Module

ShortExact_map_pushforward_thickening.lean

Nyll.JacobianVarieties.Picard.ShortExact_map_pushforward_thickening

58 declarations indexed from sourceView source ↗

Module documentation

Pushforward of a short exact sequence on a section thickening

This file proves preservation of a specified short exact sequence supported on an infinitesimal thickening of a section of a proper smooth relative curve. The comparison map is required to agree with the original adjunction-unit map, and the sequence remains short exact after pushforward to the base.

Imports33

Imported by2

def

restrictTriv

L166 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31X1.restrictTriv

Documentation

Restrict a trivialization on an open subset to a smaller open subset.

Source context 17 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31X1
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

def restrictTriv {Z : Scheme.{u}} {N : Z.Modules} {U₀ U : Z.Opens} (h : U ≤ U₀)
    (e : (Scheme.Modules.pullback U₀.ι).obj N ≅ 𝟙_ (U₀ : Scheme.{u}).Modules) :
    (Scheme.Modules.pullback U.ι).obj N ≅ 𝟙_ (U : Scheme.{u}).Modules

theorem

locallyTrivialOver

L179 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.locallyTrivialOver

Source context 17 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

theorem locallyTrivialOver {Z T : Scheme.{u}} (q : Z ⟶ T) [UniversallyClosed q]
    (hinj : Function.Injective q.base) (hsurj : Function.Surjective q.base)
    (N : Z.Modules) (hN : IsInvertible N) (t : T) :
    ∃ W : T.Opens, t ∈ W ∧
      Nonempty ((Scheme.Modules.pullback (q ⁻¹ᵁ W).ι).obj N ≅
        𝟙_ (↑(q ⁻¹ᵁ W) : Scheme.{u}).Modules)

lemma

rigSection_snd

L209 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.rigSection_snd

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))
omit [IsProper c] [SmoothOfRelativeDimension 1 c] in

Lean source · statement

lemma rigSection_snd : rigSection c t ε ≫ pullback.snd c t = 𝟙 T

instance

smooth_snd

L211 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.smooth_snd

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

instance smooth_snd : SmoothOfRelativeDimension 1 (pullback.snd c t)

instance

isClosedImmersion_rigSection

L215 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.isClosedImmersion_rigSection

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

instance isClosedImmersion_rigSection : IsClosedImmersion (rigSection c t ε)

lemma

isInvertible_sectionIdeal

L220 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.isInvertible_sectionIdeal

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

lemma isInvertible_sectionIdeal : (sectionIdeal c ε t).IsInvertible

lemma

pker

L224 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.pker

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

lemma pker (n : ℕ) :
    IsFinite ((sectionIdeal c ε t ^ n).subschemeι ≫ pullback.snd c t) ∧
      (∀ s : T, ((sectionIdeal c ε t ^ n).subschemeι ≫ pullback.snd c t).finrank s = n) ∧
      Flat ((sectionIdeal c ε t ^ n).subschemeι ≫ pullback.snd c t) ∧
      LocallyOfFinitePresentation ((sectionIdeal c ε t ^ n).subschemeι ≫ pullback.snd c t)

lemma

range_rigSection_eq_support

L233 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.range_rigSection_eq_support

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))
omit [SmoothOfRelativeDimension 1 c] in

Lean source · statement

lemma range_rigSection_eq_support :
    Set.range (rigSection c t ε) =
      (((sectionIdeal c ε t).support : Closeds ↥(pullback c t)) : Set ↥(pullback c t))

lemma

injective_q

L240 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.injective_q

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))
omit [SmoothOfRelativeDimension 1 c] in

Lean source · statement

lemma injective_q (n : ℕ) (hn : n ≠ 0) :
    Function.Injective ((sectionIdeal c ε t ^ n).subschemeι ≫ pullback.snd c t).base

lemma

surjective_q

L264 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.surjective_q

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))
omit [SmoothOfRelativeDimension 1 c] in

Lean source · statement

lemma surjective_q (n : ℕ) (hn : n ≠ 0) :
    Function.Surjective ((sectionIdeal c ε t ^ n).subschemeι ≫ pullback.snd c t).base

lemma

isInvertible_G

L279 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.isInvertible_G

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))
variable (M : RigidifiedLineBundle c ε t)

Lean source · statement

lemma isInvertible_G (r : ℕ) : IsInvertible (M.L ⊗ sectionTwist c ε t r)

theorem

lf

L282 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.lf

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))
variable (M : RigidifiedLineBundle c ε t)

