Module

exists_openChart_openImmersion_relSubPicPresheaf_algEquivZeroCut_of_fibrewise_zeroScheme.lean

Nyll.JacobianVarieties.Picard.exists_openChart_openImmersion_relSubPicPresheaf_algEquivZeroCut_of_fibrewise_zeroScheme

49 declarations indexed from sourceView source ↗

Module documentation

An open Picard chart from fibrewise zero schemes

Supported Cartier divisors and the open cohomology-vanishing locus construct a chart with the required factorization and uniqueness properties.

Imports63

Mathlib.Algebra.Category.Ring.Basic
Mathlib.AlgebraicGeometry.Limits
Mathlib.AlgebraicGeometry.Morphisms.FiniteType
Mathlib.AlgebraicGeometry.Morphisms.Flat
Mathlib.AlgebraicGeometry.Morphisms.Proper
Mathlib.AlgebraicGeometry.Morphisms.Separated
Mathlib.AlgebraicGeometry.Morphisms.Smooth
Mathlib.AlgebraicGeometry.Noetherian
Mathlib.AlgebraicGeometry.OpenImmersion
Mathlib.AlgebraicGeometry.ResidueField
Mathlib.AlgebraicGeometry.Restrict
Mathlib.AlgebraicGeometry.Scheme
Mathlib.CategoryTheory.Comma.Over.Basic
Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Defs
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Pasting
Mathlib.CategoryTheory.Yoneda
Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
Mathlib.FieldTheory.IsAlgClosed.Basic
Mathlib.LinearAlgebra.Dimension.Finrank
Mathlib.RingTheory.Noetherian.Defs
Mathlib.RingTheory.Spectrum.Prime.Topology
Mathlib.Tactic.CategoryTheory.Reassoc
Mathlib.Topology.Compactness.Compact
Mathlib.Topology.Defs.Filter
Mathlib.Topology.NoetherianSpace
Nyll.AlgebraicCurves.Geometry.Picard.Functor
Nyll.JacobianVarieties.Picard.Representation
Nyll.AlgebraicCurves.Geometry.Picard.AlgEquivZeroCut
Nyll.AlgebraicCurves.Geometry.Picard.RelSubPic.Presheaf
Nyll.SchemeTheory.CategoryTheory.OverTotalPresheaf
Nyll.JacobianVarieties.Picard.ThetaBundle
Nyll.SchemeTheory.Cohomology.Cech.TwoAffineOpenCover
Nyll.AlgebraicCurves.Cohomology.Cech.TwoChartCechSectionsOf
Nyll.AlgebraicCurves.RelativeDivisors.RelEffCartierDiv
Nyll.AlgebraicCurves.RelativeDivisors.RelEffCartierDivSupportedIn
Nyll.AlgebraicCurves.Geometry.Sheaves.IdealSheafModule
Nyll.SchemeTheory.Modules.Monoidal
Nyll.AlgebraicCurves.RelativeDivisors.RelEffCartierDivTwist2
Nyll.AlgebraicCurves.Geometry.Sheaves.ModulesSectionZeroScheme
Nyll.AlgebraicCurves.RelativeDivisors.RelEffCartierDivFunctor
Nyll.SchemeTheory.Modules.PullbackMonoidal
Nyll.AlgebraicCurves.Geometry.Sheaves.IdealSheafModuleMaps
Nyll.AlgebraicCurves.Geometry.Sheaves.ModulesRigidify
Nyll.AlgebraicCurves.Geometry.Picard.isInvertible_sectionIdeal_of_range_subset
Nyll.AlgebraicCurves.RelativeDivisors.RelEffCartierDiv_isInvertible_I_of_supportedIn
Nyll.JacobianVarieties.Picard.nonempty_pullback_sectionTwist_iso_of_range_subset
Nyll.AlgebraicCurves.RelativeDivisors.RelEffCartierDiv_nonempty_twistModule_pullbackAlong_iso_pullback_of_supportedIn
Nyll.AlgebraicCurves.RelativeDivisors.RelEffCartierDiv_isInvertible_twistModule_and_nonempty_pullback_iso_of_supportedIn
Nyll.AlgebraicCurves.RelativeDivisors.RelEffCartierDiv_exists_supportedIn_I_eq_mul_of_supportedIn
Nyll.JacobianVarieties.Picard.exists_relEffCartierDiv_supportedIn_twistModule_iso_of_subsingleton_H1_of_zeroScheme
Nyll.JacobianVarieties.Picard.relEffCartierDiv_eq_pullbackAlong_of_twistModule_iso_of_supportedIn_of_zeroScheme
Nyll.JacobianVarieties.Picard.exists_opens_range_subset_iff_forall_subsingleton_H1_fibre_of_twoAffineOpenCover
Nyll.AlgebraicCurves.Geometry.Schemes.Scheme_TwoAffineOpenCover_exists_linearEquiv_sectionsOf_of_iso
Nyll.AlgebraicCurves.Geometry.Sheaves.Scheme_IdealSheafData_IsInvertible_isIso_pullbackModuleComparison
Nyll.AlgebraicCurves.Geometry.Sheaves.Scheme_IdealSheafData_IsInvertible_pow
Nyll.AlgebraicCurves.Geometry.Sheaves.Scheme_IdealSheafData_IsInvertible_isInvertible_invModule
Nyll.AlgebraicCurves.Geometry.Sheaves.Scheme_IdealSheafData_IsInvertible_isInvertible_module
Nyll.AlgebraicCurves.Geometry.Sheaves.Scheme_Modules_IsInvertible_tensor
Nyll.AlgebraicCurves.Geometry.Sheaves.Scheme_IdealSheafData_comap_mul
Nyll.AlgebraicCurves.Geometry.Sheaves.Scheme_IdealSheafData_IsInvertible_mul_left_cancel
Nyll.JacobianVarieties.Picard.RepresentsRelSubPicPresheaf

