Module

RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal.lean

Nyll.AlgebraicCurves.RelativeDivisors.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal

13 declarations indexed from sourceView source ↗

Module documentation

Construct the effective relative Cartier divisor cut out by a section of an invertible sheaf, and compute its ideal after base change.

Imports26

Mathlib.AlgebraicGeometry.Morphisms.Proper
Mathlib.AlgebraicGeometry.Morphisms.Smooth
Mathlib.AlgebraicGeometry.Morphisms.Flat
Mathlib.AlgebraicGeometry.Morphisms.FlatRank
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite
Mathlib.AlgebraicGeometry.Morphisms.Affine
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap
Mathlib.AlgebraicGeometry.ZariskisMainTheorem
Mathlib.AlgebraicGeometry.Noetherian
Mathlib.AlgebraicGeometry.IdealSheaf.Functorial
Mathlib.AlgebraicGeometry.ResidueField
Mathlib.FieldTheory.IsAlgClosed.Basic
Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
Mathlib.RingTheory.RingHom.Flat
Mathlib.RingTheory.LocalProperties.Basic
Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra
Mathlib.Algebra.Category.Ring.Constructions
Mathlib.LinearAlgebra.TensorProduct.RightExactness
Mathlib.RingTheory.IsTensorProduct
Nyll.AlgebraicCurves.Geometry.Sheaves.ModulesSectionZeroScheme
Nyll.AlgebraicCurves.Geometry.Sheaves.IdealSheafModule
Nyll.AlgebraicCurves.RelativeDivisors.RelEffCartierDiv
Nyll.AlgebraicCurves.RelativeDivisors.Cartier
Nyll.AlgebraicCurves.Geometry.Sheaves.Scheme_Modules_IsInvertible_isInvertible_zeroSchemeIdeal_and_flat
Nyll.AlgebraicCurves.Geometry.Sheaves.Scheme_Modules_IsInvertible_exists_iso_invModule_zeroSchemeIdeal
Nyll.AlgebraicCurves.RelativeDivisors.RelEffCartierDiv_isInvertible_I

Imported by4

theorem

ker_algebraMap_eq_map_ker

L118 ↗

AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal.Algebra.IsPushout.ker_algebraMap_eq_map_ker

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
attribute [-simp]
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
open _root_.TensorProduct
open _root_.nonZeroDivisors
namespace Algebra
open _root_.Algebra
namespace IsPushout
open _root_.Algebra.IsPushout
open _root_.Algebra.IsPushout
open _root_.Algebra

Lean source · statement

theorem ker_algebraMap_eq_map_ker
    (R B C P : Type*) [CommRing R] [CommRing B] [CommRing C] [CommRing P]
    [Algebra R B] [Algebra R C] [Algebra B P] [Algebra C P] [Algebra R P]
    [IsScalarTower R B P] [IsScalarTower R C P] [Algebra.IsPushout R B C P]
    (hC : Function.Surjective (algebraMap R C)) :
    RingHom.ker (algebraMap B P) = (RingHom.ker (algebraMap R C)).map (algebraMap R B)

theorem

ker_eq_map_ker_of_isPushout

L159 ↗

AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal.CommRingCat.ker_eq_map_ker_of_isPushout

Source context 14 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
attribute [-simp]
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
open _root_.TensorProduct
open _root_.nonZeroDivisors
namespace CommRingCat
open _root_.CommRingCat
open _root_.CommRingCat

Lean source · statement

theorem ker_eq_map_ker_of_isPushout {R B C P : CommRingCat.{u}} {f : R ⟶ B} {g : R ⟶ C}
    {inl : B ⟶ P} {inr : C ⟶ P} (h : IsPushout f g inl inr)
    (hg : Function.Surjective g.hom) :
    RingHom.ker inl.hom = (RingHom.ker g.hom).map f.hom

theorem

ideal_comap_top

L184 ↗

AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal.AlgebraicGeometry.Scheme.IdealSheafData.ideal_comap_top

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
attribute [-simp]
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
open _root_.TensorProduct
open _root_.nonZeroDivisors
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData hiding ideal_comap_of_le
open _root_.AlgebraicGeometry.Scheme hiding IdealSheafData.ideal_comap_of_le
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
variable {X Y : Scheme.{u}}

