoid-closed-subspaces*)
`IsPreperfectoid.closedSubspace` takes an analytic source and a preperfectoid target,
with closed-subspace charts given by surjective quotient maps and the integral closure
of the image plus ring. It concludes preperfectoidness of the source
([SW13] Proposition 2.3.7, p. 18).

**Absolute and relative preperfectoidness.** (*preperfectoid-base-change*)
`IsPreperfectoid.baseChange` assumes preperfectoidness after every perfectoid-field
base change over `ℚ_p` and concludes it over the given characteristic-zero perfectoid
field ([SW13] Proposition 2.3.10, pp. 18–19).

### P7.8 the strong completion

**The strong completion `X̂` of a preperfectoid space and `X̂ ~ X`.** (*strong-completion*) Let
`X` be a preperfectoid analytic adic space (see *preperfectoid-space*), over no base field ([SW13]
state it over `Spa(K, O_K)`). There is a perfectoid space `X̂` with a morphism `X̂ → X` (of
Yoneda-adic spaces) such that: (i) `|X̂| → |X|` is a homeomorphism; (ii) for every open affinoid
`U = Spa(A, A⁺) ⊆ X` whose strong completion `(Â, Â⁺)` is perfectoid, the corresponding open
subspace of `X̂` is `Spa(Â, Â⁺)`; (iii) `X̂` is universal for morphisms from perfectoid spaces to
`X`: every morphism `Y → X` with `Y` perfectoid factors uniquely through `X̂`; (iv) when `X` is an
adic space in the sense of AdicSpaces layer 5 (sheafy affinoid coordinate rings), `X̂ ~ X`, i.e.
`(X̂, X̂ → X)` is a perfectoid tilde-limit of the one-object system `X` in the sense of
*perfectoid-tilde-limit* ([SW13] Proposition 2.4.4, stated there for adic spaces in their §2.1
generality). `X̂` is unique up to unique isomorphism by (iii). In `PerfectoidSpace`, define
`strongCompletion` (the perfectoid space `X̂` of a preperfectoid `X`, with its map to `X`) and
`strongCompletion.map` (a morphism of preperfectoid spaces induces one of strong completions;
identity and composition); prove `strongCompletion.homeomorph` (`|X̂| → |X|` is a homeomorphism),
`strongCompletion.affinoid` (over `U = Spa(A, A⁺)` with perfectoid strong completion, the open
subspace of `X̂` is `Spa(Â, Â⁺)`), `strongCompletion.lift` (every morphism from a perfectoid space
to `X` factors uniquely through `X̂`), `strongCompletion.isTildeLimit` (`X̂ ~ X`) and
`strongCompletion.of_perfectoid` (for `X` perfectoid the canonical map `X̂ → X` is an isomorphism)
([SW13] §2.3, Proposition 2.3.6, p. 18 (properties (i)–(iii)); [SW13] §2.4, Proposition 2.4.4,
p. 20 (property (iv))). *Needs:* *preperfectoid-space*; *perfectoid-tilde-limit*; P2.3; P2.10;
AdicEtaleGeometry A1.16.

**Checks.**

- `strongCompletionPair_test_nonreducedPoint` (computation) — the strong completion of
  `Spa(K[ε]/(ε²), O_K + Kε)` over a perfectoid field `K` is `Spa(K, O_K)`.
- `strongCompletion_test_perfectoid` (degenerate) — for `X` perfectoid, the canonical `X̂ → X` is
  an isomorphism.
- `strongCompletion_test_not_iso` (non-example) — the map `Spa(K, O_K) → Spa(K[ε]/(ε²), O_K + Kε)`
  is a homeomorphism with dense image on functions but not an isomorphism of adic spaces: `X̂ ~ X`
  does not mean `X̂ ≅ X`.
- `strongCompletion_test_universal` (characterisation) — for every perfectoid `Y`,
  `Hom(Y, X̂) → Hom(Y, X)` is bijective; for the non-reduced point this says every map from a
  perfectoid space kills `ε`.

- `strongCompletion_test_empty` (degenerate): the strong completion of an empty
  preperfectoid space has no points. The perfectoid and dual-number non-isomorphism Checks test
  the global canonical map.

- `strongCompletion_map_component_swap` takes `Y = X × {false,true}` for a
  perfectoid `X` with a specified point `x`, and completes the map exchanging
  the two components. Through `strongCompletion.homeomorph`, the lift of
  `(x,false)` maps into the true component and is not fixed. The input includes
  the preperfectoidness of `Y`; this is a finite-product computation using
  [SW13] Proposition 2.3.6, p. 18.
- `strongCompletion_map_test_id` (compatibility) — on a perfectoid object the identity remains identity.
- `strongCompletion_map_test_empty` (degenerate) — all maps out of an empty completion agree.
- `strongCompletion_map_test_nonreduced` (computation) — completing the dual-number reduction
  makes its map an isomorphism.

`strongCompletion.ι` is the canonical morphism to the original space.
`homeomorph_apply`, `affinoid_ι` and `map_ι` identify respectively its point map, its affinoid
coordinate map and its functoriality square. `lift` gives a unique factorization of every map
from a perfectoid source, which also determines `map_id` and `map_comp` ([SW13] Proposition
2.3.6, p. 18).

### P7.9 cofinality witnesses for level families

**Cofinality witnesses for families of compact open subgroups.** (*level-cofinality-witness*) Let
`G` be a Hausdorff topological group in which the compact open subgroups form a basis of
neighbourhoods of `1` (a locally profinite group: a profinite group, or `G(Q_p)` for a linear
algebraic group `G` over `Q_p`). Write `CO(G)` for the set of compact open subgroups, a cofiltered
category under inclusion (`K′ → K` when `K′ ⊆ K`; `K ∩ K′` is compact open). A level family is a
cofiltered category `J` with a functor `j ↦ K_j` to `CO(G)`. A cofinality witness for `(K_j)` is a
function `U ↦ j(U)` from the open subgroups `U` of `G` (equivalently from open neighbourhoods of
`1`) to `J` with `K_{j(U)} ⊆ U`; the witness is data. By Mathlib's criterion
`CategoryTheory.Functor.initial_of_exists_of_isCofiltered` (morphisms in `CO(G)` are unique, so the
second condition is automatic), a witness makes `j ↦ K_j` an initial functor `J → CO(G)`; hence,
for every tower `(X_K)_{K ∈ CO(G)}` of analytic honest adic spaces with qcqs transition maps (a Shimura
tower at `p`) and every perfectoid `X` with compatible maps, `X ~ lim_{K ∈ CO(G)} X_K` iff
`X ~ lim_{j ∈ J} X_{K_j}` (see *tilde-limit-cofinal-change*). A level family without a witness only
computes the tilde-limit of the subtower it indexes: no infinite-level statement for the full tower
follows from it. The existence of a witness for decreasing sequences inside a compact open subgroup is
characterised by the library: a directed family of closed subgroups of a compact group whose
intersection lies in an open subgroup has a member inside it (Tau Ceti
`Subgroup.exists_le_of_iInf_le_of_directed`), and conversely the open normal subgroups of a
profinite group are a neighbourhood basis of `1` (Mathlib
`ProfiniteGrp.exist_openNormalSubgroup_sub_open_nhds_of_one`); so a decreasing family of open
subgroups of a profinite group is cofinal iff its intersection is trivial, compactness being
essential. Hence the principal congruence subgroups `H ∩ (1 + p^n M_N(ℤ_p))` (`n ≥ 1`) of a closed
`H ⊆ GL_N(ℤ_p)` are cofinal (their intersection is trivial), `Γ_0(p^n)` and `Γ_1(p^n)` are not
(both contain the unipotent upper triangular subgroup; the `Γ_0`-, `Γ_1`- and `Γ`-level towers
are distinct towers, [BHW] §5.2, proof of Theorem 5.11, p. 22 of the arXiv manuscript), and a family below a cofinal
family is cofinal. In `PerfectoidSpace`, define `LevelFamily`
(a cofiltered `J` with a functor to the compact open subgroups of a locally profinite `G`),
`CofinalityWitness` (a choice `j(U)` with `K_{j(U)} ⊆ U` for every open subgroup `U`),
`CofinalityWitness.ofDecreasing` (a decreasing sequence of open subgroups of a compact open `K_0`
with trivial intersection has a witness), `CofinalityWitness.principalCongruence` (the family
`1 + p^(n+1) M_N(Z_p)` is witnessed in `GL_N(Q_p)`),
`CofinalityWitness.principalCongruence_closed` (for a closed subgroup `H ⊆ GL_N(Z_p)`,
its intersections with these levels are witnessed in `H`, with its subspace topology;
the entrywise bound is `|g_ij − δ_ij| ≤ p^(-(n+1))`),
`CofinalityWitness.conj` (if `(K_j)` is witnessed, so is `(gK_jg^{-1})` for `g ∈ G`),
`CofinalityWitness.mono` (if `K′_j ⊆ K_j` compatibly and `(K_j)` is witnessed, so is `(K′_j)`) and
`CofinalityWitness.openNormal` (in a profinite group the open normal subgroups are witnessed;
Mathlib `ProfiniteGrp.exist_openNormalSubgroup_sub_open_nhds_of_one`); prove
`CofinalityWitness.initial` (a witnessed family `J → CO(G)` is a Mathlib
`CategoryTheory.Functor.Initial`) and `CofinalityWitness.isTildeLimit_iff` (for a tower `(X_K)`
whose every level is analytic and honest and whose every transition is qcqs, and perfectoid `X`
with a compatible cone: `X ~ lim_{CO(G)} X_K ⟺ X ~ lim_J X_{K_j}`). Apply
`IsTildeLimit.reindex` using `CofinalityWitness.initial`; the three hypotheses concern the full
tower, including levels and arrows outside the chosen family. The qcqs condition is the one in
[SW13] Definition 2.4.1, pp. 19–20; analyticity and honesty are the additional conditions of this
roadmap's analytic Yoneda comparison with AdicSpacesPartII R0.20.1 ([Sch13E] item (1), p. 1 (reindexing a
cofiltered system along a cofinal map); [CHJ] §2, Theorem 2.8, p. 11 (the infinite-level Shimura
curve indexed by the principal congruence levels `K(p^n)`); [BHW] §5.2, proof of Theorem 5.11,
p. 22 of the arXiv manuscript (the `Γ_0`-, `Γ_1`- and `Γ`-level towers treated separately: different level families give
different level problems)). *Needs:* Mathlib `OpenSubgroup`, `ProfiniteGrp`,
`ProfiniteGrp.exist_openNormalSubgroup_sub_open_nhds_of_one`, `CategoryTheory.Functor.Initial`,
`CategoryTheory.Functor.initial_of_exists_of_isCofiltered`, `CategoryTheory.IsCofiltered`.

The records `LevelFamily J G` and `CofinalityWitness F` themselves make sense for any group
with a topology: `J` is cofiltered, `F.level j` is an open subgroup, each level is compact,
and `a : i → j` gives `F.level i ≤ F.level j`. Functor laws are automatic in this poset.
`CofinalityWitness F` chooses `levelFor U` with `F.level (levelFor U) ≤ U`. The neighbourhood-basis
interpretation and the comparison with the full tower above use the locally profinite hypotheses.
Prove `CofinalityWitness.not_nonempty_of_common_element`: if `x` belongs to every level and some
open subgroup `U` excludes `x`, there is no witness (apply its defining inclusion to `U`).

**Checks.**

- `cofinalityWitness_test_trivialGroup` (degenerate) — the one-object constant family at the
  whole trivial group `Multiplicative (ZMod 1)` has a witness.
- `cofinalityWitness_test_twoLevels` (convention) — in the two-element group, use
  the category `0 → 1` with levels `{1} → G`. The inclusions point from smaller
  subgroups to larger ones, and `levelFor U = 0` is a witness for every open `U`.
- `cofinalityWitness_test_finiteConst_not` (non-example) — the one-object constant family at
  the whole two-element discrete group `Multiplicative (ZMod 2)` has no witness, since the
  trivial open subgroup excludes its nonidentity element. This is a valid `LevelFamily`.
- `cofinalityWitness_test_principalCongruence` (computation) — in `G = GL_N(Q_p)` the family
  `K(p^n) = 1 + p^n M_N(Z_p)` (`n ≥ 1`) has the witness `U ↦` the least `n` with `K(p^n) ⊆ U`, which
  exists by `Subgroup.exists_le_of_iInf_le_of_directed` since `⋂_n K(p^n) = {1}`.
- `cofinalityWitness_closed_rankZero` — in rank zero every level is the whole trivial group.
- `cofinalityWitness_closed_trivial` — for `H = {1}` every level is the whole of `H`.
- `cofinalityWitness_closed_scalar` — for `H = GL_1(Z_p)`, membership at index `n`
  is exactly `|g_00 − 1| ≤ p^(-(n+1))`.
- `cofinalityWitness_test_discrete` (degenerate) — for `G` finite with the discrete topology, the
  one-element family `K = {1}` has the witness `U ↦ ∗` (in particular for the two-element
  group `Multiplicative (ZMod 2)`); for a tower over `CO(G)` with every level analytic and honest,
  all transitions qcqs, and `X_{1}` perfectoid, its value `X_{1}` is then a perfectoid tilde-limit.
- `cofinalityWitness_nonanalyticLevel` (non-example) — a nonanalytic level anywhere in the
  full tower prevents any cone from satisfying `PerfectoidSpace.IsTildeLimit`.
- `cofinalityWitness_nonhonestLevel` (non-example) — a nonhonest level anywhere in the full
  tower likewise prevents the full tilde-limit predicate.
- `cofinalityWitness_nonqcqsTransition` (non-example) — a transition of the full tower that
  is not qcqs prevents the full tilde-limit predicate, even if a chosen family omits that arrow.
- `cofinalityWitness_test_gamma0_not` (non-example) — in `GL_2(Q_p)` the decreasing family
  `Γ_0(p^n) = {(a b; c d) ∈ GL_2(Z_p) : c ≡ 0 mod p^n}` has no witness: every `Γ_0(p^n)` contains the
  upper-triangular unipotent `(1 1; 0 1)`, which lies outside the open subgroup `K(p)`. Its
  tilde-limit (the anticanonical `Γ_0(p^∞)` level) does not by cofinality compute the full infinite level (a constant tower can still have
  the same limit along either family).
- `cofinalityWitness_test_const_not` (non-example) — the constant family `K_j = K` with `K ≠ {1}`
  (`G` infinite profinite) has no witness; the limit of the subtower is `X_K` itself.
- `cofinalityWitness_test_initial` (compatibility) — a witness makes `j ↦ K_j` a Mathlib
  `Functor.Initial` `J → CO(G)`, so `Functor.Initial.limitIso` identifies `lim_J |X_{K_j}|` with
  `lim_{CO(G)} |X_K|`.

`CompactOpenSubgroup G` is the subtype of open subgroups with compact underlying set.
`LevelFamily.functor` uses the inclusion order on this subtype. `CofinalityWitness.initial` is
the adapter from the chosen neighbourhood inclusions to Mathlib initiality; it adds no new
categorical criterion. The matrix Check expresses `K(p^(n+1))` inside `GL_N(Q_p)` by the entrywise
bounds `|g_ij-δ_ij| ≤ p^(-(n+1))`. The `Γ₀` Check includes integrality of both the matrix and its
inverse, and the bound on its lower-left entry; it denies a witness for this family.

### P7.10 uniqueness and representability of perfectoid tilde-limits

A perfectoid tilde-limit is determined by the system: this subsection and the next prove that it
represents the inverse limit of the represented functors on perfectoid spaces, hence is unique up
to unique isomorphism, and compare it with inverse limits of represented functors; the further
comparison with inverse limits of diamonds is DiamondsAndVStacks', downstream.