Imported by2

theorem

pullbackAlong_congr_hom

L335 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.pullbackAlong_congr_hom

Documentation

Pullback of a Cartier divisor depends only on the underlying morphism.

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · statement

theorem pullbackAlong_congr_hom {𝒞 S : Scheme.{u}} {f : 𝒞 ⟶ S} {n : ℕ} {T T' : Scheme.{u}}
    {g : T ⟶ S}
    {g' : T' ⟶ S} (D : RelEffCartierDiv f n g') {φ₁ φ₂ : T ⟶ T'} (h : φ₁ = φ₂) (h₁ : φ₁ ≫ g' = g)
    (h₂ : φ₂ ≫ g' = g) : D.pullbackAlong φ₁ h₁ = D.pullbackAlong φ₂ h₂

theorem

pullbackAlong_I_eq_mul

L342 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.pullbackAlong_I_eq_mul

Documentation

Pullback preserves a product identity between Cartier ideals.

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · statement

theorem pullbackAlong_I_eq_mul {𝒞 S : Scheme.{u}} {f : 𝒞 ⟶ S} {n m l : ℕ}
    {T' : Scheme.{u}} {g' : T' ⟶ S} (D : RelEffCartierDiv f n g') (D₀ : RelEffCartierDiv f m g')
    (E : RelEffCartierDiv f l g') (hD : D.I = D₀.I * E.I) {T : Scheme.{u}} {g : T ⟶ S} (φ : T ⟶ T')
    (hφ : φ ≫ g' = g) :
    (D.pullbackAlong φ hφ).I = (D₀.pullbackAlong φ hφ).I * (E.pullbackAlong φ hφ).I

theorem

OverTotal_snd_eq_of_eq

L351 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.OverTotal_snd_eq_of_eq

Documentation

Equal elements of the total presheaf have compatible second components.