Lean source · statement

theorem lf [LocallyOfFiniteType t] (r n : ℕ) (hn : n ≠ 0) :
    IsLocallyFreeOfRank n
      ((Scheme.Modules.pushforward (pullback.snd c t)).obj
        ((Scheme.Modules.pushforward ((sectionIdeal c ε t ^ n).subschemeι)).obj
          ((Scheme.Modules.pullback ((sectionIdeal c ε t ^ n).subschemeι)).obj
            (M.L ⊗ sectionTwist c ε t r))))

def

isoUnitOfPullbackIso

L309 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.E6.isoUnitOfPullbackIso

Documentation

Recover a trivialization of a module sheaf from its trivial pullback along an isomorphism.

Source context 18 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
noncomputable section
namespace Ws31ThetaF
namespace E6

Lean source · statement

def isoUnitOfPullbackIso {X Y : Scheme.{u}} (f : X ⟶ Y) [IsIso f] (L : Y.Modules)
    (e : (Scheme.Modules.pullback f).obj L ≅ 𝟙_ X.Modules) : L ≅ 𝟙_ Y.Modules

def

pushforwardInvIsoPullback

L319 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.E6.pushforwardInvIsoPullback

Documentation

Identify pushforward along the inverse of a scheme isomorphism with pullback along the isomorphism.

Source context 18 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
noncomputable section
namespace Ws31ThetaF
namespace E6

Lean source · statement

def pushforwardInvIsoPullback {X Y : Scheme.{u}} (e : X ≅ Y) :
    Scheme.Modules.pushforward e.inv ≅ Scheme.Modules.pullback e.hom

theorem

nonempty_iso_unit_of_field

L331 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.E6.nonempty_iso_unit_of_field

Source context 18 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
noncomputable section
namespace Ws31ThetaF
namespace E6

Lean source · statement

theorem nonempty_iso_unit_of_field {K : Type u} [Field K] (L : (Spec (CommRingCat.of K)).Modules)
    (hL : Scheme.Modules.IsInvertible L) : Nonempty (L ≅ 𝟙_ (Spec (CommRingCat.of K)).Modules)

lemma

rigSection_snd

L347 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.E6.rigSection_snd

Source context 21 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
noncomputable section
namespace Ws31ThetaF
namespace E6
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))
omit [IsProper c] [SmoothOfRelativeDimension 1 c] in

Lean source · statement

lemma rigSection_snd : rigSection c t ε ≫ pullback.snd c t = 𝟙 T

instance

smooth_snd

L349 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.E6.smooth_snd

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
noncomputable section
namespace Ws31ThetaF
namespace E6
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

instance smooth_snd : SmoothOfRelativeDimension 1 (pullback.snd c t)

lemma

isInvertible_sectionIdeal

L353 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.E6.isInvertible_sectionIdeal

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
noncomputable section
namespace Ws31ThetaF
namespace E6
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

lemma isInvertible_sectionIdeal : (sectionIdeal c ε t).IsInvertible

theorem

nonempty_pullback_rigSection_sectionTwist_iso

L357 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.E6.nonempty_pullback_rigSection_sectionTwist_iso

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
noncomputable section
namespace Ws31ThetaF
namespace E6
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

theorem nonempty_pullback_rigSection_sectionTwist_iso (r : ℕ) :
    Nonempty ((Scheme.Modules.pullback (rigSection c t ε)).obj (sectionTwist c ε t r) ≅
      𝟙_ T.Modules)

theorem

nonempty_pullback_rigSection_module_iso

L375 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.E6.nonempty_pullback_rigSection_module_iso

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
noncomputable section
namespace Ws31ThetaF
namespace E6
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

theorem nonempty_pullback_rigSection_module_iso :
    Nonempty ((Scheme.Modules.pullback (rigSection c t ε)).obj (sectionIdeal c ε t).module ≅
      𝟙_ T.Modules)

theorem

nonempty_pushforward_thickening_one_iso

L392 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.E6.nonempty_pushforward_thickening_one_iso

Source context 21 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
noncomputable section
namespace Ws31ThetaF
namespace E6
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))
omit [SmoothOfRelativeDimension 1 c] in

Lean source · statement

theorem nonempty_pushforward_thickening_one_iso (H : (pullback c t).Modules) :
    Nonempty ((Scheme.Modules.pushforward (pullback.snd c t)).obj
      ((Scheme.Modules.pushforward (sectionIdeal c ε t).subschemeι).obj
        ((Scheme.Modules.pullback (sectionIdeal c ε t).subschemeι).obj H)) ≅
      (Scheme.Modules.pullback (rigSection c t ε)).obj H)

theorem

nonempty_pullback_rigSection_G_iso

L431 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.E6.nonempty_pullback_rigSection_G_iso