**A perfectoid tilde-limit represents `lim_i Hom(−, X_i)` on perfectoid spaces and is unique.**
(*tilde-limit-perfectoid-uniqueness*) Let `(X_i)_{i∈I}` be a cofiltered system of analytic adic
spaces with quasicompact quasiseparated transition maps and `X ~ lim X_i` a perfectoid tilde-limit
(see *perfectoid-tilde-limit*). Then for every perfectoid space `Y` (of any characteristic, over no
fixed base field), and more generally every analytic adic space `Y` with a basis of open affinoids
`Spa(B, B⁺)` with `B` uniform (for instance `Y` stably uniform; only uniformity of these coordinate
rings is used, not perfectoidness; [SW13] state it for perfectoid affinoid `(K, O_K)`-algebras),
the map
`Hom(Y, X) → lim_i Hom(Y, X_i)`, `g ↦ (φ_i ∘ g)_i`, is bijective. Consequently: (i) `X` represents
the functor `Y ↦ lim_i Hom(Y, X_i)` on perfectoid spaces; (ii) if `(X′, φ′)` is another perfectoid
tilde-limit of the same system, there is a unique isomorphism `X ≅ X′` compatible with the cones;
(iii) a morphism of systems `(X_i) → (X′_i)` (over a common index category) induces a unique
morphism between perfectoid tilde-limits commuting with the cones; (iv) over a base `S` (all `X_i`
over `S` compatibly), the same holds for `S`-morphisms. Injectivity is by density and existence
starts from the good affinoid perfectoids ([SW13] §2.4, Proposition 2.4.5, p. 20; [SW13] §2.4,
proof of Proposition 2.4.5, p. 21; [Survey] §2, Proposition 2.23, p. 12 (representability)).
`IsTildeLimit.lift` states the perfectoid-source case with `Y : Perfd p`.
`IsTildeLimit.lift_uniform` states the extension to an honest analytic Yoneda adic space:
its point space has a basis consisting of images of open immersions `Spa^Y(B) → Y`,
where `B : CAff` is sheafy, Tate and uniform. For every compatible cone with vertex `Y`,
there is a unique morphism to `X` commuting with all projections. The uniform-source
extension uses the completion argument in [SW13] Proposition 2.4.5, pp. 20–21;
uniformity bounds the power-bounded elements of each source chart.
*Needs:* *perfectoid-tilde-limit*; *good-affinoid-perfectoid*; *good-affinoid-perfectoid-basis*;
P1.1; P2.10; Tau Ceti `TauCeti.ValuationSpectrum.mem_of_forall_vle_one`,
`TauCeti.Huber.IsPseudoUniformizer`, `TauCeti.Huber.Pair.Hom`.

The existence step rests on the fact that good affinoid perfectoids cover a tilde-limit and are
stable under rational subsets (*good-affinoid-perfectoid-basis*): let `(X_i)` be a cofiltered
system of analytic adic spaces with qcqs transition maps and `X` a perfectoid space with a
compatible cone such that `|X| → lim |X_i|` is a homeomorphism. (i) If `X ~ lim X_i`, then `X` is
covered by good affinoid perfectoids with `V` affinoid at their level. (ii) If `(W, i_0, V)` is good
and `U ⊆ W` is a rational subset, then `U` is good at some level `j → i_0` with a quasicompact open
`V′ ⊆ V_a` defined by the descended inequalities; if the later inverse image `V_a` is
affinoid, `V′` can be taken rational in `V_a`. In general it is the quasicompact open
`{|f′_k| ≤ |g′| ≠ 0} ⊆ V_a`, as in P7.11. (iii) Hence the good
affinoid perfectoids form a basis of the topology of `X`. (iv) Conversely, if `X` is covered by
good affinoid perfectoids, then `X ~ lim X_i`. Consequently the conditions (b), (b′), (b″) of
*perfectoid-tilde-limit* are equivalent given (a) ([Survey] §2, Proposition 2.22, p. 11
(statements (i), (ii)); [Survey] §2, proof of Proposition 2.22, p. 11 (descent of the quasicompact
open to a finite level, and density on rational subsets); [Stacks] Tag 0A30, Lemma 5.24.6
(quasicompact opens of a cofiltered limit of spectral spaces come from a finite level)). *Needs:*
*perfectoid-tilde-limit*; *good-affinoid-perfectoid*; P2.3; P5; AdicSpacesPartII R0.19.5–R0.19.8 and P5.4’s perfectoid limit comparisons; Tau
Ceti `TauCeti.ValuationSpectrum.exists_mem_nhds_forall_rationalSubset_eq_of_sub_mem`,
`TauCeti.ValuationSpectrum.spaComap_preimage_rationalSubset`,
`TauCeti.ValuationSpectrum.rationalSubset`.

**Checks for the good affinoid basis.**

- In the perfectoid disc tower of `goodAffinoid_test_disc`, the rational subdisc descends
  to the displayed rational domain in an affinoid inverse image.
- The empty rational subset descends to the empty quasicompact open at any level.
- Affinoidness of `V` is not preserved by a qcqs transition: the inverse image of
  `V = Spa(K, K°)` under `P¹_K → Spa(K, K°)` is the nonaffinoid projective line.
  It therefore gives no affinoid-spectrum rational-domain description of that inverse image.

### P7.11 represented functors and finite-level descriptions

**Finite-level analytic carriers.** (*finite-level-analytic-carriers*)
`TowerGeometry.IsFinite` tests finiteness using cartesian affinoid charts and
AdicSpacesPartII R0.13’s finite pair maps. `TowerGeometry.IsOverZp` requires
topological nilpotence of `p` on each affinoid chart. These are the finite-transition
and residue-characteristic hypotheses of [HJ] p. 26 and Lemma 5.11, p. 32;
[P8](../PerfectoidTowers/README.md#layer-p8-finite-quotients-and-closed-perfectoid-loci-in-towers) uses the same carriers.

**The raw compact-open covering condition; the v-site is DiamondsAndVStacks'.**
(*perfectoid-untilt-test-sheaves*) P7 uses no topology on perfectoid spaces. The only covering
condition it refers to is the raw one: for a category `C : Type v` with `Category.{u} C` and a
functor `sp : C ⥤ TopCat.{u}`, a `Type u`-indexed family covers when its images cover each compact
open exactly by finitely many compact opens, Mathlib's `IsCompactOpenCovered` ([ECD] Definition
8.1(iii), p. 39). The v-site built on that condition belongs to DiamondsAndVStacks: the precoverage
`vPrecoverage` and the generated Grothendieck topology `vTopology` to its D2.2, subcanonicity and
v-descent to its D2.7; it is not an input here. No local spectrality is required to define either
object; nonempty spaces with no nonempty compact opens need not exclude the empty cover. Their
specializations to `Perfd p`, along the inclusion `ObjectProperty.ι (isPerfectoidSpace p)` followed
by the underlying space, are DiamondsAndVStacks' as well.

Checks of the covering condition and of the precoverage and topology that DiamondsAndVStacks D2.2
builds on it (specified in this README only):

| Check | Instance and outcome |
| --- | --- |
| `vPrecoverage_point` | The identity presieve covers the terminal object of `Fin 2`, with the constant one-point space functor. |
| `vPrecoverage_empty` | An object with empty underlying space admits the empty covering presieve. |
| `vPrecoverage_nonempty` | At the terminal object of the constant one-point `Fin 2` diagram, the empty presieve is not covering. |
| `vTopology_point` | The sieve generated by the identity covers the same one-point object. |
| `vTopology_empty` | The constant empty-space functor on `Fin 2` generates the maximal topology. |
| `vTopology_nonempty` | The empty sieve does not cover the terminal object for the constant one-point functor. |

The specialized Checks `vPrecoverage_identity`, `vPrecoverage_empty`,
`vPrecoverage_nonempty`, `vTopology_identity`, `vTopology_empty` and
`vTopology_nonempty` test the identity, empty perfectoid space, and nonempty
perfectoid space, respectively. Local spectrality makes the last case valid.
The comparisons below are stated for represented functors, not sheaves:
`VQuotient.representedFunctor p` sends an analytic Yoneda-adic space `X` to the functor
`Hom(−, X)` on perfectoid test spaces (P7.10). That these functors are v-sheaves for
DiamondsAndVStacks' topology (subcanonicity, D2.7; [ECD] Theorem 8.7, p. 43), and that they agree
there with the sheafified restricted Yoneda functor on perfectoid objects, are downstream statements
of DiamondsAndVStacks; nothing here uses them.
Checks `representedFunctor_identity`, `representedFunctor_empty` and
`representedFunctor_two_points` test the identity map, empty target with nonempty source, and a
product with two points. This is the carrier for the represented comparison below;
DiamondsAndVStacks supplies its downstream diamond interpretation.

**Tilde-limits and inverse limits of represented functors.**
(*represented-functor-comparison*) Work with the represented functors on perfectoid test spaces,
`VQuotient.representedFunctor p : YonedaAdicSpace ⥤ ((Perfd p)ᵒᵖ ⥤ Type 2)`; no topology is used.
For spaces over `Spa(Z_p, Z_p)`, these functors describe the diamond functors with the
untilt retained. The diamond interpretation, and the sheaf property on DiamondsAndVStacks' v-site,
are downstream comparisons in DiamondsAndVStacks D2.7 and D5–D6; the statements here use only the
represented functors.

(i) Let `D` be a small cofiltered system of honest analytic adic spaces over `Z_p`
with qcqs transition maps, and let `c` be a perfectoid tilde-limit cone.
`VQuotient.tildeLimit_represented_comparison` says that `representedFunctor` sends `c` to a
limit cone and gives an isomorphism from its vertex to `limit (D ⋙ representedFunctor p)`.
Its composite with each limit projection is `representedFunctor(c.π_i)`. The theorem also
gives `|c.pt| ≃ lim_i |D_i|`, with the same projections on points. Thus the functor
isomorphism and the topological comparison both use the given cone. On DiamondsAndVStacks' v-site
this becomes, downstream, `X^◇ ≅ lim_i X_i^◇`, compatibly with
`|X^◇| = |X| ≅ lim_i |X_i| = lim_i |X_i^◇|`.
`VQuotient.tildeLimit_represented_of_adic` gives the represented limit for a genuine
adic-space cone after `YonedaAdicSpace.ofAdicSpace`, with the tilde-limit hypothesis
on that Yoneda cone, a perfectoid vertex, analytic levels over `Z_p`, and qcqs
transitions. The general adic-space predicate is supplied by
`AdicSpacesPartII.AdicSpace.IsTildeLimit`.
The presheaf factorization theorem remains `IsTildeLimit.lift` in P7.10
([SW13] Proposition 2.4.5, pp. 20–21).

(ii) For a filtered diagram `B : J ⥤ CAff` of sheafy Tate pairs over `Z_p`, take a
completed-colimit cocone `b` satisfying `IsCompletedColimitOfTower`, with compatible
rings of definition and a common pseudouniformizer. Set
`U = CAff.of (Pair.uniformization b.pt.P)`, using AdicSpacesPartII R0.4.
`VQuotient.completed_colimit_uniformization_comparison` constructs the cone from
`Spa(U)` using the composites `B_j → b.pt → U`, proves that its image under
`representedFunctor` is a limit, and gives the represented form of
`lim_j Spd(B_j) ≅ Spd(U)`. The inverse isomorphism followed by
each limit projection is induced by that composite of pair maps. It also identifies
`|Spa(U)|` with the compatible finite-level points. If `U.A` is perfectoid, the same
cone is a perfectoid tilde-limit. The general completed-affinoid tilde-limit theorem
is the completed-direct-limit criterion of AdicSpacesPartII R0.20.1's *tilde-limits-affinoid-criterion*
(AdicSpacesPartII R0.20.1); P7.11 adds its
represented-functor comparison, using the uniformization universal property
([SW13] Proposition 2.4.2, p. 20; [KL15] Definition 2.8.13, p. 63).
The diamond interpretation gives `|lim_j X_j^◇| ≅ |Spa(U)|` downstream.

(iii) Suppose the levels are locally noetherian analytic over `Z_p`, the transition
maps are finite, the vertex of `c` is perfectoid, and
`IsLimit ((representedFunctor p).mapCone c)` holds. No density hypothesis is imposed.
`VQuotient.represented_limit_homeomorph` gives the homeomorphism induced by `c`.
`VQuotient.represented_limit_affinoid_basis` says that the inverse images of
finite-level affinoid opens which are affinoid perfectoid form a neighborhood basis
([HJ] Lemma 5.11, p. 32). If all levels are qcqs,
`VQuotient.represented_limit_quasicompact_open` descends each quasicompact open to
one finite level; the same statement applies after restriction over a qcqs open.
For an affinoid tower with an affinoid perfectoid representative,
`VQuotient.represented_limit_rational_subset` descends each rational subset to
finite rational data generating an open ideal. These are the finite-level
descriptions in the proof of [HJ] Lemma 5.11, p. 32, together with the spectral
quasicompact-open descent of P5.4.

The density condition remains a separate assertion.
`VQuotient.represented_uniformization_density_iff` takes the completed affinoid
system of (ii), a perfectoid pair `R`, and a map `q : b.pt → R` whose induced
`representedFunctor` map is an isomorphism. The union of the images of `B_j → R` is dense
if and only if the uniformization of `b.pt` is perfectoid. Apply this on the
finite-level affinoid inverse images from (iii) to obtain the density criterion
for `W = φ_a⁻¹(Spa(A_a, A_a⁺))`; when it holds, `W` is the spectrum of that
uniformization by its universal property and full faithfulness on perfectoid
test spaces ([SW13] Proposition 2.4.5, pp. 20–21; [KL15] Definition 2.8.13, p. 63;
[HJ] Lemma 5.11, p. 32).

**Checks.** `tildeLimit_represented_comparison_identity` gives the limit cone for a
constant perfectoid space. `tildeLimit_represented_comparison_empty` gives no
sections of the empty affinoid over a perfectoid field test space.
`represented_uniformization_density_iff_wrong_cone` excludes `Spa(Cp) → Spa(Qp)`
from the isomorphism hypothesis on represented functors, even though their underlying spaces each
have one point. These three Checks are specified in this README only.

*Needs:* P7.10 *tilde-limit-perfectoid-uniqueness*;
*tilde-limit-from-completed-colimit*; *rational-subsets-from-finite-level*;
*perfectoid-tilde-limit*; AdicSpacesPartII R0.20.1 (the completed-affinoid criterion);
AdicSpacesPartII R0.4 (uniformization) and R0.13 (finite pair maps); AdicEtaleGeometry A1.16
(`CAff`, `YonedaAdicSpace` and `ofAdicSpace`); P2; P5; `VQuotient.representedFunctor` (the
represented-functor carrier of P7.11).

The completed-direct-limit supplier in AdicSpacesPartII R0.20.1 gives tilde-limits
(*tilde-limit-from-completed-colimit*), with P7 using its Tate corollary:
let `(A_i, A_i⁺)_{i∈I}` be a filtered direct system of complete Huber pairs (Tau Ceti `Huber.Pair`)
with continuous transition maps, and assume there are rings of definition `A_{i,0} ⊆ A_i` mapped
into one another and finitely generated ideals of definition `I_i ⊆ A_{i,0}` with
`I_j = I_i A_{j,0}` for `j ≥ i`. Let `A := colim_i A_i` with the topology making
`A_0 := colim_i A_{i,0}` an open subring with the `I`-adic topology, `I := colim_i I_i`; let `A⁺ :=`
the integral closure in `A` of the image of `colim_i A_i⁺`, and `(Â, Â⁺)` the completion. Then (i)
`(A, A⁺)` is a Huber pair and the maps `(A_i, A_i⁺) → (A, A⁺)` are continuous; (ii) a valuation on
`A` is a continuous valuation with `|A⁺| ≤ 1` iff its restrictions to all `A_i` are, so the
restriction map `Spa(Â, Â⁺) = Spa(A, A⁺) → lim_i Spa(A_i, A_i⁺)` is a bijection, and it is a
homeomorphism; (iii) every rational subset of `Spa(A, A⁺)` is the preimage of a rational subset of
some `Spa(A_i, A_i⁺)` — in the Tate case (ii) and (iii) use
AdicSpacesPartII R0.19.7 (*spa-of-filtered-colimit-of-tate-pairs*), and the Huber-pair generality with ideals of definition
`I_i` belongs to AdicSpacesPartII R0.20.1; (iv) `colim_i A_i → Â` has dense image. Consequently, if the
`X_i = Spa(A_i, A_i⁺)` are adic spaces (`A_i` sheafy) and `Â` is perfectoid (a hypothesis of this
clause, supplied by P7.13, by P7.14, or by the application), then `Spa(Â, Â⁺) ~ lim_i X_i`. In the
Tate case with compatible pseudouniformiser `ϖ ∈ A_{i_0,0}`, the hypotheses hold with
`I_i = ϖA_{i,0}` ([SW13] §2.4, Proposition 2.4.2, p. 20; [SW13] §2.4, proof of Proposition 2.4.2,
p. 20 (step (ii))). *Needs:* *perfectoid-tilde-limit*; P5; Mathlib `Ring.DirectLimit`; Tau Ceti
`TauCeti.Huber.Pair`, `TauCeti.ValuationSpectrum.spa`, `TauCeti.ValuationSpectrum.spa_integralClosure`,
`TauCeti.ValuationSpectrum.spaComap`, `TauCeti.ValuationSpectrum.spaComap_preimage_rationalSubset`,
`TauCeti.Huber.IsRingOfIntegralElements`.