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · full declaration

theorem OverTotal_snd_eq_of_eq {Cat : Type*} [Category Cat] {S : Cat} {G : (Over S)ᵒᵖ ⥤ Type*}
    {T : Cat}
    {a b : G.OverTotal T} (h : a = b) :
    b.2 = G.map (Over.homMk (𝟙 T)
      (by subst h; exact Category.id_comp _) : Over.mk b.1 ⟶ Over.mk a.1).op a.2 := by
  subst h
  exact (Functor.map_op_apply_eq_self _ rfl _).symm

theorem

RigidifiedLineBundle_pullbackAlong_L_eq

L360 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.RigidifiedLineBundle_pullbackAlong_L_eq

Documentation

Pullbacks with equal underlying parameter maps have equal modules.

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · statement

theorem RigidifiedLineBundle_pullbackAlong_L_eq {R : Type u} [CommRing R] {C : Scheme.{u}}
    {c : C ⟶ Spec (CommRingCat.of R)} {ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c}
    {T T' : Scheme.{u}} {t : T ⟶ Spec (CommRingCat.of R)} {t' : T' ⟶ Spec (CommRingCat.of R)}
    (M : RigidifiedLineBundle c ε t) {ψ ψ' : SchemeHomOver t' t} (h : ψ.1 = ψ'.1) :
    (M.pullbackAlong ψ).L = (M.pullbackAlong ψ').L

theorem

rigSection_snd'

L378 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.rigSection_snd'

Documentation

The rigidifying section projects to the identity.

Source context 26 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]
omit [IsSeparated c] [SmoothOfRelativeDimension 1 (U.ι ≫ c)] in

Lean source · statement

theorem rigSection_snd' {V : Scheme.{u}} (u : V ⟶ Spec (CommRingCat.of R)) :
    rigSection c u ε ≫ pullback.snd c u = 𝟙 V

theorem

isInvertible_sectionIdeal

L384 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.isInvertible_sectionIdeal

Documentation

A section supported in the smooth open has an invertible ideal.

Source context 26 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]
include hεU in

Lean source · statement

theorem isInvertible_sectionIdeal {V : Scheme.{u}} (u : V ⟶ Spec (CommRingCat.of R)) :
    (sectionIdeal c ε u).IsInvertible

theorem

isInvertible_sectionTwist

L390 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.isInvertible_sectionTwist

Documentation

Every power of the inverse section ideal is invertible.

Source context 26 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]
include hεU in

Lean source · statement

theorem isInvertible_sectionTwist {V : Scheme.{u}} (u : V ⟶ Spec (CommRingCat.of R)) (n : ℕ) :
    Scheme.Modules.IsInvertible (sectionTwist c ε u n)

abbrev

Mγ

L395 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.Mγ

Documentation

The chart module twisted by the section and the fixed Cartier ideal.

Source context 25 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]

Lean source · statement

abbrev Mγ {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
    {V : Scheme.{u}}
    (u : V ⟶ Spec (CommRingCat.of R)) (LV : RigidifiedLineBundle c ε u) : (pullback c u).Modules

theorem

isInvertible_Dγ

L401 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.isInvertible_Dγ

Documentation

The pulled-back fixed Cartier ideal is invertible.

Source context 25 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]

Lean source · statement

theorem isInvertible_Dγ {e : ℕ} (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
    (hDγ : Dγ.SupportedIn U)
    {V : Scheme.{u}} (u : V ⟶ Spec (CommRingCat.of R)) :
    (Dγ.pullbackAlong u (Category.comp_id u)).I.IsInvertible

theorem

isInvertible_Mγ

L409 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.isInvertible_Mγ

Documentation

The twisted chart module is invertible.

Source context 26 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]
include hεU in

Lean source · statement

theorem isInvertible_Mγ {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
    (hDγ : Dγ.SupportedIn U)
    {V : Scheme.{u}} (u : V ⟶ Spec (CommRingCat.of R)) (LV : RigidifiedLineBundle c ε u) :
    Scheme.Modules.IsInvertible (Mγ R c ε n Dγ u LV)

def

fibreIso

L426 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.fibreIso

Documentation

The comparison of fibres before and after changing the base, induced by pullback associativity.

