{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "b1c661e0",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## The abelian category of coherent sheaves on $\\mathbb{P}^1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "6192d9c4",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CapAndHomalg v\u001b[32m1.6.1\u001b[39m\n",
      "Imported OSCAR's components GAP and Singular_jll\n",
      "Type: ?CapAndHomalg for more information\n"
     ]
    }
   ],
   "source": [
    "using CapAndHomalg"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "356d35c8",
   "metadata": {},
   "outputs": [],
   "source": [
    "LoadPackage( \"GradedModulePresentationsForCAP\" )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "1bb40c8b",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: Q"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ℚ = HomalgFieldOfRationalsInSingular()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "ffbda3a8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: Q[x,y]\n",
       "(weights: yet unset)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S = GradedRing( ℚ[\"x,y\"] )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e7e3c83a",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### The category $S$-grmod of finitely presented graded modules over $S=\\mathbb{Q}[x,y]$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "3270d600",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sgrmod = GradedLeftPresentations( S )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "c2550caf",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "38 primitive operations were used to derive 228 operations for this category which algorithmically\n",
      "* IsMonoidalCategory\n",
      "* IsAbelianCategoryWithEnoughProjectives\n",
      "and not yet algorithmically\n",
      "* IsSymmetricClosedMonoidalCategory\n"
     ]
    }
   ],
   "source": [
    "InfoOfInstalledOperationsOfCategory( Sgrmod )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "53a1f73f",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "#ListPrimitivelyInstalledOperationsOfCategory( Sgrmod )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ca8ff7c2",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Create two objects:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "faa40326",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: <An object in The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = GradedFreeLeftPresentation( 2, S, @gap([ 1, 1 ]) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "46435fd7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: <An object in The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = GradedFreeLeftPresentation( 1, S, @gap([ 0 ]) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2556efdc",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Create a morphism:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "78426f88",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: <A 2 x 1 matrix over a graded ring>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mat = HomalgMatrix( \"[x,y]\", 2, 1, S )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "5e90211c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x,\n",
      "y \n",
      "(over a graded ring)\n"
     ]
    }
   ],
   "source": [
    "Display( mat )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "face3f01",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: <A morphism in The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ϕ = GradedPresentationMorphism( M, mat, N )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "3b6196ad",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IsWellDefined( ϕ )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "8d1a4aaf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IsMonomorphism( ϕ )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "ae5ff011",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IsEpimorphism( ϕ )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "0d8aedbe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: <A monomorphism in The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ι = ImageEmbedding( ϕ )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "f4191e98",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IsMonomorphism( ι )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "1cbbcece",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IsIsomorphism( ι )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "bbaf33bf",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: <An object in The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coker_mod = CokernelObject( ϕ )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "813e04b9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x,\n",
      "y \n",
      "(over a graded ring)\n",
      "\n",
      "An object in The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ])\n",
      "\n",
      "(graded, degree of generator:[ 0 ])\n"
     ]
    }
   ],
   "source": [
    "Display( coker_mod )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "ecf1cfaa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IsZero( coker_mod )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dfaff2da",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### The Serre subcategory of modules supported on the irrelevant locus, i.e., the finite length modules:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "51a73779",
   "metadata": {},
   "outputs": [],
   "source": [
    "is_artinian = GapObj( M -> AffineDimension( M ) <= 0 );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "d47d6672",
   "metadata": {},
   "outputs": [],
   "source": [
    "SetNameFunction( is_artinian, g\"is_artinian\" )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "0e0842a4",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: <Subcategory of The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by is_artinian>"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = FullSubcategoryByMembershipFunction( Sgrmod, is_artinian )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b6ee7e07",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### The Serre quotient of a constructive abelian category modulo a Serre category with decidable membership is again constructive abelian [[BLH14](https://arxiv.org/abs/1409.2028),[Gut17](http://dokumentix.ub.uni-siegen.de/opus/volltexte/2017/1241/)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "766586fe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: The Serre quotient category of The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by test function with name: is_artinian"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CohP1 = Sgrmod / C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "22e4e0c1",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "21 primitive operations were used to derive 187 operations for this category which algorithmically\n",
      "* IsAbelianCategory\n"
     ]
    }
   ],
   "source": [
    "InfoOfInstalledOperationsOfCategory( CohP1 )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "62da0c5b",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### The sheafification functor:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "761f0638",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: Localization functor of The Serre quotient category of The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by test function with name: is_artinian"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sh = CanonicalProjection( CohP1 )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "2edc5429",
   "metadata": {},
   "outputs": [],
   "source": [
    "InstallFunctor( Sh, g\"Sheafification\" )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d64a49c4",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Interpret $\\phi$ as a morphism $\\psi$ of sheaves:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "87cef2f5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: <A morphism in The Serre quotient category of The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by test function with name: is_artinian>"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ψ = ApplyFunctor( Sh, ϕ )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "91e5351a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IsMonomorphism( ψ )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "6c94a6ef",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IsEpimorphism( ψ )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "60289215",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: <A zero object in The Serre quotient category of The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by test function with name: is_artinian>"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coker_shv = CokernelObject( ψ )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "a88841b8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IsZero( coker_shv )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "ce01cf5a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: <A morphism in The Serre quotient category of The category of graded left f.p. modules over Q[x,y] (with weights [ 1, 1 ]) by test function with name: is_artinian>"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ϵ = ApplyFunctor( Sh, ι )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "f55d95f0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IsIsomorphism( ϵ )"
   ]
  }
 ],
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