Lean source · statement

theorem ideal_comap_top [IsAffine X] [IsAffine Y] (I : X.IdealSheafData) (f : Y ⟶ X) :
    (I.comap f).ideal ⟨⊤, isAffineOpen_top Y⟩ =
      (I.ideal ⟨⊤, isAffineOpen_top X⟩).map f.appTop.hom

theorem

ideal_comap_of_le

L195 ↗

AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal.AlgebraicGeometry.Scheme.IdealSheafData.ideal_comap_of_le

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
attribute [-simp]
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
open _root_.TensorProduct
open _root_.nonZeroDivisors
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData hiding ideal_comap_of_le
open _root_.AlgebraicGeometry.Scheme hiding IdealSheafData.ideal_comap_of_le
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
variable {X Y : Scheme.{u}}

Lean source · statement

theorem ideal_comap_of_le (I : X.IdealSheafData) (f : Y ⟶ X) (U : X.affineOpens)
    (V : Y.affineOpens) (hVU : (V : Y.Opens) ≤ f ⁻¹ᵁ (U : X.Opens)) :
    (I.comap f).ideal V = (I.ideal U).map (f.appLE U V hVU).hom

theorem

ideal_comap_of_isAffineHom

L248 ↗

AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal.AlgebraicGeometry.Scheme.IdealSheafData.ideal_comap_of_isAffineHom

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
attribute [-simp]
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
open _root_.TensorProduct
open _root_.nonZeroDivisors
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData hiding ideal_comap_of_le
open _root_.AlgebraicGeometry.Scheme hiding IdealSheafData.ideal_comap_of_le
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
variable {X Y : Scheme.{u}}

Lean source · statement

theorem ideal_comap_of_isAffineHom (I : X.IdealSheafData) (f : Y ⟶ X) [IsAffineHom f]
    (U : X.affineOpens) :
    (I.comap f).ideal ⟨f ⁻¹ᵁ U, U.2.preimage f⟩ = (I.ideal U).map (f.app U).hom

theorem

mem_nonZeroDivisors_of_injective

L258 ↗

AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal.ZDFlatG.mem_nonZeroDivisors_of_injective

Source context 12 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
attribute [-simp]
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
open _root_.TensorProduct
open _root_.nonZeroDivisors
namespace ZDFlatG

Lean source · statement

theorem mem_nonZeroDivisors_of_injective {R S F : Type*} [CommRing R] [CommRing S]
    [FunLike F R S] [MulHomClass F R S] [ZeroHomClass F R S] (e : F) (he : Function.Injective e)
    {r : R} (h : e r ∈ nonZeroDivisors S) : r ∈ nonZeroDivisors R

theorem

finite_preimage_singleton_of_isPullback

L269 ↗

AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal.ZDHeadG.finite_preimage_singleton_of_isPullback

Source context 12 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
attribute [-simp]
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
open _root_.TensorProduct
open _root_.nonZeroDivisors
namespace ZDHeadG

Lean source · statement

theorem finite_preimage_singleton_of_isPullback {P X Y Z : Scheme.{u}} {fst : P ⟶ X} {snd : P ⟶ Y}
    {f : X ⟶ Z} {g : Y ⟶ Z} (h : IsPullback fst snd f g) [Finite P] {z : Z}
    (hz : z ∈ Set.range g.base) : (f.base ⁻¹' {z}).Finite

theorem

finite_of_isFinite

L280 ↗

AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal.ZDHeadG.finite_of_isFinite

Source context 12 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
attribute [-simp]
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
open _root_.TensorProduct
open _root_.nonZeroDivisors
namespace ZDHeadG

Lean source · statement

theorem finite_of_isFinite {P : Scheme.{u}} {k : Type u} [Field k] (p : P ⟶ Spec (CommRingCat.of k))
    [IsFinite p] : Finite P

theorem

pullbackLeftPullbackSndIso_hom_comp_eq

L294 ↗

AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal.ZDHeadG.pullbackLeftPullbackSndIso_hom_comp_eq

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
attribute [-simp]
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
open _root_.TensorProduct
open _root_.nonZeroDivisors
namespace ZDHeadG
section Square
variable {𝒞 S T W : Scheme.{u}} (f : 𝒞 ⟶ S) (g : T ⟶ S) (x : W ⟶ T)
  (F : pullback f (x ≫ g) ⟶ pullback f g)
  (hF1 : F ≫ pullback.fst f g = pullback.fst f (x ≫ g))
  (hF2 : F ≫ pullback.snd f g = pullback.snd f (x ≫ g) ≫ x)
include hF1 hF2