Source context 27 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
  Dγ.SupportedIn U)
  {T T' : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (t' : T' ⟶ Spec (CommRingCat.of R))
  (ψ : SchemeHomOver t' t) (L : RigidifiedLineBundle c ε t)
  {k : Type u} [Field k] (s' : Spec (CommRingCat.of k) ⟶ T')

Lean source · statement

def fibreIso : pullback (pullback.snd c t) (s' ≫ ψ.1) ≅ pullback (pullback.snd c t')
  s'

theorem

fibreIso_hom_fibreAt

L433 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.fibreIso_hom_fibreAt

Documentation

The fibre comparison commutes with the structure morphism.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
  Dγ.SupportedIn U)
  {T T' : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (t' : T' ⟶ Spec (CommRingCat.of R))
  (ψ : SchemeHomOver t' t) (L : RigidifiedLineBundle c ε t)
  {k : Type u} [Field k] (s' : Spec (CommRingCat.of k) ⟶ T')
omit [IsSeparated c] in

Lean source · statement

theorem fibreIso_hom_fibreAt :
    (fibreIso R c t t' ψ s').hom ≫ fibreAt c t' s' = fibreAt c t (s' ≫ ψ.1)

theorem

fibreIso_hom_fst_baseChangeSnd

L441 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.fibreIso_hom_fst_baseChangeSnd

Documentation

The fibre comparison commutes with the projection to the original family.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
  Dγ.SupportedIn U)
  {T T' : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (t' : T' ⟶ Spec (CommRingCat.of R))
  (ψ : SchemeHomOver t' t) (L : RigidifiedLineBundle c ε t)
  {k : Type u} [Field k] (s' : Spec (CommRingCat.of k) ⟶ T')
omit [IsSeparated c] in

Lean source · statement

theorem fibreIso_hom_fst_baseChangeSnd :
    (fibreIso R c t t' ψ s').hom ≫ pullback.fst (pullback.snd c t') s' ≫ baseChangeSnd c ψ =
      pullback.fst (pullback.snd c t) (s' ≫ ψ.1)

theorem

nonempty_Mγ_iso_pullback

L459 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.nonempty_Mγ_iso_pullback

Documentation

The chart module commutes with base change.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
  Dγ.SupportedIn U)
  {T T' : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (t' : T' ⟶ Spec (CommRingCat.of R))
  (ψ : SchemeHomOver t' t) (L : RigidifiedLineBundle c ε t)
  {k : Type u} [Field k] (s' : Spec (CommRingCat.of k) ⟶ T')
include hεU hDγ in

Lean source · statement

theorem nonempty_Mγ_iso_pullback :
    Nonempty (Mγ R c ε n Dγ t' (L.pullbackAlong ψ) ≅
      (Scheme.Modules.pullback (baseChangeSnd c ψ)).obj (Mγ R c ε n Dγ t L))

theorem

nonempty_fibreModule_iso

L485 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.nonempty_fibreModule_iso

Documentation

The two fibre descriptions of the chart module are isomorphic.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
  Dγ.SupportedIn U)
  {T T' : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (t' : T' ⟶ Spec (CommRingCat.of R))
  (ψ : SchemeHomOver t' t) (L : RigidifiedLineBundle c ε t)
  {k : Type u} [Field k] (s' : Spec (CommRingCat.of k) ⟶ T')
include hεU hDγ in