The Tate data predicate `IsCompletedColimitOfTower` uses a filtered category `J`, Huber pairs `P_j`,
a target Huber pair `Q` on `R`, transition maps `f` and a cone `g`, with identity, composition
and cone laws. It does not require the finite rings to be complete or perfectoid. Its additional
data are rings of definition `A_{j,0}` preserved by `f` and `ϖ ∈ R`; it requires `R` Hausdorff
and complete, `ϖ` a pseudouniformiser lifted from one finite ring of definition, and the closure
`R₀` of the subring generated by `⋃ g_j(A_{j,0})` to be an open ring of definition with ideal
`(ϖ)`. The union `⋃ g_j(A_j)` is dense, `Q.plus` is the closure of the integral closure of the
subring generated by `⋃ g_j(P_j.plus)`, and divisibility by `ϖ^n` in `R₀` is detected at a finite
stage for each finite-stage element and lift of `ϖ`. This predicate records the Tate
tower data used by `FrobeniusTower.isCompletedDirectLimit`; it does not define another
tilde-limit relation. The ideals are indexed by `n ∈ ℕ`; at `n = 0` divisibility by `1`
imposes no restriction.

`IsCompletedColimitOfTower.isTildeLimit_affinoid` is the corollary for a small filtered
`B : J ⥤ CAff` with cocone `b`, sheafy levels and sheafy apex, and the predicate above
on `B.map` and `b.ι.app`. Completeness and Hausdorffness come from `CAff`;
perfectoidness is not assumed. Apply the completed-direct-limit criterion of AdicSpacesPartII R0.20.1's
*tilde-limits-affinoid-criterion* after restricting to the
cofinal category under a stage carrying a pseudouniformiser `w` mapping to `ϖ`.
At each later stage use its image as the sole generator (`m = 1`), and use `ϖ` at
the apex. The existing rings of definition are retained with these principal ideals.
Density supplies approximation modulo powers, `exists_dvd_of_dvd` supplies finite-stage
detection, and `plus_eq` supplies the closed integral closure defining the plus ring.
The supplier gives the point-space homeomorphism; cofinality transports it to the
original system and `h.dense` supplies the single-affinoid density clause.
AdicSpacesPartII R0.20.1 supplies the criterion on complete sheafy carriers ([SW13] Proposition 2.4.2, p. 20).

For `B : J ⥤ CAff` and its cocone `b`,
`PerfectoidSpace.IsCompletedColimitOfTower.isTildeLimit` assumes the same completed-colimit
predicate on `B.map`, `b.ι.app`, `B₀` and `ϖ`, with every level sheafy and Tate and
`b.pt.A` a perfectoid Tate ring at the fixed prime `p`. It concludes
`PerfectoidSpace.IsTildeLimit p (B.op ⋙ S) (S.mapCone b.op)`, with the same `S` as above.
In particular, this states `Spa(Â, Â⁺) ~ lim_j Spa(A_j, A_j⁺)` on the global carrier;
sheafiness of the apex follows from perfectoidness (P2.6). It combines
`IsCompletedColimitOfTower.isTildeLimit_affinoid` with the preceding bridge, as the
perfectoid specialization of AdicSpacesPartII R0.20.1's completed-direct-limit criterion
([SW13] Proposition 2.4.2, p. 20).

**Checks for `IsCompletedColimitOfTower`.**

- `completedColimit_perfectoid` (compatibility) — the completed-colimit hypotheses with
  sheafy Tate levels and a perfectoid apex give the global perfectoid tilde-limit on the
  induced cone, by `IsCompletedColimitOfTower.isTildeLimit`.

- `completedColimit_test_const` (degenerate) — a complete perfectoid Tate pair `(R, R⁺)`
  with `ϖ ∈ R⁺` a pseudouniformiser is its own completed constant system, taking `A₀ = R⁺`.
- `completedColimit_test_Qp` (computation) — the constant pair `(ℚ_p, ℤ_p)` with
  `A₀ = ℤ_p` and `ϖ = p` satisfies the predicate, although `ℚ_p` is not perfectoid.
- `completedColimit_test_not_dense` (non-example) — the constant `(ℚ_p, ℤ_p)` system with
  cone to `(ℂ_p, O_{ℂ_p})` fails density, even though both spectra are one point.
- `completedColimit_test_zeroParameter` (non-example) — the constant `(ℚ_p, ℤ_p)` system
  with `ϖ = 0` fails the pseudouniformiser condition.

Quasicompact opens and rational subsets of a tilde-limit come from finite level
(*rational-subsets-from-finite-level*): let `X ~ lim X_i` be a perfectoid tilde-limit. Clauses
(i) and (ii) use AdicSpacesPartII R0.19.8 (*quasicompact-opens-in-limits-of-spectral-spaces*)
and P5.4's *descent-of-quasicompact-opens*, applied to the homeomorphism `|X| ≅ lim |X_i|`: (i) every
quasicompact open `U ⊆ X` is `φ_i^{-1}(U_i)` for some `i` and a quasicompact open `U_i ⊆ X_i`; if
`U_i, U′_i ⊆ X_i` are quasicompact open with `φ_i^{-1}(U_i) ⊆ φ_i^{-1}(U′_i)`, then
`f_a^{-1}(U_i) ⊆ f_a^{-1}(U′_i)` for some `a : j → i`, and likewise for equalities, finite unions
and finite intersections; (ii) a finite cover of a quasicompact open `U ⊆ X` by quasicompact opens
is the preimage of a finite cover at some finite level; what is added is (iii) if `(W, i_0, V)` is good with
`V = Spa(A, A⁺)` affinoid and `U = W ∩ {|f_k| ≤ |g| ≠ 0}` is a rational subset of `W`, there are
`j → i_0` and `f′_k, g′ ∈ O(V_a)` with `U = φ_j^{-1}(U_a(f′/g′))` for the rational subset
`U_a(f′/g′)` of `V_a` when `V_a` is affinoid; in general `U = φ_j^{-1}` of the quasicompact open
`{|f′_k| ≤ |g′| ≠ 0} ⊆ V_a` ([CHJ] §2, proof of Theorem 2.17, p. 16; [Survey] §2, proof of
Proposition 2.22 (ii), p. 11; [Stacks] Tag 0A30, Lemma 5.24.6 (part (i))). *Needs:*
*perfectoid-tilde-limit*; *good-affinoid-perfectoid*; *good-affinoid-perfectoid-basis*;
AdicSpacesPartII R0.19.5–R0.19.8 and P5.4’s perfectoid limit comparisons; P5; Tau Ceti
`TauCeti.ValuationSpectrum.exists_mem_nhds_forall_rationalSubset_eq_of_sub_mem`,
`TauCeti.ValuationSpectrum.spaComap_preimage_rationalSubset`.

Tilde-limits are invariant under initial changes of index category (*tilde-limit-cofinal-change*):
let `u : J → I` be an initial functor between small cofiltered categories (Mathlib
`CategoryTheory.Functor.Initial`; for directed sets, a coinitial subset). For a system
`(X_i)_{i∈I}` of analytic adic spaces with qcqs transition maps, a perfectoid space `X` and a
compatible cone `(φ_i)`, `X ~ lim_{i∈I} X_i` iff `X ~ lim_{j∈J} X_{u(j)}` for the restricted cone
`(φ_{u(j)})`. In particular: (a) for every `i_0 ∈ I` the forgetful functor `I/i_0 → I` is initial,
so a tilde-limit depends only on the tail of the system; (b) for `I = ℕ` (as an inverse system) any
infinite subsequence may be used; (c) for level families of compact open subgroups with a
cofinality witness (see *level-cofinality-witness*), the full tower and the witnessed subtower have the
same perfectoid tilde-limits ([Sch13E] item (1), p. 1; [CHJ] §2, Theorem 2.8 and the remarks after
it, p. 11). *Needs:* *perfectoid-tilde-limit*; Mathlib `CategoryTheory.Functor.Initial`,
`CategoryTheory.Functor.Initial.limitIso`, `CategoryTheory.Functor.initial_of_exists_of_isCofiltered`,
`TopCat.limitCone`.

### P7.12 locality and gluing of perfectoid tilde-limits

Perfectoid tilde-limits are local on a finite level and glue; the restriction, finite étale and
étale base-change results used in the gluing are stated with it.

**Locality and gluing of perfectoid tilde-limits.** (*tilde-limit-gluing*) Let `(X_i)_{i∈I}` be a
cofiltered system of analytic adic spaces with qcqs transition maps, `i_0 ∈ I`, and `(V^α)_α` an
open cover of `X_{i_0}` (a cover by opens of one finite level), with preimages
`V^α_a := f_a^{-1}(V^α) ⊆ X_j` (`a : j → i_0`). (i) Locality: a perfectoid space `X` with a
compatible cone satisfies `X ~ lim X_i` iff for every `α` the open subspace
`U^α := φ_{i_0}^{-1}(V^α)` satisfies `U^α ~ lim_a V^α_a`. (ii) Gluing: if for every `α` the system
`(V^α_a)_a` has a perfectoid tilde-limit `U^α`, then there is a perfectoid space `X` with a
compatible cone and `X ~ lim X_i`; `X` is glued from the `U^α` along the open subspaces
`U^{αβ} ⊆ U^α` that are the restrictions of `U^α` to `V^α ∩ V^β` (see *tilde-limit-rational-restriction*),
identified by the unique isomorphisms of *tilde-limit-perfectoid-uniqueness* (which also give the
cocycle condition). (iii) Torsion-paper form: suppose that for all large `m` the `X_m` are the
generic fibres of a tower and that `|lim_m X_m|` is covered by subsets `W` that are preimages of
affinoid opens `V_m = Spa(R_m, R_m⁺) ⊆ X_m` for all sufficiently large `m` (the level at which the
affinoid description starts may depend on `W`; compatible with the transition maps) such that,
with `R_∞⁺` the `ϖ`-adic completion of `colim_m R_m⁺` and `R_∞ = R_∞⁺[1/ϖ]`, `(R_∞, R_∞⁺⁺)` is
perfectoid (`R_∞⁺⁺` the integral closure of `R_∞⁺`); then there is a unique perfectoid space `X_∞`
with `|X_∞| = lim |X_m|` and `X_∞ ~ lim X_m`, whose restriction to each `W` is `Spa(R_∞, R_∞⁺⁺)`
([Torsion] §III.3, Definition III.3.5 (ii), p. 56 (perfectoid subsets of the topological limit are
those covered by affinoid perfectoid subsets); [Torsion] §III.3, after Definition III.3.5, p. 57
(the perfectoid structure is obtained by gluing the affinoid pieces `Spa(R_∞, R_∞⁺)`); [Torsion]
§III.2.3, proof of Corollary III.2.19, p. 43 (existence on affinoids by the completed direct limit
and uniqueness by representability, which is what makes gluing possible)). *Needs:*
*perfectoid-tilde-limit*; *tilde-limit-perfectoid-uniqueness*;
*tilde-limit-from-completed-colimit*; *tilde-limit-rational-restriction*; P2.10; P2.

Restriction of a tilde-limit to compatible open and rational subsets
(*tilde-limit-rational-restriction*): let `X ~ lim X_i` be a perfectoid tilde-limit, `i_0 ∈ I`,
`U_{i_0} ⊆ X_{i_0}` an open subset at one finite level (in particular a rational subset of an
affinoid open), `U_a := f_a^{-1}(U_{i_0}) ⊆ X_j` for `a : j → i_0` (the transition maps restricted
to the `U_a` stay qcqs), and `U := φ_{i_0}^{-1}(U_{i_0})`, an open perfectoid subspace of `X` (P2).
Then `U ~ lim_{a ∈ I/i_0} U_a` (the slice category `I/i_0`, which is initial in `I` by
*tilde-limit-cofinal-change*). Moreover, if `(W, i_0, V)` is good with `V = Spa(A, A⁺)` affinoid and
`U_{i_0} = V(f/g)` is a rational subset of `V`, then `W ∩ U = Spa(R⟨f/g⟩, R⟨f/g⟩⁺)` is the rational
localisation of `(R, R⁺) = (O(W), O⁺(W))` at the pulled-back parameters (P2.3), it is good at level
`i_0` with the rational subset `V(f/g)` (*good-affinoid-perfectoid-basis* (ii), P7.10), and
`colim_a O_{X_j}(V_a(f/g)) → R⟨f/g⟩` has dense image
([BHW] §5.2, proof of Theorem 5.11, p. 22 of the arXiv manuscript (restriction of a perfectoid tilde-limit to an open
subset pulled back from finite level); [Survey] §2, proof of Proposition 2.22, p. 11 (density on
rational subsets)). *Needs:* *perfectoid-tilde-limit*; *good-affinoid-perfectoid*;
*good-affinoid-perfectoid-basis*; *tilde-limit-cofinal-change*; P2; P2.3.

Finite étale base change of perfectoid tilde-limits (*tilde-limit-finite-etale-base-change*): let
`X ~ lim X_i` be a perfectoid tilde-limit, `i_0 ∈ I` and `g : Y_{i_0} → X_{i_0}` a finite étale
morphism of analytic adic spaces (AdicEtaleGeometry A1.3). Put
`Y_a := Y_{i_0} ×_{X_{i_0}} X_j` for `a : j → i_0` (finite étale over `X_j`) and
`Y := Y_{i_0} ×_{X_{i_0}} X`. Then `Y` exists in adic spaces, is a perfectoid space, `Y → X` is
finite étale (P3.16, P3.10), and `Y ~ lim_{a ∈ I/i_0} Y_a`. Affinoid formula: if `(W, i_0, V)` is
good with `V = Spa(A, A⁺)` affinoid and `g^{-1}(V) = Spa(B, B⁺)` with `B ∈ FÉt(A)` and `B⁺` the
integral closure of `A⁺` (AdicEtaleGeometry A1.2), then the preimage of `W` in `Y` is
`Spa(B ⊗_A R, (B ⊗_A R)⁺)` with `(B ⊗_A R)⁺` the integral closure of `R⁺`, and
`colim_a (B ⊗_A O(V_a)) → B ⊗_A R` has dense image; it is good at level `i_0` with
`V′ = g^{-1}(V)` ([Sch12] §7, Lemma 7.3 (iii), p. 40 (existence and perfectoidness of the base
change); [Sch12] §7, Proposition 7.10, p. 43 (characteristic `0`); [Torsion] §III.2.4, before Lemma
III.2.25, p. 49; [BHW] §5.2, proof of Theorem 5.11, p. 22 of the arXiv manuscript (a finite étale tower over a perfectoid
tilde-limit); [Sch13] §4, Lemma 4.5, p. 23 (the analogous base change for affinoid perfectoid
pro-étale presentations)). *Needs:* *perfectoid-tilde-limit*; *good-affinoid-perfectoid*;
*good-affinoid-perfectoid-basis*; *good-affinoid-uniform-completion*;
*tilde-limit-rational-restriction*; *tilde-limit-gluing*; AdicEtaleGeometry A1.2 and A1.3; P3.16;
P3.10; P3.17; P3.