Lean source · statement

theorem pullbackLeftPullbackSndIso_hom_comp_eq :
    (pullbackLeftPullbackSndIso f g x).hom ≫ F = pullback.fst (pullback.snd f g) x

theorem

isPullback_mapOnProd

L300 ↗

AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal.ZDHeadG.isPullback_mapOnProd

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
attribute [-simp]
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
open _root_.TensorProduct
open _root_.nonZeroDivisors
namespace ZDHeadG
section Square
variable {𝒞 S T W : Scheme.{u}} (f : 𝒞 ⟶ S) (g : T ⟶ S) (x : W ⟶ T)
  (F : pullback f (x ≫ g) ⟶ pullback f g)
  (hF1 : F ≫ pullback.fst f g = pullback.fst f (x ≫ g))
  (hF2 : F ≫ pullback.snd f g = pullback.snd f (x ≫ g) ≫ x)
include hF1 hF2

Lean source · statement

theorem isPullback_mapOnProd :
    IsPullback F (pullback.snd f (x ≫ g)) (pullback.snd f g) x

theorem

isPullback_snd_comap

L310 ↗

AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal.ZDHeadG.isPullback_snd_comap

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
attribute [-simp]
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
open _root_.TensorProduct
open _root_.nonZeroDivisors
namespace ZDHeadG
section Square
variable {𝒞 S T W : Scheme.{u}} (f : 𝒞 ⟶ S) (g : T ⟶ S) (x : W ⟶ T)
  (F : pullback f (x ≫ g) ⟶ pullback f g)
  (hF1 : F ≫ pullback.fst f g = pullback.fst f (x ≫ g))
  (hF2 : F ≫ pullback.snd f g = pullback.snd f (x ≫ g) ≫ x)
include hF1 hF2

Lean source · statement

theorem isPullback_snd_comap (Z : (pullback f g).IdealSheafData) :
    IsPullback (pullback.snd F Z.subschemeι) (pullback.fst F Z.subschemeι ≫ pullback.snd f (x ≫ g))
      (Z.subschemeι ≫ pullback.snd f g) x

theorem

isInvertible_comap_of_isIso

L317 ↗

AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal.ZDHeadG.isInvertible_comap_of_isIso

Source context 12 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
attribute [-simp]
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
open _root_.TensorProduct
open _root_.nonZeroDivisors
namespace ZDHeadG

Lean source · statement

theorem isInvertible_comap_of_isIso {X Y : Scheme.{u}} {J : Y.IdealSheafData}
    (hJ : J.IsInvertible) (e : X ⟶ Y) [IsIso e] : (J.comap e).IsInvertible

theorem

exists_I_eq_zeroSchemeIdeal

L343 ↗

AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_zeroSchemeIdeal

Source context 11 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.RelativeDivisors.Proofs.RelEffCartierDiv_exists_I_eq_zeroSchemeIdeal
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
attribute [-simp]
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry hiding Scheme.IdealSheafData.ideal_comap_of_le
open _root_.TensorProduct
open _root_.nonZeroDivisors

Lean source · statement

theorem _root_.AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_zeroSchemeIdeal
    {𝒞 S : Scheme.{u}} {f : 𝒞 ⟶ S} [IsProper f] [SmoothOfRelativeDimension 1 f]
    {T : Scheme.{u}} {g : T ⟶ S} [IsLocallyNoetherian T]
    {M : (pullback f g).Modules} (hM : Scheme.Modules.IsInvertible M)
    (s : 𝟙_ (pullback f g).Modules ⟶ M) (r : ℕ)
    (hfib : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T),
      ∃ Dx : RelEffCartierDiv f r (x ≫ g),
        Dx.I = (Scheme.Modules.zeroSchemeIdeal s).comap (mapOnProdOver f x rfl)) :
    ∃ D : RelEffCartierDiv f r g,
      D.I = Scheme.Modules.zeroSchemeIdeal s ∧
        ∃ e : M ≅ D.lineBundle, s ≫ e.hom = D.I.invModuleSection

Module documentation

Declarations