Lean source · statement

theorem nonempty_fibreModule_iso :
    Nonempty (fibreModule c t (s' ≫ ψ.1) (Mγ R c ε n Dγ t L) ≅
      (Scheme.Modules.pullback (fibreIso R c t t' ψ s').hom).obj
        (fibreModule c t' s' (Mγ R c ε n Dγ t' (L.pullbackAlong ψ))))

theorem

fibre_transport_to_t

L501 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.fibre_transport_to_t

Documentation

Transport first cohomology from the changed base to the original parameter.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
  Dγ.SupportedIn U)
  {T T' : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (t' : T' ⟶ Spec (CommRingCat.of R))
  (ψ : SchemeHomOver t' t) (L : RigidifiedLineBundle c ε t)
  {k : Type u} [Field k] (s' : Spec (CommRingCat.of k) ⟶ T')
include hεU hDγ in

Lean source · statement

theorem fibre_transport_to_t (𝒲 : (pullback (pullback.snd c t') s').TwoAffineOpenCover) :
    ∃ 𝒱 : (pullback (pullback.snd c t) (s' ≫ ψ.1)).TwoAffineOpenCover,
      Nonempty ((𝒱.sectionsOf (fibreAt c t (s' ≫ ψ.1)) (fibreModule c t (s' ≫ ψ.1) (Mγ R c ε n Dγ t
        L))).H1 ≃ₗ[k]
        (𝒲.sectionsOf (fibreAt c t' s') (fibreModule c t' s' (Mγ R c ε n Dγ t' (L.pullbackAlong
          ψ)))).H1)

theorem

fibre_transport_to_t'

L515 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.fibre_transport_to_t'

Documentation

Transport first cohomology from the original parameter to the changed base.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
  Dγ.SupportedIn U)
  {T T' : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (t' : T' ⟶ Spec (CommRingCat.of R))
  (ψ : SchemeHomOver t' t) (L : RigidifiedLineBundle c ε t)
  {k : Type u} [Field k] (s' : Spec (CommRingCat.of k) ⟶ T')
include hεU hDγ in

Lean source · statement

theorem fibre_transport_to_t' (𝒲 : (pullback (pullback.snd c t) (s' ≫ ψ.1)).TwoAffineOpenCover) :
    ∃ 𝒱 : (pullback (pullback.snd c t') s').TwoAffineOpenCover,
      Nonempty ((𝒱.sectionsOf (fibreAt c t' s') (fibreModule c t' s' (Mγ R c ε n Dγ t'
        (L.pullbackAlong ψ)))).H1 ≃ₗ[k]
        (𝒲.sectionsOf (fibreAt c t (s' ≫ ψ.1)) (fibreModule c t (s' ≫ ψ.1) (Mγ R c ε n Dγ t
          L))).H1)

theorem

forall_subsingleton_H1_pullbackAlong

L535 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.forall_subsingleton_H1_pullbackAlong

Documentation

Vanishing of first cohomology is preserved by parameter pullback.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
  Dγ.SupportedIn U)
  {T T' : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (t' : T' ⟶ Spec (CommRingCat.of R))
  (ψ : SchemeHomOver t' t) (L : RigidifiedLineBundle c ε t)
  {k : Type u} [Field k] (s' : Spec (CommRingCat.of k) ⟶ T')
include hεU hDγ in

Lean source · statement

theorem forall_subsingleton_H1_pullbackAlong
    (h : ∀ 𝒲 : (pullback (pullback.snd c t) (s' ≫ ψ.1)).TwoAffineOpenCover,
      Subsingleton (𝒲.sectionsOf (fibreAt c t (s' ≫ ψ.1)) (fibreModule c t (s' ≫ ψ.1) (Mγ R c ε n
        Dγ t L))).H1)
    (𝒲 : (pullback (pullback.snd c t') s').TwoAffineOpenCover) :
    Subsingleton (𝒲.sectionsOf (fibreAt c t' s') (fibreModule c t' s' (Mγ R c ε n Dγ t'
      (L.pullbackAlong ψ)))).H1

theorem

forall_subsingleton_H1_of_pullbackAlong

L548 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.forall_subsingleton_H1_of_pullbackAlong