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
noncomputable section
namespace Ws31ThetaF
namespace E6
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

theorem nonempty_pullback_rigSection_G_iso (M : RigidifiedLineBundle c ε t) (r : ℕ) :
    Nonempty ((Scheme.Modules.pullback (rigSection c t ε)).obj (M.L ⊗ sectionTwist c ε t r) ≅
      𝟙_ T.Modules)

lemma

surjective_of_comp_of_injective

L449 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.E6.surjective_of_comp_of_injective

Source context 18 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
noncomputable section
namespace Ws31ThetaF
namespace E6

Lean source · statement

lemma surjective_of_comp_of_injective {A B C : Ab.{u}} (f : A ⟶ B) (g : B ⟶ C)
    (hg : Function.Injective g) (h : Function.Surjective (f ≫ g)) : Function.Surjective f

lemma

surjective_comp

L456 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.E6.surjective_comp

Source context 18 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
noncomputable section
namespace Ws31ThetaF
namespace E6

Lean source · statement

lemma surjective_comp {A B C : Ab.{u}} (f : A ⟶ B) (g : B ⟶ C)
    (hf : Function.Surjective f) (hg : Function.Surjective g) : Function.Surjective (f ≫ g)

lemma

unit_app_unit_comp_map_pullbackObjUnitToUnit

L463 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.E6.unit_app_unit_comp_map_pullbackObjUnitToUnit

Source context 18 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
noncomputable section
namespace Ws31ThetaF
namespace E6

Lean source · statement

lemma unit_app_unit_comp_map_pullbackObjUnitToUnit {X Y : Scheme.{u}} (f : X ⟶ Y) :
    (Scheme.Modules.pullbackPushforwardAdjunction f).unit.app (𝟙_ Y.Modules) ≫
        (Scheme.Modules.pushforward f).map (Scheme.Modules.pullbackUnitIso f).hom =
      f.unitToPushforwardUnit

theorem

surjective_unit_app_of_isAffineOpen_of_trivial

L475 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.E6.surjective_unit_app_of_isAffineOpen_of_trivial

Source context 18 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
noncomputable section
namespace Ws31ThetaF
namespace E6

Lean source · statement

theorem surjective_unit_app_of_isAffineOpen_of_trivial {Z' Z : Scheme.{u}}
    (k : Z' ⟶ Z) [IsClosedImmersion k]
    (N : Z.Modules) (W : Z.Opens) (hW : IsAffineOpen W)
    (e : (Scheme.Modules.pullback W.ι).obj N ≅ 𝟙_ (↑W : Scheme.{u}).Modules) :
    Function.Surjective (((Scheme.Modules.pullbackPushforwardAdjunction k).unit.app N).app W)

theorem

subsingleton_sections_of_isEmpty

L567 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.BaseCase.subsingleton_sections_of_isEmpty

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
namespace BaseCase
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.CategoryTheory.MonoidalCategory
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

theorem subsingleton_sections_of_isEmpty {Z : Scheme.{u}} [IsEmpty Z] (N : Z.Modules)
    (U : Z.Opens) :
    Subsingleton Γ(N, U)

theorem

isZero_of_subsingleton

L575 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.BaseCase.isZero_of_subsingleton

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
namespace BaseCase
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.CategoryTheory.MonoidalCategory
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

theorem isZero_of_subsingleton {X : Scheme.{u}} (M : X.Modules)
    (h : ∀ U : X.Opens, Subsingleton Γ(M, U)) :
    IsZero M

theorem

isZero_of_isEmpty

L583 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.BaseCase.isZero_of_isEmpty

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
namespace BaseCase
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.CategoryTheory.MonoidalCategory
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

theorem isZero_of_isEmpty {Z : Scheme.{u}} [IsEmpty Z] (N : Z.Modules) : IsZero N

theorem

isZero_free_fin_zero

L586 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.BaseCase.isZero_free_fin_zero

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
namespace BaseCase
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.CategoryTheory.MonoidalCategory
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

theorem isZero_free_fin_zero (X : Scheme.{u}) :
    IsZero (SheafOfModules.free.{u} (R := X.ringCatSheaf) (ULift.{u} (Fin 0)))

theorem

isLocallyFreeOfRank_zero_of_isZero

L593 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.BaseCase.isLocallyFreeOfRank_zero_of_isZero