**General base-change supplier.** (*tilde-limit-base-change-comparison*)
`AdicSpacesPartII.AdicSpace.IsTildeLimit.baseChange` is the R0.20.2 theorem for
the intrinsic adic-space relation. Its hypotheses include qcqs spaces, adic
transitions and cone maps, and a base-change map locally of finite type on charts.
For analytic étale base change, apply it on qcqs affinoid charts, transport by
`AdicSpace.IsTildeLimit.ofAdicSpace_iff`, and use P3.15 to prove perfectoidness
of the apex. The resulting Yoneda theorem is
`PerfectoidSpace.IsTildeLimit.etale_baseChange`; its hypotheses specify pullback
squares at every level of `Over i` and at the apex, and compatibility with both
projections. `PerfectoidSpace.IsTildeLimit.finiteEtale_baseChange` and
`PerfectoidSpace.IsTildeLimit.open_baseChange` are its finite étale and open-immersion
corollaries. They use AdicEtaleGeometry's `YonedaAdicSpace.IsFiniteEtale`,
`IsEtaleLocalDescription` and `IsOpenImmersion`; only preservation of perfectoidness
and carrier transport are added here ([SW13] Proposition 2.4.3, p. 20).

Étale base change of perfectoid tilde-limits (*tilde-limit-etale-base-change*): let `X ~ lim X_i` be a
perfectoid tilde-limit, `i_0 ∈ I` and `g : Y_{i_0} → X_{i_0}` an étale morphism of analytic adic
spaces in the sense of AdicEtaleGeometry A1.3 (locally an open immersion into a space finite
étale over an open subset). Put `Y_a := Y_{i_0} ×_{X_{i_0}} X_j` and `Y := Y_{i_0} ×_{X_{i_0}} X`.
Then `Y` exists, is a perfectoid space étale over `X` (P3.15), and `Y ~ lim_{a ∈ I/i_0} Y_a`. The
adic-space statement — a tilde-limit is preserved by base change along a morphism locally of weakly
finite type, [SW13] Proposition 2.4.3 by reference to Huber's Remark 2.4.3 — is AdicSpacesPartII
R0.20.2 (*tilde-limit-base-change*); what is proved here is the case in which the base change of
the perfectoid space `X` is again perfectoid, the sectionwise version of [Sch12] Proposition 7.16; for non-étale `g` the fibre
product `Y` need not be perfectoid and is not considered here (Zariski-closed base changes are
[P8.21](../PerfectoidTowers/README.md#layer-p8-finite-quotients-and-closed-perfectoid-loci-in-towers)); an open immersion `U_{i_0} ⊆ X_{i_0}` recovers *tilde-limit-rational-restriction* ([SW13]
§2.4, Proposition 2.4.3, p. 20; [Sch12] §7, Proposition 7.16, p. 44; [BHW] §5.2, proof of Theorem
5.11, p. 23 of the arXiv manuscript (étale base change)). *Needs:*
*perfectoid-tilde-limit*; *tilde-limit-finite-etale-base-change*;
*tilde-limit-rational-restriction*; *tilde-limit-gluing*; AdicEtaleGeometry A1.3;
AdicSpacesPartII R0.20.2 (*tilde-limit-base-change*); P3.16; P3.15; P3.

### P7.13 the perfectoid Frobenius criterion

The criterion makes the completed generic fibre of a Frobenius-controlled tower a perfectoid
tilde-limit; its three ingredients are the Frobenius bijection with the uniformity bound, the tilt
of the limit, and the perfection of a characteristic-`p` space as the tilde-limit along relative
Frobenius.

**The perfectoid Frobenius criterion: Frobenius-controlled towers have perfectoid tilde-limits.**
(*perfectoid-frobenius-criterion-for-towers*) Let `(A_•⁺, ϖ, m_0, w)` be a Frobenius-controlled
integral tower (*frobenius-controlled-integral-tower*; the eventual Frobenius lifting modulo `ϖ^p`
is the only input beyond `ϖ`-torsion-freeness and `ϖ`-adic completeness) with generic fibres `(A_m, A_m⁺⁺)` and
completion `(A_∞, A_∞⁺⁺)` (see *frobenius-tower-completion*). Then: (i) `A_∞` is a perfectoid Tate ring
in the sense of [ECD] Definition 3.1 (complete, uniform, with the pseudouniformiser `ϖ` satisfying
`ϖ^p | p` in `A_∞°` and `Φ : A_∞°/ϖ ≅ A_∞°/ϖ^p`), when `A_∞ ≠ 0`, of characteristic `p` exactly when `p = 0` in
`A_∞` (so whenever some `A_m⁺` has characteristic `p`; a non-injective `t_m` may kill `p` at a
finite level even if `A_0⁺` has characteristic `0`), and `(A_∞, A_∞⁺⁺)` is a perfectoid Tate
pair; (ii) the restriction map identifies
`|Spa(A_∞, A_∞⁺⁺)|` with `lim_m |Spa(A_m, A_m⁺⁺)|`, and, when the `A_m` are sheafy (for instance
strongly noetherian, as for affinoids of rigid spaces), `X_∞ := Spa(A_∞, A_∞⁺⁺) ~ lim_m X_m` in the
sense of *perfectoid-tilde-limit*, `X_∞` is unique up to unique isomorphism, and it represents
`lim_m Hom(−, X_m)` on perfectoid spaces; (iii) global form: let `(𝔛_m)` be a tower of flat
`ϖ`-adic formal schemes (over a `ϖ`-adically complete base) with generic fibres `X_m` analytic adic
spaces, and suppose `𝔛_{m_0}` has an affine open cover by `Spf A_{m_0}⁺` whose preimages `Spf A_m⁺`
(`m ≥ m_0`) form Frobenius-controlled towers compatibly with restriction; then the generic fibres
have a perfectoid tilde-limit `X_∞ ~ lim X_m`, glued from the `Spa(A_∞, A_∞⁺⁺)`
(see *tilde-limit-gluing*), which is the generic fibre of the formal scheme `lim 𝔛_m` (limit in formal
schemes, affine-locally `Spf A_∞⁺`). This theorem concerns towers whose finite levels are not
perfectoid; the case of perfectoid finite levels is P7.14 ([Torsion] §III.2.3, proof of Corollary
III.2.19, p. 42 (the eventual Frobenius lifting condition: transition maps equal to relative
Frobenius modulo `p^{1−ε} = ϖ^p` with `ϖ = p^{(1−ε)/p}`); [Torsion] §III.2.3, proof of Corollary
III.2.19, p. 43 (the tilde-limit and its uniqueness from the completed limit; the completed limit is
a perfectoid integral algebra in the sense of [Sch12] Definition 5.1(ii), reached here over no base
field via [BMS] Lemmas 3.10 and 3.21); [BHW] §5.2, proof of Theorem 5.11, p. 22 of the arXiv manuscript (the
Hilbert-modular instance: relative Frobenius modulo `p^{1−δ}`); [Sch13] §4, Example 4.4, p. 23 (the
torus tower over a perfectoid field)). *Needs:* *frobenius-controlled-integral-tower*;
*frobenius-tower-completion*; *frobenius-tower-uniformity-bound*;
*tilde-limit-from-completed-colimit*; *tilde-limit-perfectoid-uniqueness*; *tilde-limit-gluing*;
*perfectoid-tilde-limit*; P1; P1.1; P2.6; P2.10.

Frobenius on the completed tower is bijective modulo `ϖ`, and the uniformity bound
`ϖA_∞° ⊆ A_∞⁺` holds (*frobenius-tower-uniformity-bound*): let `(A_•⁺, ϖ, m_0, w)` be a
Frobenius-controlled integral tower with completion `(A_∞⁺, A_∞, A_∞⁺⁺)`
(see *frobenius-tower-completion*); only the terms `m ≥ m_0` matter. Then: (i)
`Φ : A_∞⁺/ϖ → A_∞⁺/ϖ^p`, `x ↦ x^p`, is a bijection (in fact `colim_m A_m⁺/ϖ → colim_m A_m⁺/ϖ^p`
is, and these are the quotients of `A_∞⁺`); (ii) `A_∞⁺` is a `ϖ`-torsion-free, `ϖ`-adically
complete ring with `p ∈ ϖ^pA_∞⁺` and bijective Frobenius modulo `ϖ`, hence an integral perfectoid
ring in the sense of [BMS] Definition 3.5 (Lemma 3.10(ii)); (iii) uniformity bound: `A_∞⁺ ⊆ A_∞°`
and `ϖ·A_∞° ⊆ A_∞⁺`; more precisely, after replacing `ϖ` by a unit multiple admitting a compatible
system of `p`-power roots ([BMS] Lemma 3.9), the cokernel of `A_∞⁺ → A_∞°` is killed by every
`ϖ^{1/p^n}`, i.e. `A_∞⁺ → A_∞°` is an almost isomorphism for the ideal `(ϖ^{1/p^∞})` (P1);
consequently `A_∞°` is bounded, `A_∞` is uniform, and the `ϖ`-adic topology on `A_∞⁺` is the
topology of the spectral norm of `A_∞`; (iv) `A_∞⁺⁺` satisfies
`A_∞⁺ ⊆ A_∞⁺⁺ ⊆ A_∞° ⊆ ϖ^{-1}A_∞⁺` ([Torsion] §III.2.3, proof of Corollary III.2.19, pp. 42–43
(the colimit Frobenius isomorphism `R_∞/ϖ ≅ R_∞/ϖ^p`); [BMS] §3.2, Lemma 3.10 (ii), p. 22 (step
(ii)); [BMS] §3.2, Lemma 3.21, p. 27 (passage to the Tate ring and the uniformity bound)). *Needs:*
*frobenius-controlled-integral-tower*; *frobenius-tower-completion*; P1 (P1.24, P1.25); Mathlib
`frobenius`, `AdicCompletion`, `Ring.DirectLimit`.

The tilt of the limit of a Frobenius-controlled tower (*frobenius-tower-tilt*): let
`(A_•⁺, ϖ, m_0, w)` be a Frobenius-controlled integral tower and `(S_•⁺, t, m_0, w′)` a
Frobenius-controlled integral tower of characteristic `p` (`p = 0` in `S_0⁺`), together with ring
isomorphisms `α_m : A_m⁺/ϖ^p ≅ S_m⁺/t^p` for `m ≥ m_0` compatible with the transition maps (and
with the `w`, `w′`), and quotient isomorphisms `β_m : A_m⁺/ϖ ≅ S_m⁺/t`
induced by the `α_m`. The quotient map uses `(t^p) ⊆ (t)`, from `1 ≤ p`;
parameter-preserving tower morphisms likewise carry `(ϖ^p)` into `(t^p)`. Require `α_m ∘ w_m = w′_m ∘ β_{m+1}`; the comparisons need not lift
to the integral rings. The completed tilt theorem `completed_tilt` states that the perfectoid
Tate ring `S_∞ = S_∞⁺[1/t]` of *perfectoid-frobenius-criterion-for-towers* is perfect and that
there is a unique topological ring isomorphism `e : A_∞♭ ≅ S_∞` preserving the Huber-pair
plus rings and satisfying the following characteristic equation. Write
`c : A_∞⁺/ϖ ≅ S_∞⁺/t` for the colimit of the `β_m`: for finite-level representatives with
`β_m([a]) = [b]`, require `c([ι_m(a)]) = [ι_m(b)]`. For every compatible integral root sequence
`x ∈ lim_{z ↦ z^p} A_∞⁺`, its image under `e` lies in `S_∞⁺`, and
`c([x⁽⁰⁾]) = [e(x)]`. Thus the square with sharp on the source and reduction modulo `t`
on the target commutes on the completed integral models, before taking integral closures.
The inverse Frobenius limits of these quotients determine the integral comparison uniquely
([BMS] Lemma 3.2(i), p. 19).
For every such `ϖ♭` with `(ϖ♭)⁽⁰⁾ = ϖu`, `u ∈ (A_∞⁺)ˣ`, also require
`e(ϖ♭) = tv` for some `v ∈ (S_∞⁺)ˣ`; such a choice of `ϖ♭` exists by [BMS] Lemma 3.9,
p. 22, and the sharp quotient has kernel `(ϖ♭)` by Lemma 3.10, pp. 22–23.
Under this isomorphism, `Spa(S_∞, S_∞⁺⁺)` is the tilt of
`Spa(A_∞, A_∞⁺⁺)`, and in the global form of the criterion the tilt of `X_∞` is the limit of the
characteristic-`p` tower ([Torsion] §III.2.3, proof of Corollary III.2.19, p. 43 (the tilt
identification from the congruence of the two towers); [Torsion] §III.2.3, before Definition
III.2.18, p. 42 (the characteristic-`p` side is a perfection)). *Needs:*
*perfectoid-frobenius-criterion-for-towers*; *frobenius-tower-completion*;
*frobenius-tower-uniformity-bound*; P1; P1.27.

Checks for the completed tilt theorem `completed_tilt`:

- `completed_tilt_identity_reduction`: for a characteristic-`p` tower compared with itself,
  the canonical tilt isomorphism sends an integral root sequence to its zeroth coordinate,
  so the quotient comparison is the identity.
- `completed_tilt_zero_reduction`: when both completed integral models are zero, their quotient
  rings admit exactly one isomorphism and every representative equation holds.
- `completed_tilt_swap_reduction`: on the split characteristic-two quotient `𝔽₂ × 𝔽₂`,
  swapping the factors sends `(1,0)` to `(0,1)`. It cannot satisfy the identity comparison;
  preservation of integral elements alone would not detect this change.

The perfection of a characteristic-`p` adic space is a tilde-limit along relative Frobenius
(*perfection-tilde-limit*): let `k` be a perfect complete nonarchimedean field of characteristic `p`
with pseudouniformiser `t` (more generally a perfect complete Tate ring of characteristic `p` with
`t ∈ k°` admitting all `p`-power roots), `(S, S⁺)` a complete Tate pair over `(k, k°)` with `S⁺`
flat over `k°` and `t`-adically complete. Put `S_m⁺ := S⁺ ⊗_{k°, F^{-m}} k°`, with the `k°`-algebra
structure from the right factor, and use the transitions `s ⊗ c ↦ s^p ⊗ c`. These are well defined:
`(as)^p ⊗ c = s^p ⊗ F^{-(m+1)}(a^p)c = s^p ⊗ F^{-m}(a)c`.
No identification of a Frobenius twist with `S` as a `k`-algebra is required. (i) This tower is
Frobenius-controlled with `ϖ = 1 ⊗ t` and `w_m` the base-change isomorphisms
of `FrobeniusTower.ofRelativeFrobenius`, specified for `FrobeniusTower.perfection` by its lift
specification (P7.5): `w_m([s ⊗ c]) = [s ⊗ c^p]`;
(ii) its completion is `S^{perf⁺} :=` the `t`-adic
completion of `colim_m S_m⁺` with
`S^{perf} = S^{perf⁺}[1/t]`, a perfect, hence perfectoid, Tate ring; (iii) if `S` is sheafy,
`Y^{perf} := Spa(S^{perf}, S^{perf⁺⁺}) ~ lim_m Y_m`, where
`Y_m = Spa(S_m⁺[1/t], S_m⁺)` and `Y = Y_0 = Spa(S, S⁺)`, the maps `|Y^{perf}| → |Y|`
are homeomorphisms, and `Y^{perf}` is the unique perfectoid space over `k` with this property; (iv)
for an analytic adic space `Y` over `k` covered by such affinoids, the `Y^{perf}` glue to a
perfectoid space `Y^{perf} ~ lim_Φ Y` (see *tilde-limit-gluing*). Here `lim_Φ Y` denotes this tower of base twists. The corresponding statement for flat
`t`-adic formal schemes `𝔜`: `lim_Φ 𝔜` is representable by the perfect flat formal scheme
`𝔜^{perf}`, locally `Spf(R^{perf})`, compatibly with generic fibres ([Torsion] §III.2.3,
Definition III.2.18 (ii), p. 42 (statements (ii)–(iv) over the base of the source; the tilde-limit
is that of Scholze–Weinstein); [Torsion] §III.2.3, Definition III.2.18 (i), p. 42 (the
formal-scheme version)). *Needs:* *frobenius-controlled-integral-tower*;
*frobenius-tower-completion*; *perfectoid-frobenius-criterion-for-towers*; *tilde-limit-gluing*;
P1; P1.1.