Documentation

Recover first-cohomology vanishing from the parameter pullback.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
  Dγ.SupportedIn U)
  {T T' : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (t' : T' ⟶ Spec (CommRingCat.of R))
  (ψ : SchemeHomOver t' t) (L : RigidifiedLineBundle c ε t)
  {k : Type u} [Field k] (s' : Spec (CommRingCat.of k) ⟶ T')
include hεU hDγ in

Lean source · statement

theorem forall_subsingleton_H1_of_pullbackAlong
    (h : ∀ 𝒲 : (pullback (pullback.snd c t') s').TwoAffineOpenCover,
      Subsingleton (𝒲.sectionsOf (fibreAt c t' s') (fibreModule c t' s' (Mγ R c ε n Dγ t'
        (L.pullbackAlong ψ)))).H1)
    (𝒲 : (pullback (pullback.snd c t) (s' ≫ ψ.1)).TwoAffineOpenCover) :
    Subsingleton (𝒲.sectionsOf (fibreAt c t (s' ≫ ψ.1)) (fibreModule c t (s' ≫ ψ.1) (Mγ R c ε n Dγ
      t L))).H1

theorem

forall_subsingleton_H1_of_iso

L563 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.forall_subsingleton_H1_of_iso

Documentation

First-cohomology vanishing is invariant under a module isomorphism.

Source context 26 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Separated
variable [IsSeparated c]
omit [IsSeparated c] in

Lean source · statement

theorem forall_subsingleton_H1_of_iso {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R))
    {F F' : (pullback c t).Modules} (i : F ≅ F') {k : Type u} [Field k] (s : Spec (CommRingCat.of
      k) ⟶ T)
    (h : ∀ 𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover,
      Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s F)).H1)
    (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover) :
    Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s F')).H1

theorem

uliftYonedaEquiv_uliftYoneda_map_comp

L587 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.uliftYonedaEquiv_uliftYoneda_map_comp

Documentation

The lifted Yoneda equivalence is natural in the source.

Source context 26 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
section Generic
omit [SmoothOfRelativeDimension 1 (U.ι ≫ c)]

Lean source · statement

theorem uliftYonedaEquiv_uliftYoneda_map_comp {F : Scheme.{u}ᵒᵖ ⥤ Type (u + 1)} {T T' : Scheme.{u}}
    (ψ : T' ⟶ T)
    (f : uliftYoneda.{u + 1}.obj T ⟶ F) :
    uliftYonedaEquiv.{u + 1, u, u + 1} (uliftYoneda.{u + 1}.map ψ ≫ f) =
      F.map ψ.op (uliftYonedaEquiv.{u + 1, u, u + 1} f)

def

pointElem

L595 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.pointElem

Documentation

The total-presheaf element defined by a rigidified algebraically trivial bundle.

Source context 26 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
section Generic
omit [SmoothOfRelativeDimension 1 (U.ι ≫ c)]

Lean source · statement

def pointElem {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R))
    (L : RigidifiedLineBundle c ε t)
    (hL : FibrewiseAlgEquivZero L) :
    (relSubPicPresheaf c ε (algEquivZeroCut c ε)).overTotal.obj (op T)

theorem

pointElem_fst

L602 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.pointElem_fst

Documentation

The first component of the presheaf point is the structure morphism.

Source context 26 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
section Generic
omit [SmoothOfRelativeDimension 1 (U.ι ≫ c)]

Lean source · statement

theorem pointElem_fst {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R))
    (L : RigidifiedLineBundle c ε t)
    (hL : FibrewiseAlgEquivZero L) : (pointElem R c ε t L hL).1 = t

def

LY

L662 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.LY

Documentation

The rigidified bundle defined by the universal twisted Cartier divisor.

Source context 25 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)

Lean source · statement

def LY : RigidifiedLineBundle c ε y

theorem

LY_L

L669 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.LY_L

Documentation

The underlying module of the universal rigidified bundle.