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
namespace BaseCase
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.CategoryTheory.MonoidalCategory
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

theorem isLocallyFreeOfRank_zero_of_isZero {X : Scheme.{u}} {F : X.Modules} (hF : IsZero F) :
    IsLocallyFreeOfRank 0 F

theorem

nonempty_det_zero_iso_of_isZero

L598 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.BaseCase.nonempty_det_zero_iso_of_isZero

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
namespace BaseCase
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.CategoryTheory.MonoidalCategory
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

theorem nonempty_det_zero_iso_of_isZero {X : Scheme.{u}} {F : X.Modules} (hF : IsZero F) :
    Nonempty (Scheme.Modules.det 0 F ≅ SheafOfModules.unit X.ringCatSheaf)

theorem

support_pow_zero

L603 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.BaseCase.support_pow_zero

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
namespace BaseCase
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.CategoryTheory.MonoidalCategory
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

theorem support_pow_zero {X : Scheme.{u}} (I : X.IdealSheafData) : (I ^ 0).support = ⊥

theorem

isEmpty_subscheme_of_support_eq_bot

L608 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.BaseCase.isEmpty_subscheme_of_support_eq_bot

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
namespace BaseCase
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.CategoryTheory.MonoidalCategory
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

theorem isEmpty_subscheme_of_support_eq_bot {X : Scheme.{u}} (I : X.IdealSheafData)
    (hI : I.support = ⊥) :
    IsEmpty I.subscheme

theorem

surjective_pushforward_map_app

L624 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.surjective_pushforward_map_app

Source context 17 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

theorem surjective_pushforward_map_app {Z Y : Scheme.{u}} (f : Z ⟶ Y) {M N : Z.Modules}
    (φ : M ⟶ N) (U : Y.Opens)
    (h : Function.Surjective (φ.app (f ⁻¹ᵁ U))) :
    Function.Surjective (((Scheme.Modules.pushforward f).map φ).app U)

theorem

surjective_comp_app

L631 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.surjective_comp_app

Source context 17 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

theorem surjective_comp_app {X : Scheme.{u}} {M N P : X.Modules} (a : M ⟶ N) (b : N ⟶ P)
    (U : X.Opens)
    (ha : Function.Surjective (a.app U)) (hb : Function.Surjective (b.app U)) :
    Function.Surjective ((a ≫ b).app U)

theorem

surjective_hom_app

L641 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.surjective_hom_app

Source context 17 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

theorem surjective_hom_app {X : Scheme.{u}} {M N : X.Modules} (e : M ≅ N) (U : X.Opens) :
    Function.Surjective (e.hom.app U)

theorem

surjective_inv_app

L645 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.surjective_inv_app

Source context 17 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

theorem surjective_inv_app {X : Scheme.{u}} {M N : X.Modules} (e : M ≅ N) (U : X.Opens) :
    Function.Surjective (e.inv.app U)

theorem

epi_unit_of_iso

L649 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.epi_unit_of_iso

Source context 17 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

theorem epi_unit_of_iso {C D : Type*} [Category C] [Category D] {L : C ⥤ D} {R : D ⥤ C}
    (adj : L ⊣ R)
    {A B : C} (φ : A ≅ B) (h : Epi (adj.unit.app A)) : Epi (adj.unit.app B)

def

cmpIso

L664 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.cmpIso

Documentation

Compare the nested pullback and pushforward with the direct pullback and pushforward along the composite inclusion.

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section unitfactor
variable {ZA ZB X : Scheme.{u}} (iA : ZA ⟶ X) (iB : ZB ⟶ X) (k : ZA ⟶ ZB) (hk : k ≫ iB = iA)

Lean source · statement

def cmpIso (H : X.Modules) :
    (Scheme.Modules.pushforward iB).obj ((Scheme.Modules.pushforward k).obj
      ((Scheme.Modules.pullback k).obj ((Scheme.Modules.pullback iB).obj H))) ≅
      (Scheme.Modules.pushforward iA).obj ((Scheme.Modules.pullback iA).obj H)

theorem

unit_app_eq_comp_unit

L674 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.unit_app_eq_comp_unit

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section unitfactor
variable {ZA ZB X : Scheme.{u}} (iA : ZA ⟶ X) (iB : ZB ⟶ X) (k : ZA ⟶ ZB) (hk : k ≫ iB = iA)

Lean source · statement

theorem unit_app_eq_comp_unit (H : X.Modules) :
    (Scheme.Modules.pullbackPushforwardAdjunction iA).unit.app H =
      (Scheme.Modules.pullbackPushforwardAdjunction iB).unit.app H ≫
        (Scheme.Modules.pushforward iB).map
          ((Scheme.Modules.pullbackPushforwardAdjunction k).unit.app
            ((Scheme.Modules.pullback iB).obj H)) ≫
        (cmpIso iA iB k hk H).hom

def

ΦGen

L697 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.ΦGen

Documentation

The morphism induced by the adjunction unit for the intermediate inclusion and the comparison isomorphism.