Lean: `FrobeniusTower.isIntegralPerfectoid_completedPlus` (the completion `A_∞⁺` of a
Frobenius-controlled tower is integral perfectoid, the integral form of (i) of
*perfectoid-frobenius-criterion-for-towers*, [BMS] Lemma 3.10(ii)).

### P7.14 limits of towers of perfectoid spaces

The easy limit theorem: when every finite level is already perfectoid, the limit exists in
perfectoid spaces and is a tilde-limit.

**Towers of perfectoid spaces with affinoid transition maps have perfectoid tilde-limits.**
(*limit-of-perfectoid-tower*) Let `(X_i)_{i∈I}` be a cofiltered system of perfectoid spaces whose
transition maps are affinoid (for every `a : j → i` and every affinoid open `V ⊆ X_i`, `f_a^{-1}(V)`
is affinoid; hence qcqs; as for finite étale towers of affinoids). Then: (i) the limit
`X := lim_i X_i` exists in the category of perfectoid spaces, glued from the affinoid limits of
P5.3; (ii) `X ~ lim_i X_i` (see *perfectoid-tilde-limit*), and `X` represents `lim_i Hom(−, X_i)`
on perfectoid spaces (P7.10 (i) once (ii) is known); (iii) for `i_0 ∈ I` and an affinoid perfectoid
open `V = Spa(R_{i_0}, R_{i_0}⁺) ⊆ X_{i_0}` with preimages `V_a = Spa(R_a, R_a⁺)`, the preimage of
`V` in `X` is the completed direct limit `Spa(R, R⁺)` of P5.1–P5.2 (`R⁺` the `ϖ`-adic completion
of `colim_a R_a⁺`, `R = R⁺[1/ϖ]`, perfectoid); (iv) `|X| ≅ lim_i |X_i|` (P5.4); (v) if `I` is
`ω_1`-cofiltered, `colim R_a → R` and `colim R_a⁺ → R⁺` are bijective (P5.5 applied to the
affinoid pieces). Every finite level is already perfectoid here; the theorem is
distinct from *perfectoid-frobenius-criterion-for-towers*, whose finite levels are not perfectoid
([BHW] §5.2, proof of Theorem 5.11, p. 23 of the arXiv manuscript; [ECD] §6,
Proposition 6.5, p. 29 (the affinoid case); [ECD] §6, proof of Proposition 6.5, p. 29 (the universal
property on affinoids)). *Needs:* P5.3; P1.25; *frobenius-tower-uniformity-bound*;
*tilde-limit-from-completed-colimit*; *tilde-limit-perfectoid-uniqueness*; *tilde-limit-gluing*;
*perfectoid-tilde-limit*; P2.9; P5.2; P1.

Lean: `PerfectoidSpace.exists_tildeLimit_of_affinoid_transitions` states both the categorical
limit in `Perfd` and the supplied AdicSpacesPartII `AdicSpace.IsTildeLimit` relation on its
underlying adic cone, for an arbitrary small cofiltered index category.

**Checks.**

- For a constant system on any perfectoid space, the identity cone gives the limit and the
  tilde-limit; no affinoidness of that space is required.
- For a perfectoid field `K` and a cofiltered system of finite sets `S_i`, the spaces
  `Spa(K, K°) × S_i` have limit `Spa(C⁰(lim S_i, K), C⁰(lim S_i, K°))`.
- A constant system of empty perfectoid spaces has empty limit and satisfies both the
  topological and section-density conditions.

### P7.15 noetherian tilde-limits and the comparison of étale topoi

**Tilde-limits `X ~ lim X_i` of noetherian adic spaces and the comparison of étale topoi.**
(*tilde-limits-and-etale-topos-comparison*) Let `(X_i)_{i∈I}` be a cofiltered system of
quasicompact quasiseparated locally noetherian analytic adic spaces (AdicSpacesPartII R0.6; no
perfectoid base field is assumed, and the `X_i` may lie over `Spa(Q_p, Z_p)` while `X` lies over a
perfectoid field), `X` a perfectoid space (its étale site is that of [Sch12] Definition 7.11 /
[ECD] Definition 6.2) and `X ~′ lim X_i` (*residue-field-tilde-limit*; this holds whenever
`X ~ lim X_i`, *tilde-limit-implies-residue-field-tilde-limit*). Then: (i) `|X|` is a spectral space
and `X` is qcqs ([Sch12] Remark 7.15); (ii) (Theorem 7.17) the étale topos `X_ét^~` of the
perfectoid space `X` (P3.18) is the projective limit of the fibred topos `(X_{i,ét}^~)_i` of the
étale topoi of the `X_i` (AdicEtaleGeometry A1.6, equal to Huber's): `X_ét^~` is equivalent to
the topos of sheaves on the colimit site `colim_i (X_i)_{ét,qcqs}` (objects qcqs étale over some
`X_i`, morphisms the filtered colimits of Hom sets, coverings the families that are coverings at
some finite level), compatibly with the pullback morphisms `X_ét^~ → X_{i,ét}^~`; (iii) (Corollary
7.18) for `i_0 ∈ I` and a sheaf of abelian groups `F_{i_0}` on `X_{i_0,ét}`, with pullbacks `F_j` to
`X_{j,ét}` (`j → i_0`) and `F` to `X_ét`, the natural map `colim_j H^n(X_{j,ét}, F_j) → H^n(X_ét, F)`
is bijective for all `n ≥ 0` (for sheaves of sets, `n = 0`; for sheaves of groups, `n ≤ 1`). When
`X` is itself locally noetherian (and not assumed perfectoid), the corresponding statements are
Huber's Proposition 2.4.4 and Corollary 2.4.6 ([Huber96], §2.4, pp. 127–129); the perfectoid extension uses
local factorisation of étale maps and finite étale descent by henselian approximation (AdicEtaleGeometry A1.0.5).
On the good affinoid charts of P7.4, `GoodAffinoid.finiteEtaleEquiv` supplies this descent
by composing AdicEtaleGeometry A1.0.5's `finiteEtale_completedColimit` ([Berkeley] Theorem 7.4.8 and
its corollary, printed p. 54) with AdicEtaleGeometry A4.4's
`Huber.Pair.uniformization.baseChange_finiteEtale_equivalence`
([KL15] Proposition 2.8.16(a), p. 64). Corollary 7.19
is *etale-topos-invariance-under-inseparable-towers* ([Sch12] §7, proof of Theorem 7.17, p. 44;
[Sch12] §7, Corollary 7.18, p. 44 (statement (iii)); [Sch12] §7, before Definition 7.14, p. 43 (the
purpose of the comparison); [SW13] §2.4, Theorem 2.4.7, p. 21 (restatement of (ii) for the
sectionwise tilde-limit); [Sch13] §4, proof of Theorem 4.9, p. 26 (the application over `Q_p`)).
*Needs:* *residue-field-tilde-limit*; *tilde-limit-implies-residue-field-tilde-limit*;
*residue-field-tilde-limit-etale-base-change*; *tilde-limit-qcqs-etale-finite-stage-descent*;
P3.20; P3.21; P3.18; P3.15; AdicEtaleGeometry A1.0.5; AdicEtaleGeometry A1.3, A1.6 and A4.4; AdicSpacesPartII R0.6;
AdicSpacesPartII R0.19.5–R0.19.8 and P5.4’s perfectoid limit comparisons; P5; Mathlib `CategoryTheory.Limits.limit`, `TopCat.Presheaf`.

The colimit site's `residueEtalePrecoverage` consists of finite families represented
at a common finite stage and jointly surjective there. `residueEtaleTopology` is
the topology it generates. `residueEtaleTopos` identifies sheaves on this site
with sheaves on the perfectoid étale site; its presheaf equation specifies
restriction to every finite-stage object ([Sch12] Theorem 7.17).

For abelian coefficients, `residueEtaleSheafPullback` and
`residueEtaleLimitSheaf` are the inverse-image functors on A1's finite-stage
`YonedaAdicSpace.smallEtaleTopology` and on `smallEtaleTopology p ⟨c.pt, h.perfectoid⟩`,
respectively. `residueEtaleSheafPullback_universal` and
`residueEtaleLimitSheaf_universal` characterize them by Hom equivalences natural
in both sheaves, using geometric base change on the qcqs bases. Coefficients are
in `AddCommGrpCat.{2}` and cohomology in universe 3; these size choices allow
Mathlib's sheafification and `HasExt.standard` without a small skeleton.
For `h : IsResidueFieldTildeLimit p D c`, an initial stage `i`, a sheaf `F` there,
and any `n : ℕ`, `residueEtaleCohomologyDiagram` is indexed by `(Over i)ᵒᵖ`:
its value at `a : j → i` is `Sheaf.H (a* F) n`. The arrows and the legs of
`residueEtaleCohomologyCocone` are induced by exact inverse image on Ext,
as specified by `residueEtaleCohomologyDiagram_map` and
`residueEtaleCohomologyCocone_map`; the natural coefficient comparisons are
chosen independently of `F` and `n`. The cocone's vertex is
`Sheaf.H ((residueEtaleLimitSheaf p D c h i).obj F) n` on the perfectoid site.
The cohomology comparison (*etale-cohomology-along-residue-field-towers*)
`residueEtaleCohomologyColimit` asserts its universal property, and
`residueEtaleCohomologyColimit_isIso` states that its canonical map from the
filtered colimit is an isomorphism of abelian groups ([Sch12] §7, Corollary 7.18,
p. 44). This includes degrees zero and one and all higher nonnegative degrees;
no torsion or constructibility condition is imposed on the abelian sheaf.

**Checks.** `residueEtaleSheafPullback_identity`, `_zero`, and `_constant`
identify identity pullback, preservation of the zero sheaf, and pullback of
constant abelian coefficients. `residueEtaleLimitSheaf_zero`, `_constant`, and
`_transition` check zero coefficients, constant coefficients, and compatibility
with a finite-stage transition. `residueEtaleCohomologyDiagram_identity`, `_zero`,
and `_degreeZero` check identity arrows, zero cohomology for the zero sheaf in
every degree, and the identification of degree zero with sections at a terminal
étale object. `residueEtaleCohomologyCocone_zero`, `_degreeZero`, and `_degreeOne`
check the zero vertex, the same sections convention on the perfectoid site, and
bijectivity of the degree-one colimit comparison.

**Checks.** For both `residueEtalePrecoverage` and `residueEtaleTopology`,
`_identity` covers an object by itself, `_empty` covers an empty finite-stage
object by the empty family, and `_nonempty` excludes the empty covering of an
object whose pullback to the limit has a point.

A perfectoid tilde-limit over qcqs locally noetherian spaces is a residue-field tilde-limit
(*tilde-limit-implies-residue-field-tilde-limit*): let `X ~ lim X_i` be a perfectoid tilde-limit
(see *perfectoid-tilde-limit*) with every `X_i` quasicompact, quasiseparated and locally noetherian
analytic (so that `~′` is defined). Then `X ~′ lim X_i` (see *residue-field-tilde-limit*): for every
`x ∈ X` with images `x_i`, `colim_i k(x_i)^ → k(x)^` has dense image ([SW13] §2.4, after Theorem
2.4.7, p. 21). *Needs:* *perfectoid-tilde-limit*; *residue-field-tilde-limit*;
*good-affinoid-perfectoid-basis*; P2.5.

Étale base change of residue-field tilde-limits, and bijectivity suffices
(*residue-field-tilde-limit-etale-base-change*): let `X ~′ lim X_i` be a residue-field tilde-limit,
`i_0 ∈ I` and `g : Y → X_{i_0}` an étale morphism of qcqs locally noetherian analytic adic spaces
(Huber's étale, equal to the local description of AdicEtaleGeometry A1.3). Put
`Y_a := Y ×_{X_{i_0}} X_j` (`a : j → i_0`) and `Y_X := Y ×_{X_{i_0}} X`, a perfectoid space étale
over `X`. Then (i) `Y_X ~′ lim_a Y_a`; (ii) (the remark in the proof) in Definition 7.14, if `X`
is quasicompact and quasiseparated, `|X| → lim |X_i|` is bijective and the residue-field condition
(c) holds, then `|X| → lim |X_i|` is already a homeomorphism (a bijective, generalizing, spectral
map of spectral spaces), i.e. `X ~′ lim X_i`; quasicompactness cannot be dropped: for a perfectoid field `K`, let
`X_n = ⨿_{a ∈ ℤ/p^nℤ} Spa(K, K°)` with transitions induced by reduction of the index. The space
`X = ⨿_{a ∈ ℤ_p} Spa(K, K°)` maps compatibly to this tower, bijectively onto the underlying limit
`ℤ_p`, with all completed residue fields equal to `K`. Its underlying space is discrete, whereas
`ℤ_p` has its compact nondiscrete topology; thus the map is not a homeomorphism; for an open immersion `Y ⊆ X_{i_0}`, `Y_X` is the open
subspace `φ_{i_0}^{-1}(Y)` ([Sch12] §7, Proposition 7.16, p. 44 (statement (i)); [Sch12] §7, proof
of Proposition 7.16, p. 44 (statement (ii)); [Stacks] Tag 0A2Z, Lemma 5.24.5 (spectrality of
`lim |X_i|` used in (ii))). *Needs:* *residue-field-tilde-limit*; P3.16; AdicEtaleGeometry A1.0.5; P2.5;
AdicEtaleGeometry A1.3; AdicSpacesPartII R0.19.5–R0.19.8 and P5.4’s perfectoid limit comparisons; P3.

Qcqs étale spaces over a residue-field tilde-limit descend to a finite level
(*tilde-limit-qcqs-etale-finite-stage-descent*): let `X ~′ lim X_i` be a residue-field tilde-limit
with all `X_i` qcqs locally noetherian analytic and `X` perfectoid. Then: (i) base change induces an
equivalence of categories `2-colim_i (X_i)_{ét,qcqs} → X_{ét,qcqs}` between qcqs étale adic spaces
over the `X_i` (AdicEtaleGeometry A1.6) and qcqs étale perfectoid spaces over `X` (P3.18), and
likewise `2-colim_i (X_i)_{fét} → X_{fét}`; (ii) a finite family of morphisms `(Z^k_i → Y_i)_k` in
`(X_i)_{ét,qcqs}` is jointly surjective after base change to `X` if and only if it is so after base
change to some `X_j`. This is the perfectoid-limit analogue of [ECD] Proposition 6.4(i)–(ii)
(P5.12), with the `X_i` locally noetherian instead of affinoid perfectoid ([Sch12] §7, proof of
Theorem 7.17, p. 44 (finite étale maps descend by henselian approximation, AdicEtaleGeometry A1.0.5); [ECD] §6, proof of
Proposition 6.4, p. 28 (the graph argument for fullness, and essential surjectivity); [Berkeley]
Lecture 7, after Theorem 7.4.8, p. 54 (finite étale algebras over completed colimits of complete
Tate rings)). *Needs:* *residue-field-tilde-limit*; *residue-field-tilde-limit-etale-base-change*;
AdicEtaleGeometry A1.0.5; P3.18; P3.15; P2.5; AdicEtaleGeometry A1.3 and A1.6; P5.12; AdicSpacesPartII R0.19.5–R0.19.8 and P5.4’s perfectoid limit comparisons; P5; P3.

`analyticQcqsEtale` is the qcqs full subcategory of A1’s
`YonedaAdicSpace.smallEtale`. Its perfectoid target `qcqsEtale p X` uses
`smallEtale p X`, the restriction of A1’s `IsEtaleLocalDescription` to
`Perfd p`, with site comparison `smallEtaleAdicEquiv`. Quasi-compactness
of its structural map and diagonal is R0’s `AdicSpace.QuasiCompact`,
applied through `toAdicSpace`. Thus the perfectoid carrier retains its
objects and commuting triangles while using the supplied morphism predicates.
Checks `qcqsEtale_identity`, `qcqsEtale_empty`, and `qcqsEtale_noncompact`
include the identity and empty open immersion and exclude an étale arrow
whose adic representative fails R0 quasi-compactness.
`analyticEtalePullback_spec` and `analyticEtalePullback_map` specify geometric
pullback and its maps in this same A1 site. Étaleness is A1's local description
around each point, as in `IsResidueFieldTildeLimit.etale_baseChange`.
`IsResidueFieldTildeLimit.level_isAnalytic` extracts each stage’s analytic Tate charts
from its locally noetherian hypothesis ([Sch12] §7, p. 39).
`residueEtaleDiagram` is the corresponding finite-stage pseudofunctor;
`residueEtaleComparison_spec` identifies its comparison with pullback to the
perfectoid limit. Under the residue-field tilde-limit hypotheses above,
`residueEtaleComparison_isEquivalence` gives the equivalence,
`residueEtaleComparison_finite` detects A1's `YonedaAdicSpace.IsFiniteEtale`
at a finite stage, and `residueEtaleComparison_cover` detects finite covering
families after passage to a later stage. On the perfectoid carrier,
`IsFiniteEtale p` is the restriction of this same A1 predicate.
These are the finite-stage assertions in the proof of [Sch12] Theorem 7.17, p. 44.

**Checks.** `analyticQcqsEtale_suppliedEtale` and
`analyticEtalePullback_suppliedEtale` extract A1's local étale condition.
`analyticQcqsEtale_nonqcqs` excludes a supplied étale object whose structural
map is not qcqs; `analyticQcqsEtale_fullHom` retains every A1 morphism between
qcqs objects. `residueEtaleComparison_suppliedFiniteEtale` expresses the
finite-stage criterion with A1's finite étale predicate on both sides.
`residueEtaleDiagram_suppliedCover` identifies finite-stage covering sieves
with A1's pointwise covering condition.

The cartesian projections are chosen simultaneously and naturally:
`analyticEtalePullbackProjection_exists` supplies this family,
`analyticEtalePullbackProjection` selects it, and
`analyticEtalePullbackProjection_isPullback` and
`analyticEtalePullbackProjection_naturality` specify its cartesian squares
and its compatibility with every map of étale objects.
`analyticEtalePullbackIdIso_exists` and
`analyticEtalePullbackCompIso_exists` give unique comparison isomorphisms
preserving these projections, for analytic bases and arbitrary composable
base maps. `analyticEtalePullbackIdIso_projection` identifies the forward
identity comparison with the projection; `analyticEtalePullbackCompIso_projection`
identifies the inverse composition comparison followed by the direct
projection with the two successive projections. The selected
`analyticEtalePullbackIdIso` and `analyticEtalePullbackCompIso` supply the
identity and composition fields of `residueEtaleDiagram`, after transport
along the diagram's identity and composition equations. For the cofiltered
system `D` and `h : IsResidueFieldTildeLimit p D c`,
`residueEtaleDiagram_mapId_hom_app` identifies each identity component
with the chosen cartesian projection over `D.map (𝟙 i)`.
For `a : j → i`, `b : k → j` and a qcqs étale object `U` over `D_i`,
`residueEtaleDiagram_mapComp_hom_app` says that the forward comparison
from direct to iterated pullback, followed by the projections to the
pullback over `D_j` and then to `U`, equals the direct projection to `U`.
The component already lies over `D_k`, so these equations determine it
by the pullback universal property. Mathlib's
`pseudofunctorOfIsLocallyDiscrete` supplies `map₂` by equality transport;
there is no additional choice of a transformation on 2-morphisms.
`IsResidueFieldTildeLimit.level_isAnalytic` supplies all analytic-base
hypotheses. The underlying universal property is geometric base change
in [Sch12] Proposition 7.16 and the proof of Theorem 7.17, p. 44.

**Checks.** `residueEtaleDiagram_identity_projection` fixes the unit
component, `residueEtaleDiagram_two_steps` fixes the order of the two
successive projections, and `residueEtaleDiagram_nontrivial_automorphism`
shows that postcomposing the unit component with a nonidentity automorphism
changes it. Thus a permutation of an étale object's components cannot be
inserted into the unit comparison.

**Checks.** `analyticEtalePullbackProjection_square` fixes the structural
map equation, `_empty` has empty source over an empty étale object, and
`_isomorphism` makes the projection invertible along an invertible base
map. For each of `analyticEtalePullbackIdIso` and
`analyticEtalePullbackCompIso`, `_projection` fixes the comparison,
`_empty` retains the empty object, and `_not_twisted` excludes any map
that changes the projection to the original étale object.

**Checks.** For `analyticQcqsEtale` and `analyticEtalePullback`, `_identity`
recovers the given object, `_empty` stays empty, and `_split` pulls back an
isomorphism to an isomorphism. For `residueEtaleDiagram` and
`residueEtaleComparison`, `_identity` fixes identity maps, `_empty` stays empty,
and `_sections` detects equality of maps at some later finite stage.

`IsResidueFieldTildeLimit.level_isAnalytic` supplies analyticity of each
noetherian analytic finite stage for A1’s `YonedaAdicSpace.smallEtaleTopology`
in `inseparable_tower_etale_topos` ([Sch12] Definition 7.14, pp. 43–44).

Towers with homeomorphic, purely inseparable transition maps do not change the étale topos
(*etale-topos-invariance-under-inseparable-towers*): in the situation of
*tilde-limits-and-etale-topos-comparison* (`X ~′ lim X_i`, `X_i` qcqs locally noetherian analytic,
`X` perfectoid), assume that every transition map `X_j → X_i` induces a homeomorphism `|X_j| → |X_i|`
and purely inseparable extensions of completed residue fields (a universal-homeomorphism condition,
typically relative Frobenius in characteristic `p`). Then for every `i` the pullback
`X_ét^~ → X_{i,ét}^~` is an equivalence of topoi; in particular `H^n(X_ét, F) = H^n(X_{i,ét}, F_i)`
for every abelian sheaf `F_i` on `X_{i,ét}` with pullback `F` ([Sch12] §7, Corollary 7.19, p. 44;
[Sch12] §7, proof of Corollary 7.19, p. 44 (by reference to Huber)). *Needs:*
*tilde-limits-and-etale-topos-comparison*; *perfection-tilde-limit*;
*tilde-limit-qcqs-etale-finite-stage-descent*.

<a id="p7-16-rigid-proetale-site"></a>

### P7.16 the pro-étale site of a rigid space

**Pro-étale site of a rigid space; affinoid perfectoid objects.** (*proetale-site*) Let `X` be a
locally noetherian adic space over `Spa(k, k°)` for a complete nonarchimedean field `k ⊇ ℚ_p`. Its
pro-étale site `X_proét` — objects the pro-objects `lim U_i` of `X_ét` along cofiltered diagrams
with finite étale surjective transition maps from some index on, coverings the families with the
corrected transfinite covering condition ([Sch13E], correction (1), p. 1), and the morphism of sites
`ν : X_proét → X_ét` — is AdicEtaleGeometry layer A1 ([*pro-etale-morphism*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.10),
[*pro-etale-site-corrected*](../AdicEtaleGeometry/README.md) (AdicEtaleGeometry A1.12), API `AdicSpace.ProEtale.ProEt`, `IsProEtale`, `ProEtalePresentation`),
with its site and carrier interfaces supplied by AdicEtaleGeometry A1.10–A1.12; it is not built again here. An
object `U = lim U_i ∈ X_proét` with each `U_i = Spa(R_i, R_i⁺)` affinoid is affinoid perfectoid if
the pair obtained by completing `colim R_i⁺` for the `p`-adic topology and then
inverting `p`, with `R⁺ = (colim R_i⁺)^∧` and `R = R⁺[1/p]`, is a perfectoid
Tate pair (the objects compared in [P6.14](README.md), [*scholze-2013-affinoid-perfectoid-comparison*](README.md), which also
supplies the functoriality of `U ↦ Û` and the fact that `Û` is affinoid pro-étale over the
completion of a smaller object); the associated affinoid perfectoid space `Û = Spa(R, R⁺)`
satisfies `Û ~ lim U_i` ([P7.2](README.md)). What is proved here: affinoid perfectoid objects form a basis
of `X_proét` ([Sch13] Proposition 4.8 proves this over a perfectoid field `K` of characteristic
`0` and Remark 4.11 extends it to any such `k` by base change), products of an affinoid perfectoid
object with a profinite set are affinoid perfectoid ([Sch13] Corollary 6.6), and the points of
`Spa(R, R⁺)` (not all continuous valuations of `R`: `R⁺` need not be `R°`) correspond to the
points of `lim |U_i|`. In the perfectoid-base comparison consumed by DiamondsAndVStacks D2, a constant
pro-object `U_i = U = Spa(R₀, R₀⁺)` is affinoid perfectoid iff
`(R₀, R₀⁺)` is a perfectoid Tate pair. Define `ProEtaleSite.IsAffinoidPerfectoid` (the predicate
on `U = lim U_i` of AdicEtaleGeometry's site with its perfectoid pair `(R, R⁺)`); prove
`ProEtaleSite.affinoidPerfectoid_basis` (the affinoid perfectoid objects form a basis) ([Sch13]
§3, Definition 3.9, p. 15; §4, Definition 4.3 and Lemma 4.5(i), p. 23,
Proposition 4.8, p. 25, Lemma 4.10, final clause, and Remark 4.11, p. 26;
§6, Corollary 6.6, p. 36). *Needs:* AdicEtaleGeometry layer A1
(*pro-etale-site-corrected*); [P6.14](README.md); [P7.1](README.md); [P7.2](README.md); [P5.1](README.md).


**Checks.**

- The `ℂ_p` root-torus object has completed pair `ℂ_p⟨T±¹/p^∞⟩` with its power-bounded plus ring.
- The constant disc over a complete discretely valued mixed-characteristic field fails the perfectoid-object predicate.
- The empty object over an allowed p-adic base has the zero completed pair and satisfies the affinoid-perfectoid condition.

**Coverings on the supplied site.** (*proetale-site-coverings*) Its Checks
apply `AdicEtaleGeometry.AdicSpace.ProEtaleSite.mem_topology_iff` and
`isCovering_of_isFetTower` directly; the corrected topology belongs to AdicEtaleGeometry A1.


**Checks.**

- The identity family is a covering in the corrected topology.
- A supplied finite étale surjective tower gives a covering by `isCovering_of_isFetTower`.
- A family omitting a point of the pro-object cannot pass the supplied jointly-surjective covering condition.

*Needs:* AdicEtaleGeometry A1.10–A1.12 (*pro-etale-site-corrected*),
with the transfinite covering convention of [Sch13E], correction (1), p. 1.

**Affinoid perfectoid objects.** (*proetale-affinoid-perfectoid-objects*)
`ProEtaleSite.IsAffinoidPerfectoid` is existence of an `AffinoidPresentation`
with the completed perfectoid pair specified above ([Sch13] Definition 4.3(i), p. 23).

*Needs:* P5.1; P6.14; P7.1–P7.2; AdicEtaleGeometry A1.12's site carrier
and affinoid presentations ([Sch13] Definition 4.3(i), p. 23, Remark 4.11, p. 26).

**Checks.**

- `proEtale_torus_tower` (computation) — over `ℂ_p`, with compatible primitive
  `p`-power roots of unity and pseudouniformizer roots, the root-torus site object
  has an `AffinoidPresentation` whose stages are the relative torus, whose
  transitions raise its coordinate to the `p`th power, and whose pair is
  `ℂ_p⟨T^{±1/p^∞}⟩` with its power-bounded subring. The Check explicitly concludes
  `ProEtaleSite.IsAffinoidPerfectoid`. The general perfectoid-base and discrete-base
  cases are `proEtale_torus_tower_general` and `proEtale_torus_tower_discrete` below
  ([Sch13] Example 4.4, p. 23).
- `prodProfinite_Zp_sections` (computation) — supplied by [P6.8](README.md)
  on its perfectoid-space carrier: `O(Spa(ℂ_p, O_{ℂ_p}) × ℤ_p) = C⁰(ℤ_p, ℂ_p)`, with integral
  sections `C⁰(ℤ_p, O_{ℂ_p})`.
- `proEtale_not_affinoid_perfectoid` (non-example) — a constant object `Spa k⟨T⟩`, with `k`
  complete discretely valued of mixed characteristic `(0,p)`, is not affinoid perfectoid:
  `k⟨T⟩` is not perfectoid.

`ProEtaleSite.proEtale_rootTorus_exists` constructs this site object over a
complete nonarchimedean perfectoid field `K` of characteristic zero with
`p` topologically nilpotent and a chosen system of pseudouniformizer roots.
It requires no roots of unity in `K`. Its stages are AdicEtaleGeometry’s
`relativeTorus`, with structural maps the iterates of the coordinate p-power
endomorphism; its `AffinoidPresentation.pair` is the [P1](README.md) completed Laurent pair.
The Check `proEtale_torus_tower_general` uses this statement ([Sch13] Example 4.4,
p. 23). The `proEtale_torus_tower_discrete` Check excludes the same coordinate-root
presentation over a complete mixed-characteristic field with discrete valuation:
its coefficient values remain discrete, so adjoining only variable roots cannot
supply the pseudouniformizer required for perfectoidness. The constant-disc negative Check applies
`ProEtaleSite.IsAffinoidPerfectoid` to the constant object of this site.

`ProEtaleSite.AffinoidPresentation.baseMap` is the structure map
`X → Spa(ℚ_p, ℤ_p)`, obtained by composing the map to `Spa(k, k°)` with the
base-field map. `AffinoidPresentation` also carries an affinoid presentation in the actual
pro-étale site, an isomorphism from the `p`-adic completion of its integral section
colimit to the plus ring of a perfectoid pair, and the localization property after
inverting `p`. `IsAffinoidPerfectoid` means existence of all this data, including
`baseMap`. The presentation and predicate follow [Sch13] Definition 4.3, p. 23,
with the base-field scope of Remark 4.11, p. 26.
`AffinoidPresentation.points` identifies the spectrum of the completed pair, with its
specified plus ring, homeomorphically with the pro-object's space ([Sch13] Lemma 4.10,
final clause, p. 26; Lemma 4.5(i), p. 23, supplies the étale base-change compatibility).

**Checks.** Both carriers have `_empty` (the empty object), `_nonperfect` (the
constant rational `2`-adic point is excluded), and `_profinite` (products with a
profinite set preserve the condition) examples.
The `_padic_base` Checks extract the map to `Spa(ℚ_p, ℤ_p)` from the predicate.
The `_char_p` Checks exclude every site object over the spectrum of a complete
nontrivially valued characteristic-`p` field, including the empty site object:
that nonempty base admits no map to `Spa(ℚ_p, ℤ_p)`. This predicate is scoped
to a `p`-adic base; characteristic-`p` perfectoid spaces use the supplied P6 predicate.

Lean: README only.