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section unitfactor
variable {ZA ZB X : Scheme.{u}} (iA : ZA ⟶ X) (iB : ZB ⟶ X) (k : ZA ⟶ ZB) (hk : k ≫ iB = iA)

Lean source · statement

def ΦGen (H : X.Modules) :
    (Scheme.Modules.pushforward iB).obj ((Scheme.Modules.pullback iB).obj H) ⟶
      (Scheme.Modules.pushforward iA).obj ((Scheme.Modules.pullback iA).obj H)

theorem

unit_app_comp_ΦGen

L705 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.unit_app_comp_ΦGen

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section unitfactor
variable {ZA ZB X : Scheme.{u}} (iA : ZA ⟶ X) (iB : ZB ⟶ X) (k : ZA ⟶ ZB) (hk : k ≫ iB = iA)

Lean source · statement

theorem unit_app_comp_ΦGen (H : X.Modules) :
    (Scheme.Modules.pullbackPushforwardAdjunction iB).unit.app H ≫ ΦGen iA iB k hk H =
      (Scheme.Modules.pullbackPushforwardAdjunction iA).unit.app H

theorem

ΦGen_app_surjective

L711 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.ΦGen_app_surjective

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section unitfactor
variable {ZA ZB X : Scheme.{u}} (iA : ZA ⟶ X) (iB : ZB ⟶ X) (k : ZA ⟶ ZB) (hk : k ≫ iB = iA)

Lean source · statement

theorem ΦGen_app_surjective (H : X.Modules) (W : X.Opens)
    (h : Function.Surjective
      (((Scheme.Modules.pullbackPushforwardAdjunction k).unit.app
        ((Scheme.Modules.pullback iB).obj H)).app
        (iB ⁻¹ᵁ W))) :
    Function.Surjective ((ΦGen iA iB k hk H).app W)

theorem

nonempty_iso_unit_of_field

L723 ↗

Ws31ThetaF.nonempty_iso_unit_of_field

Source context 17 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules

Lean source · statement

theorem _root_.Ws31ThetaF.nonempty_iso_unit_of_field {K : Type u} [Field K]
    (L : (Spec (CommRingCat.of K)).Modules)
    (hL : IsInvertible L) : Nonempty (L ≅ 𝟙_ (Spec (CommRingCat.of K)).Modules)

theorem

nonempty_pullback_rigSection_module_iso

L735 ↗

Ws31ThetaF.nonempty_pullback_rigSection_module_iso

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

theorem _root_.Ws31ThetaF.nonempty_pullback_rigSection_module_iso :
    Nonempty ((Scheme.Modules.pullback (rigSection c t ε)).obj (sectionIdeal c ε t).module ≅
      𝟙_ T.Modules)

theorem

nonempty_pushforward_thickening_one_iso

L742 ↗

Ws31ThetaF.nonempty_pushforward_thickening_one_iso

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))
omit [SmoothOfRelativeDimension 1 c] in

Lean source · statement

theorem _root_.Ws31ThetaF.nonempty_pushforward_thickening_one_iso
    (H : (pullback c t).Modules) :
    Nonempty ((Scheme.Modules.pushforward (pullback.snd c t)).obj
      ((Scheme.Modules.pushforward (sectionIdeal c ε t).subschemeι).obj
        ((Scheme.Modules.pullback (sectionIdeal c ε t).subschemeι).obj H)) ≅
      (Scheme.Modules.pullback (rigSection c t ε)).obj H)

abbrev

Fm

L753 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.Fm

Documentation

Push forward the given module sheaf restricted to the subscheme cut out by the m-th power of the section ideal.