### P7.16a finite flat regular towers with algebraically closed residue field

**Finite flat regular towers with algebraically closed residue field.**
(*regular-finite-flat-residue-tower*) Let `(R, m)` be a complete regular local ring (Mathlib
`IsRegularLocalRing`, of any characteristic) with residue field `k`. There is a filtered direct
system of finite flat `R`-algebras `R_i` such that each `(R_i, mR_i)` is a regular local ring and
`(lim→ R_i, m·lim→ R_i)` is a regular local ring with algebraically closed residue field. If `k` is
separably closed of characteristic `p > 0`, the `R_i` may be chosen of `p`-power rank over `R`
(then `k̄/k` is purely inseparable and its finite subextensions have `p`-power degree). The colimit
is Noetherian (a filtered colimit of noetherian local rings along flat local homomorphisms
`R_i → R_j` with `m_iR_j = m_j` is a noetherian local ring with maximal ideal `m_i` extended); it
is not claimed to be complete (`𝔽̄_p⟦x⟧ ⊋ colim 𝔽_{pⁿ}⟦x⟧`). The proof uses the three Cohen
presentations:
`R ≅ k⟦x_1, …, x_n⟧` (equal characteristic); `R ≅ W⟦x_1, …, x_n⟧` with `W` a complete discrete
valuation ring with uniformizer `p` (`p ∈ m∖m²`); `R ≅ W⟦x_1, …, x_n⟧/(p − f)` with
`f ∈ (p, x_1, …, x_n)²` (`p ∈ m²`), together with the regularity, normality, power-series,
quotient and filtered-colimit comparison theorems these presentations need ([Čes] §5, Lemma 5.1,
p. 11; [Čes] §5, proof of Lemma 5.1, pp. 11–12 (the ramified case of the construction)). *Needs:*
Mathlib `IsAdicComplete`, `Module.Flat`, `IsRegularLocalRing`.

**Cohen coefficient rings and regular presentations.** (*regular-cohen-presentations*)
`RegularTower.CohenDVR p k` is a characteristic-zero complete DVR with maximal ideal `(p)` and
specified residue isomorphism with `k`. This subsection owns the shared labelled carrier
and `RegularTower.exists_cohenDVR`: for every prime `p` and every field `k` with `CharP k p`,
there exists a `RegularTower.CohenDVR p k`, with no perfectness assumption on `k`
([Stacks, Lemma 10.160.6, Tag 0328](https://stacks.math.columbia.edu/tag/0328)).
Coefficient-ring adapters use this carrier and existence theorem; truncated formal smoothness
and compatible coefficient lifts are additional properties of the chosen carrier.
`cohen_equalCharacteristic` assumes completeness and
regularity of `R` and equality of its characteristic with that of its residue field.
`cohen_unramified` and `cohen_ramified` assume characteristic zero and residue characteristic
`p`, splitting according to whether `p ∉ m²` or `p ∈ m²`. Their conclusions are precisely the
three displayed presentations. The forms used here are stated in [Čes], proof of Lemma 5.1,
pp. 11–12. The coefficient-ring existence theorem is also
[Stacks, Theorem 10.160.8, Tag 032A](https://stacks.math.columbia.edu/tag/032A).

`powerSeries_regular_field` and `powerSeries_regular_cohen` assert regularity for finitely many
variables over a field and over a Cohen DVR, respectively
([Stacks, Remark 10.160.9, Tag 032C](https://stacks.math.columbia.edu/tag/032C)).
`ramified_quotient_regular` assumes `f ∈ (p,x_1,…,x_n)²`: then `p-f` is part of a regular system
of parameters, and its quotient is regular
([Stacks, Lemma 10.106.3](https://stacks.math.columbia.edu/tag/00NN)).
`regular_normal` asserts integral closedness of a regular local domain
([Stacks, Lemma 10.157.5, Tag 0567](https://stacks.math.columbia.edu/tag/0567)).
`powerSeries_finite_flat` extends a finite flat local map of Cohen coefficient rings to their
finite-variable power-series rings ([Čes], proof of Lemma 5.1, pp. 11–12).
`exists_residueTower` names the finite flat regular system and the algebraically closed residue
field of its regular colimit. In the equal-characteristic construction its maximal-adic
completion is `k̄⟦x_1,…,x_n⟧`; the mixed-characteristic construction uses the completed coefficient
DVR and then the indicated quotient. Noetherianity in this construction follows by faithful-flat
descent from these completions ([Čes], proof of Lemma 5.1, pp. 11–12;
[Stacks, Tag 033E](https://stacks.math.columbia.edu/tag/033E),
[Tag 05UU](https://stacks.math.columbia.edu/tag/05UU),
[Tag 00MK](https://stacks.math.columbia.edu/tag/00MK)).

Checks:

- `cohen_test_equalChar_zeroVariables` (degenerate) — applying
  `cohen_equalCharacteristic` to a field `k` chooses `n=0`; its resulting presentation is
  `k ≃ (ResidueField k)⟦Fin 0⟧`. The Check retains the theorem's regularity, completeness
  and equal-characteristic hypotheses and tests its chosen number of variables.
- `cohen_test_ramified_two` (computation) — `W⟦X⟧/(p-X²)` is regular, including at `p=2`.
- Coefficient existence gives `ℤ_p` with its labelled `𝔽_p` residue in the prime-field case;
  for imperfect `k = 𝔽_p(t)` it still gives a Cohen DVR with residue `k`.
- `cohen_test_parameter_nonunit` (non-example) — the marked `p` in a Cohen DVR is not a unit.

- `cohen_test_parameter_not_square` (non-example) — the coefficient parameter of
  `CohenDVR` satisfies `p ∉ m²`.
- `cohen_test_wrong_characteristic` (non-example) — the characteristic-zero ring of
  `CohenDVR` cannot be isomorphic to a characteristic-`p` residue field.
  Together with `cohen_test_parameter_nonunit`, these test the coefficient-ring carrier.

The coefficient-extension interface `coefficient_extension` supplies an integral DVR extension
of a Cohen DVR with uniformizer `p`, algebraically closed residue field, and a finite flat DVR
subextension containing each element. It does not assert that this integral extension is
complete. `powerSeriesDiagram` applies finite-variable power series coefficientwise.
`powerSeries_colimit_field` completes the union of finite coefficient-field extensions at the
ideal generated by the variables. `powerSeries_colimit_cohen` completes at `(p,x_1,…,x_n)` and
uses the `p`-adic completion of the coefficient colimit on the right. The latter assumes
complete DVR stages with uniformizer `p`, finite flat over a fixed Cohen DVR. These are the
completion comparisons in [Čes], proof of Lemma 5.1, pp. 11–12.
`exists_residueTower_primePower` states the rank refinement with separably closed residue
field of characteristic `p`; its stages have rank `p^e` for some `e≥0`.

`powerSeries_colimit_regular` asserts regularity of the same finite-flat coefficient-extension
colimit before completion. `ramified_colimit_completion` starts with `f` in the square of the
maximal ideal at a fixed stage: quotienting the colimit by `p-f` gives a regular local ring,
and its maximal-adic completion is the quotient by the coefficientwise image of `p-f` over the
completed coefficient DVR ([Čes], proof of Lemma 5.1, pp. 11–12).

### P7.17 finite flat regular towers with perfectoid completion

**Finite flat regular towers with perfectoid completion.** (*regular-finite-flat-perfectoid-tower*)
Let `(R, m)` be a complete regular local ring of mixed characteristic `(0, p)` with perfect residue
field `k` (so `W ≅ W(k)` in the Cohen presentations of *regular-finite-flat-residue-tower*), and
`W := W(k)`. There is a tower `{R_m}_{m≥0}` of finite flat `R`-algebras `R_m` of `p`-power rank
over `R`, each a regular local ring, such that the `p`-adic completion `R̂_∞` of
`R_∞ := lim→_m R_m` is perfectoid (integral perfectoid in the sense of [BMS] Definition 3.5, so
`R̂_∞[1/p]` is a perfectoid Tate ring). Explicitly: if `p ∈ m∖m²`, then `R ≅ W⟦x_1, …, x_n⟧` and
`R_m := W[p^{1/p^m}]⟦x_1^{1/p^m}, …, x_n^{1/p^m}⟧`; if `p ∈ m²`, then `R ≅ W⟦x_1, …, x_n⟧/(p − f)`
with `f ∈ (p, x_1, …, x_n)²` and `R_m := W⟦x_1^{1/p^m}, …, x_n^{1/p^m}⟧/(p − f)`. The finite levels
are not perfectoid, and `R_∞` is not complete. The criterion used is [Čes] Lemma 4.7, in the
instance proved here: for a `p`-torsion-free ring `S` with `(ϖ^p) = (p)` for some `ϖ ∈ S`, if `S`
is integrally closed in `S[1/p]` then `x ↦ x^p : S/ϖ → S/p` is injective, and if moreover every
element of `S/p` is a `p`-th power then the `p`-adic completion of `S` is perfectoid ([Čes] §5,
Lemma 5.2, p. 12; [Čes] §5, proof of Lemma 5.2, p. 12 (the reduction of the ramified case to a unit
multiple of `p`); [Čes] §4, Lemma 4.7, p. 8 (the criterion)). *Needs:*
*regular-finite-flat-residue-tower*; P1; Mathlib `WittVector`, `IsIntegrallyClosedIn`,
`IsAdicComplete`, `IsRegularLocalRing`.

`RegularTower.exists_perfectoidTower` names the sequence of finite flat regular algebras,
the `p`-power rank condition, and integral perfectoidness of the `p`-adic completion of
its ring colimit. The perfect residue-field hypothesis is an explicit `PerfectRing` assumption;
completeness of the initial regular local ring is maximal-adic ([Čes] Lemma 5.2, p. 12).

### Examples

The compatible-root disc and torus run through the whole layer: for a perfectoid Tate ring `R`
whose pseudouniformiser `ϖ` has compatible `p`-power roots,
`Spa(R⟨T^{1/p^∞}⟩, R⁺⟨T^{1/p^∞}⟩) ~ lim_n Spa(R⟨T^{1/p^n}⟩, R⁺⟨T^{1/p^n}⟩)` along
`T^{1/p^n} ↦ T^{1/p^n}`, and likewise for `R⟨T^{±1/p^∞}⟩` ([Sch13] Example 4.4 for the torus over a
perfectoid field); the whole disc is good at level `0` with `V = X_0`, and so is its rational subset
`{|T| ≤ |ϖ|}`, every rational subset of the perfectoid disc being the preimage of a rational subset
at a finite level; the Huber pairs `(R⟨T^{1/p^n}⟩, R⁺⟨T^{1/p^n}⟩)` with `A_{n,0} = R⁺⟨T^{1/p^n}⟩`,
`I_n = ϖA_{n,0}` have completed direct limit `R⟨T^{1/p^∞}⟩`, already uniform, and every finite étale
`R⟨T^{1/p^∞}⟩`-algebra is defined over some `R⟨T^{1/p^n}⟩`; the integral tower `R⁺⟨T^{1/p^m}⟩`
with `w_m(r T^{a/p^{m+1}}) = Φ(r) T^{a/p^m}` is Frobenius-controlled with `A_∞⁺ = R⁺⟨T^{1/p^∞}⟩`,
`A_∞° = R°⟨T^{1/p^∞}⟩` and the bound `ϖR°⟨T^{1/p^∞}⟩ ⊆ R⁺⟨T^{1/p^∞}⟩` (because
`ϖR° ⊆ R°° ⊆ R⁺`); the characteristic-`p` tower `R♭⁺⟨T^{1/p^m}⟩` with `α_m` induced by
`R⁺/ϖ^p ≅ R♭⁺/(ϖ♭)^p` gives the tilt `R♭⟨T^{1/p^∞}⟩`; two perfectoid tilde-limits of the tower (for
instance `R⟨T^{1/p^∞}⟩` and its presentation through `T ↦ T^p` transition maps) are uniquely
isomorphic; the preimage of the torus `{|T| ≥ 1}` is `~ lim` of the tori `{|T^{1/p^n}| ≥ 1}`; over a
perfectoid field `K` the Gauss point `x` has `k(x_n)^` the completion of `K(T^{1/p^n})` for the Gauss
norm, whose union is dense in `k(x)^`, so the tower is a residue-field tilde-limit; every finite
étale cover of the perfectoid disc is pulled back from some `K⟨T^{1/p^n}⟩`; and for `p ≠ 2` the
Kummer cover `S ↦ S² = T` of the perfectoid torus pulls back to
`Spa(K⟨S^{±1/p^∞}⟩) ~ lim Spa(K⟨S^{±1/p^n}⟩)`.

The Shimura towers are the geometric instances. [Torsion] Theorem III.1.2 and Corollary III.2.19:
the Siegel anticanonical tower `X(p^{-m}ε)` (`ε < 1/2`) has, over an affine `Spf R_{m_0}`,
preimages `Spf R_m` forming a Frobenius-controlled integral tower with `ϖ = p^{(1−ε)/p}`, `R_∞` the
`p`-adic completion of `colim_m R_m`, absolute Frobenius inducing `R_∞/p^{(1−ε)/p} ≅ R_∞/p^{1−ε}`,
whence unique perfectoid spaces `X_{Γ_0(p^∞)}(ε)_a ~ lim_m X_{Γ_0(p^m)}(ε)_a` (and similarly `X*`,
`A`) over `Q_p^cycl` whose tilt is the open subset of the perfection `X′*perf` where `|Ha| ≥ |t|^ε`
(using `X′*(p^{-m}ε)/t^{1−ε} = X*(p^{-m}ε)/p^{1−ε}`, Theorem III.2.15(i)), `X′*perf` over
`F_p((t^{1/(p−1)p^∞}))` being the tilde-limit of the Siegel space along relative Frobenius
(Definition III.2.18); `|X*_{Γ(p^∞)}|` is glued from the `Spa(R_∞, R_∞⁺)` over affinoid perfectoid
subsets (Definition III.3.5) with `X*_{Γ(p^∞)} ~ lim_{K_p} X*_{K_p}`, unique up to unique
isomorphism, the principal congruence subgroups `K(p^n) = ker(GSp_{2g}(Z_p) → GSp_{2g}(Z/p^n))`
having a witness in `GSp_{2g}(Q_p)` so that the tower may be indexed by `n`, while the anticanonical
levels `Γ_0(p^m)` have none and their tilde-limit is the `Γ_0(p^∞)`-level; and the finite étale
`D_m → X_{Γ_0(p^m)}(ε)_a` pulls back to the perfectoid `D_{mΓ_0(p^∞)}` (§III.2.4). [CHJ] Theorem 2.8:
`X_∞` is glued from the two good affinoids `V_i = π_HT^{-1}(V_i)` pulled back from level `K(p^N)`,
indexed by the cofinal `K(p^n)`, and the rational subset `X_{∞,w}` of `V_1` is the preimage of an
affinoid `S ⊆ X_{K(p^N)}` (proof of Theorem 2.17). [BHW] Theorem 5.11: `X_{Γ_0*(p^∞)}(ε)_a ~ lim_n X_{Γ_0*(p^n)}(ε)_a` for Hilbert modular varieties via relative Frobenius modulo `p^{1−δ}` (Lemma
5.12), restricted from `X*(p^{-∞}ε) ~ lim X*(p^{-n}ε)` to the open modular locus; the finite étale
`X_{Γ_1*(p^n)} → X_{Γ_0*(p^n)}` pulls back to a tower of finite étale perfectoid spaces whose limit
is the perfectoid tilde-limit `X_{Γ_1*(p^∞)}(ε)_a`; the `Γ_0`-, `Γ_1`- and `Γ`-towers are distinct
towers, not cofinal reindexings. [HJ] Lemma 5.11 and its proof ([P8](../PerfectoidTowers/README.md#layer-p8-finite-quotients-and-closed-perfectoid-loci-in-towers)): for locally noetherian `X_i` with finite
transition maps and `X = lim X_i^◇` perfectoid, `X` has a basis of affinoid perfectoid opens pulled
back from finite levels, and for affinoid `X_i` every rational subset of the limit is pulled back
from a finite level.