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

abbrev Fm (m : ℕ) (H : (pullback c t).Modules) : T.Modules

theorem

lf_gen

L758 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.lf_gen

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

theorem lf_gen [LocallyOfFiniteType t] (H : (pullback c t).Modules) (hH : IsInvertible H)
    (m : ℕ) (hm : m ≠ 0) :
    IsLocallyFreeOfRank m (Fm k c ε t m H)

lemma

pow_succ_le

L773 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.pow_succ_le

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))
omit [IsProper c] [SmoothOfRelativeDimension 1 c] in

Lean source · statement

lemma pow_succ_le (m : ℕ) : sectionIdeal c ε t ^ (m + 1) ≤ sectionIdeal c ε t

def

kIncl

L781 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.kIncl

Documentation

The inclusion of the section subscheme into the thickening cut out by the (m + 1)-st power of its ideal.

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

def kIncl (m : ℕ) :
    (sectionIdeal c ε t).subscheme ⟶ (sectionIdeal c ε t ^ (m + 1)).subscheme

lemma

kIncl_comp

L787 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.kIncl_comp

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))
omit [IsProper c] [SmoothOfRelativeDimension 1 c] in

Lean source · statement

lemma kIncl_comp (m : ℕ) :
    kIncl k c ε t m ≫ (sectionIdeal c ε t ^ (m + 1)).subschemeι =
      (sectionIdeal c ε t).subschemeι

instance

(anonymous instance)

L792 ↗

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

instance (m : ℕ) : IsClosedImmersion (kIncl k c ε t m)

def

νT

L798 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.νT

Documentation

The adjunction unit restricting a module sheaf on the thickening to the section subscheme.

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

def νT (H : (pullback c t).Modules) (m : ℕ)

lemma

Φ_app_surjective

L803 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.Φ_app_surjective

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))
omit [IsProper c] [SmoothOfRelativeDimension 1 c] in

Lean source · statement

lemma Φ_app_surjective (H : (pullback c t).Modules) (m : ℕ) (W : (pullback c t).Opens)
    (h : Function.Surjective
      ((νT k c ε t H m).app (((sectionIdeal c ε t ^ (m + 1)).subschemeι) ⁻¹ᵁ W))) :
    Function.Surjective
      ((ΦGen (sectionIdeal c ε t).subschemeι ((sectionIdeal c ε t ^ (m + 1)).subschemeι)
        (kIncl k c ε t m) (kIncl_comp k c ε t m) H).app W)

theorem

g_eq_of_pin

L810 ↗

JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening.Ws31ThetaF.g_eq_of_pin

Source context 19 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ShortExact_map_pushforward_thickening
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module
  TwoChartCech.Cover.A01_algebra
  TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing
  TwoChartCech.Cover.A1_algebra
  TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup
  TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup
  TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower
  TwoChartCech.Sections.M01_moduleA
  TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA
  TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing
  TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module
  CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module
  CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free
  CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite
  CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  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
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0
  TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq
  TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec
  TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply
  TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply
  CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  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
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.CategoryTheory.MonoidalCategory
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.TensorProduct
namespace Ws31ThetaF
open AlgebraicGeometry.Scheme.Modules
section setting2
variable (k : Type u) [Field k] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of k))
  [IsProper c] [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of k))) c)
  {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of k))

Lean source · statement

theorem g_eq_of_pin (H : (pullback c t).Modules) (hH : IsInvertible H) (m : ℕ)
    (S : ShortComplex (pullback c t).Modules)
    (e₂ : S.X₂ ≅ (Scheme.Modules.pushforward ((sectionIdeal c ε t ^ (m + 1)).subschemeι)).obj
        ((Scheme.Modules.pullback ((sectionIdeal c ε t ^ (m + 1)).subschemeι)).obj H))
    (e₃ : S.X₃ ≅ (Scheme.Modules.pushforward (sectionIdeal c ε t).subschemeι).obj
        ((Scheme.Modules.pullback (sectionIdeal c ε t).subschemeι).obj H))
    (hpin :
      (Scheme.Modules.pullbackPushforwardAdjunction
        ((sectionIdeal c ε t ^ (m + 1)).subschemeι)).unit.app H ≫
        e₂.inv ≫ S.g ≫ e₃.hom
      = (Scheme.Modules.pullbackPushforwardAdjunction (sectionIdeal c ε t).subschemeι).unit.app H) :
    S.g = e₂.hom ≫
      ΦGen (sectionIdeal c ε t).subschemeι ((sectionIdeal c ε t ^ (m + 1)).subschemeι)
        (kIncl k c ε t m) (kIncl_comp k c ε t m) H ≫ e₃.inv

theorem