The pro-finite-étale and formal examples: the universal pro-finite-étale cover
`X_∞ = Spa(Â_∞, Â_∞⁺)` of a connected affinoid noetherian `X = Spa(A, A⁺)` over `Spa(Q_p, Z_p)`
satisfies `X_∞ ~ lim X_i` for the finite étale `X_i = Spa(A_i, A_i⁺) → X` in both the sectionwise
and the residue-field sense, has no nonsplit finite étale covers, and
`H^j(X_{∞,ét}, L) = colim_i H^j(X_{i,ét}, L)` for all `j ≥ 0` ([Sch13] Theorem 4.9, the form used downstream in
`p`-adic Hodge theory); for the universal finite étale `G_i`-torsor tower `A → A_i` of a Tate
`Z_p`-algebra, the uniform completion of `colim A_i` is perfectoid,
`Spd(Â_∞, Â_∞⁺) = lim Spd(A_i, A_i⁺)` and `Spa(Â_∞, Â_∞⁺) ~ lim Spa(A_i, A_i⁺)` ([ECD] Lemma 15.3,
Proposition 15.4); `T(G)^{ad} ~ lim G[p^n]^{ad}` ([SW13] §3, proof of Proposition 3.3.2); the
generic fibre of `Spf Z_p⟦X_1^{1/p^∞}, …, X_n^{1/p^∞}⟧` is preperfectoid relative to
`Spa(Q_p, Z_p)` in [SW13] Definition 2.3.9: after each base change to a characteristic-zero
perfectoid field it has perfectoid strong completions locally ([SW13] Example 2.3.12,
Proposition 2.3.13 and the paragraph following it, p. 19). This relative condition does not
imply P7.7's `IsPreperfectoid` before base change; the zero-variable case is
`Spa(Q_p, Z_p)`, which fails that absolute condition. For a perfectoid field `K` the strong
completion of
`Spa(K[ε]/(ε²), O_K + Kε)` is `Spa(K, O_K)`, with `Spa(K, O_K) ~ Spa(K[ε]/(ε²), O_K + Kε)` although
the two spaces differ, while for `X` perfectoid `X̂ = X`; the perfected projective space
`((P^n)^{ad})^{perf} ~′ lim_φ (P^n)^{ad}` over a perfectoid field of characteristic `p` ([Sch12] Theorem 8.5)
has equal étale topoi at every level, whereas the Frobenius tower `T ↦ T^p` of `P^n` in
characteristic `0` has non-inseparable residue extensions and only the limit statement of P7.15
applies; for `Y = Spa(k⟨T⟩, k°⟨T⟩)`, `Y^{perf} = Spa(k⟨T^{1/p^∞}⟩, k°⟨T^{1/p^∞}⟩)` with
`Y^{perf}_ét ≃ Y_ét`; and a pro-étale `G`-tower of perfectoid spaces `X_j → X` has
`X_∞ = lim X_j` ([P9.4](../PerfectoidTowers/README.md#layer-p9-continuous-torsor-descent-with-coefficients) in the perfectoid case).

The regular towers: for `R = 𝔽_p⟦x⟧` the colimit `lim→ 𝔽_{p^n}⟦x⟧` is not complete, its completion
`𝔽̄_p⟦x⟧` being strictly larger, and for `k = 𝔽_p` the levels `𝔽_{p^n}⟦x⟧` have rank `n`, not a
`p`-power; for `R = ℤ_p⟦x⟧/(p − x²) = ℤ_p[√p]`, `p ∈ m²`, `R_i = W(𝔽_{p^i})[√p]` with `i` ordered by
divisibility, and the colimit is regular with parameter `x = √p`; for `R = ℤ_p` (`n = 0`),
`R_m = ℤ_p[p^{1/p^m}]` and `R̂_∞` is the `p`-adic completion of `ℤ_p[p^{1/p^∞}]`, a perfectoid ring,
while every finite level fails the perfectoid criterion: for `m ≥ 1`, `R_m/p ≅ 𝔽_p[T]/(T^{p^m})` and
`T` is not a `p`-th power, and at `m = 0` Frobenius on `𝔽_p` is surjective but no `ϖ ∈ ℤ_p` has
`(ϖ^p) = (p)`; for these displayed towers the residue field stays `k`, so perfectoidness of their completion
  forces `k` perfect, since Frobenius surjectivity descends to `k`; and the
completion is `p`-adic, not `m_∞`-adic.

The proved Lean example for this layer shows that integral-tower stages cannot be fields,
so `𝔽_p` has no constant tower with a regular adic parameter.

### Dependencies

Within this roadmap: P1 (perfectoid Tate rings, tilting, uniformity and the almost-isomorphism
`A⁺ → A°`; P1.1, P1.3, P1.25, P1.27), P2 (perfectoid spaces, affinoid perfectoids, rational
localisation, open subspaces and fibre products; P2.3, P2.5, P2.6, P2.9, P2.10), P3 (finite étale and étale morphisms, the étale site and the tilting equivalence;
P3.10, P3.15–P3.18, P3.22), P5 (the limit of affinoid perfectoids, qcqs étale descent; P5.2, P5.3,
P5.12); AdicSpacesPartII R0.19.5–R0.19.8 and P5.4’s perfectoid limit comparisons.
Other roadmaps, by subsection: P7.1 uses AdicSpacesPartII R0.20.1's intrinsic tilde-limit
relation and completed-affinoid criterion and AdicSpaces layer 5's analytic adic spaces;
P7.2 uses AdicSpacesPartII R0.1; P7.3 and P7.15 use R0.6; P7.4 uses R0.4,
AdicEtaleGeometry A1.0.3–A1.0.5 (finite étale colimits and completion) and A4.4 (uniformization).
P7.1, P7.2, P7.4, P7.7, P7.8 and P7.11 use AdicEtaleGeometry A1.16's analytic Yoneda
carrier, `CAff` and global functions. P7.9 and P7.11 use AdicSpacesPartII R0.20.1;
P7.11 also uses R0.4 and R0.13. P7.12 uses AdicEtaleGeometry A1.2–A1.3 and
AdicSpacesPartII R0.20.2's base change along morphisms locally of weakly finite type.
P7.15 uses AdicEtaleGeometry A1.0.5, A1.3, A1.6 and A4.4; P7.16's rigid pro-étale basis
uses AdicEtaleGeometry A1.10–A1.12 and local P5.1, P6.14 and P7.1–P7.2.
Libraries: Mathlib `CategoryTheory.IsCofiltered`,
`CategoryTheory.Functor.Initial`, `CategoryTheory.Functor.Initial.limitIso`,
`CategoryTheory.Functor.initial_of_exists_of_isCofiltered`, `CategoryTheory.Limits.limit`,
`TopCat.limitCone`, `TopCat.Presheaf`, `Dense`, `SpectralSpace`, `IsSpectralMap`, `IsAdicComplete`,
`AdicCompletion`, `Ideal.span`, `frobenius`, `integralClosure`, `Ring.DirectLimit`, `OpenSubgroup`,
`ProfiniteGrp`, `ProfiniteGrp.exist_openNormalSubgroup_sub_open_nhds_of_one`,
`ProfiniteGrp.closedSubgroup_eq_sInf_open`, `Module.Flat`, `IsRegularLocalRing`, `WittVector`,
`IsIntegrallyClosedIn`; Tau Ceti `Subgroup.exists_le_of_iInf_le_of_directed`,
`TauCeti.Huber.Pair`, `TauCeti.Huber.Pair.Hom`,
`TauCeti.Huber.IsTateRing`, `TauCeti.Huber.IsPseudoUniformizer`,
`TauCeti.Huber.IsRingOfIntegralElements`, `TauCeti.ValuationSpectrum.spa`,
`TauCeti.ValuationSpectrum.spa_integralClosure`, `TauCeti.ValuationSpectrum.spaComap`,
`TauCeti.ValuationSpectrum.spaComap_preimage_rationalSubset`,
`TauCeti.ValuationSpectrum.rationalSubset`,
`TauCeti.ValuationSpectrum.exists_mem_nhds_forall_rationalSubset_eq_of_sub_mem`,
`TauCeti.ValuationSpectrum.mem_of_forall_vle_one`.

## Downstream consumers

DiamondsAndVStacks D0–D6 consumes P0–P7's perfectoid algebra, spaces, tilting,
pro-étale calculus and completion bound: D0 uses `ProKappa` and `completion_cardinality_bound`
(P6.10, P6.2) and Cartan's lemma on acyclic bases, the classical case of P0.29; D1–D6 use perfectoid
spaces, tilting, almost purity, finite-stage descent and the `κ`-small and pro-étale calculus of
P0–P6; D2–D3 use the Amitsur complex and descent of P0.25; D3 restricts the separatedness of P4.19
to perfectoid spaces; and D5 and D3 use the spectrum comparison of P1.12 and the mod-`ϖ` chain of
P1.28. DiamondsAndVStacks owns the v-site (precoverage and topology in D2.2, subcanonicity and
v-descent in D2.7), built on the compact-open covering condition that P7 uses without any topology.
It owns `Diamond`, `IsSpatial` and spatiality, and the comparison of its characteristic-`p` diamond
functor with the represented functors of P7.11. PerfectoidTowers P8–P9 imports both roadmaps for
finite quotients, spatial presentations, closed loci and continuous torsor descent; its P8.14
(Zariski-closed embeddings over every affinoid perfectoid open) and P8.15 (analytic separatedness,
which uses the diagonal over an analytic base of P2.17) are distinct from P4.16 and P4.19.

AdicEtaleGeometry supplies generic henselian approximation and completed-colimit finite étale
comparisons in A1.0.4–A1.0.5, and the algebraic torsor-tower and completion theory in
A4.2–A4.5 and A4.7. Its finite-group module descent in A4.2a uses the Galois tensor
isomorphism and Mathlib's faithfully flat descent over arbitrary commutative rings;
PerfectoidTowers P9 applies it to coefficient descent. P3.21 provides the integral almost
cohomological specialization of AdicEtaleGeometry A1.6a's stable-basis vector-bundle theory.
Sousperfectoid frames and spaces use P1.1's predicate
`Perfectoid.IsPerfectoidTateRing` in P3.17a. Every perfectoid space is sousperfectoid:
apply `Huber.IsSousperfectoid.of_isPerfectoidTateRing` to its affinoid perfectoid charts.
AdicSpacesPartIV R5.5 consumes this class and P3.17b's polydisc calculation to construct
products with smooth rigid spaces. Perfectoid torsor presentations belong to P6.16–P6.17;
their v-sheaf and diamond interpretations belong to DiamondsAndVStacks D6.
Perfectoid Shimura varieties and period
spaces use these foundations.

## References

Locators are theorem and section numbers of the linked versions, with page numbers of the linked manuscripts where they are stable.

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- [ECD] Peter Scholze, Étale cohomology of diamonds, arXiv:1709.07343v4. https://arxiv.org/abs/1709.07343.
- [Bhatt] Bhargav Bhatt, Lecture notes for a class on perfectoid spaces (2017). http://www-personal.umich.edu/~bhattb/teaching/mat679w17/lectures.pdf.
- [Berkeley] Peter Scholze and Jared Weinstein, Berkeley lectures on p-adic geometry, Annals of Math. Studies 207 (2020); author manuscript. https://people.mpim-bonn.mpg.de/scholze/Berkeley.pdf.
- [KL15] Kiran Kedlaya and Ruochuan Liu, Relative p-adic Hodge theory: foundations, Astérisque 371 (2015); arXiv:1301.0792. https://arxiv.org/abs/1301.0792.
- [AWS] Kiran Kedlaya, Sheaves, stacks, and shtukas (Arizona Winter School 2017 notes). https://kskedlaya.org/papers/aws-notes.pdf.
- [KL16] Kiran Kedlaya and Ruochuan Liu, Relative p-adic Hodge theory, II: imperfect period rings, arXiv:1602.06899v3. https://arxiv.org/abs/1602.06899.
- [KBF] Kiran Kedlaya, On commutative nonarchimedean Banach fields, Doc. Math. 23 (2018); arXiv:1602.09004. https://arxiv.org/abs/1602.09004.
- [BMS] Bhargav Bhatt, Matthew Morrow and Peter Scholze, Integral p-adic Hodge theory, Publ. IHÉS 128 (2018); arXiv:1602.03148v3. https://arxiv.org/abs/1602.03148.
- [Stacks] The Stacks Project authors, The Stacks Project (stable tags). https://stacks.math.columbia.edu.
- [Sch13] Peter Scholze, p-adic Hodge theory for rigid-analytic varieties, Forum Math. Pi 1 (2013); arXiv:1205.3463v2. https://arxiv.org/abs/1205.3463.
- [Sch13E] Peter Scholze, Erratum to 'p-adic Hodge theory for rigid-analytic varieties'. https://www.math.uni-bonn.de/people/scholze/pAdicHodgeErratum.pdf.
- [HK] David Hansen and Kiran Kedlaya, Sheafiness criteria for Huber rings, version dated 6 August 2026. https://kskedlaya.org/papers/criteria.pdf.
- [Wedhorn] Torsten Wedhorn, Adic spaces, arXiv:1910.05934v1. https://arxiv.org/abs/1910.05934.
- [Elkik] Renée Elkik, Solutions d'équations à coefficients dans un anneau hensélien, Ann. Sci. ÉNS 6 (1973), pp. 553–603. [Numdam article](https://www.numdam.org/item/ASENS_1973_4_6_4_553_0/), [original PDF](https://www.numdam.org/item/ASENS_1973_4_6_4_553_0.pdf). §0.2, printed p. 555 (PDF p. 4), defines the Jacobian ideal; §II, Théorèmes 2 and 2 bis, printed p. 560 (PDF p. 9), states polynomial and section approximation.
- [Torsion] Peter Scholze, On torsion in the cohomology of locally symmetric varieties, Ann. of Math. 182 (2015); arXiv:1306.2070v2. https://arxiv.org/abs/1306.2070.
- [SW13] Peter Scholze and Jared Weinstein, Moduli of p-divisible groups, Camb. J. Math. 1 (2013); arXiv:1211.6357v2. https://arxiv.org/abs/1211.6357.
- [Survey] Peter Scholze, Perfectoid spaces: a survey, arXiv:1303.5948. https://arxiv.org/abs/1303.5948.
- [BHW] Christopher Birkbeck, Ben Heuer and Chris Williams, Overconvergent Hilbert modular forms via perfectoid modular varieties, Ann. Inst. Fourier (2023); arXiv:1902.03985. https://arxiv.org/abs/1902.03985.
- [CHJ] Przemysław Chojecki, David Hansen and Christian Johansson, Overconvergent modular forms and perfectoid Shimura curves, Doc. Math. 22 (2017); arXiv:1507.04875. https://arxiv.org/abs/1507.04875.
- [Čes] Kęstutis Česnavičius, Purity for the Brauer group, Duke Math. J. 168 (2019); arXiv:1711.06456v4. https://arxiv.org/pdf/1711.06456v4.
- [HJ] David Hansen and Christian Johansson, Perfectoid Shimura varieties and the Calegari–Emerton conjectures, arXiv:2011.03951. https://arxiv.org/abs/2011.03951.
- [BS22] Bhargav Bhatt and Peter Scholze, Prisms and prismatic cohomology, Ann. of Math. 196 (2022); arXiv:1905.08229v4. https://arxiv.org/abs/1905.08229.
- [SGA1] Alexander Grothendieck et al., Revêtements étales et groupe fondamental (SGA 1), arXiv:math/0206203. https://arxiv.org/abs/math/0206203.
- [Illusie] Luc Illusie, Complexe cotangent et déformations I, LNM 239 (1971). https://doi.org/10.1007/BFb0059052.
- [Huber96] Roland Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Math. E30 (1996). https://doi.org/10.1007/978-3-663-09991-8.

- [FS] Laurent Fargues, Peter Scholze, Geometrization of the local Langlands correspondence.
  https://arxiv.org/abs/2102.13459. arXiv:2102.13459v4.

- [dJvP] Johan de Jong, Marius van der Put, Étale cohomology of rigid analytic spaces.
  https://ems.press/content/serial-article-files/25781. Documenta Mathematica 1 (1996), 1–56.