Source context 26 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
omit [IsNoetherianRing R] [LocallyOfFiniteType y] in

Lean source · statement

theorem LY_L : (LY R c ε U hεU g e y DY hDYU).L = DY.twistModule c ε

def

hulift

L679 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.hulift

Documentation

The unique parameter morphism classifying the supported Cartier divisor.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
section HunivAPI
omit [SmoothOfRelativeDimension 1 (U.ι ≫ c)] [IsNoetherianRing R] [IsProper c]
  [LocallyOfFiniteType y]
variable {T₁ : Scheme.{u}} {g₁ : T₁ ⟶ Spec (CommRingCat.of R)} (D₁ : RelEffCartierDiv c g g₁) (hD₁
  : D₁.SupportedIn U)

Lean source · statement

def hulift : T₁ ⟶ Y

theorem

hulift_comp

L682 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.hulift_comp

Documentation

The classifying morphism is a morphism over the base.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
section HunivAPI
omit [SmoothOfRelativeDimension 1 (U.ι ≫ c)] [IsNoetherianRing R] [IsProper c]
  [LocallyOfFiniteType y]
variable {T₁ : Scheme.{u}} {g₁ : T₁ ⟶ Spec (CommRingCat.of R)} (D₁ : RelEffCartierDiv c g g₁) (hD₁
  : D₁.SupportedIn U)

Lean source · statement

theorem hulift_comp : hulift R c U g y 𝒟 huniv D₁ hD₁ ≫ y = g₁

theorem

pullsBackOver_hulift

L686 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.pullsBackOver_hulift

Documentation

The classifying morphism pulls back the universal Cartier divisor.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
section HunivAPI
omit [SmoothOfRelativeDimension 1 (U.ι ≫ c)] [IsNoetherianRing R] [IsProper c]
  [LocallyOfFiniteType y]
variable {T₁ : Scheme.{u}} {g₁ : T₁ ⟶ Spec (CommRingCat.of R)} (D₁ : RelEffCartierDiv c g g₁) (hD₁
  : D₁.SupportedIn U)

Lean source · statement

theorem pullsBackOver_hulift : PullsBackOver 𝒟 (hulift R c U g y 𝒟 huniv D₁ hD₁) (hulift_comp R c U
  g y 𝒟 huniv D₁ hD₁) D₁

theorem

pullbackAlong_hulift

L691 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.pullbackAlong_hulift

Documentation

Pullback along the classifying morphism recovers the divisor.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
section HunivAPI
omit [SmoothOfRelativeDimension 1 (U.ι ≫ c)] [IsNoetherianRing R] [IsProper c]
  [LocallyOfFiniteType y]
variable {T₁ : Scheme.{u}} {g₁ : T₁ ⟶ Spec (CommRingCat.of R)} (D₁ : RelEffCartierDiv c g g₁) (hD₁
  : D₁.SupportedIn U)

Lean source · statement

theorem pullbackAlong_hulift :
    𝒟.pullbackAlong (hulift R c U g y 𝒟 huniv D₁ hD₁) (hulift_comp R c U g y 𝒟 huniv D₁ hD₁) = D₁

theorem

eq_hulift

L696 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.eq_hulift

Documentation

A morphism classifying the same divisor equals the universal lift.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
section HunivAPI
omit [SmoothOfRelativeDimension 1 (U.ι ≫ c)] [IsNoetherianRing R] [IsProper c]
  [LocallyOfFiniteType y]
variable {T₁ : Scheme.{u}} {g₁ : T₁ ⟶ Spec (CommRingCat.of R)} (D₁ : RelEffCartierDiv c g g₁) (hD₁
  : D₁.SupportedIn U)

Lean source · statement

theorem eq_hulift {φ : T₁ ⟶ Y} (hφ : φ ≫ y = g₁) (h : PullsBackOver 𝒟 φ hφ D₁) :
    φ = hulift R c U g y 𝒟 huniv D₁ hD₁

theorem

fibrewiseAlgEquivZero_LY_X

L720 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.fibrewiseAlgEquivZero_LY_X

Documentation

The universal bundle is algebraically trivial over the intersection of the two opens.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
omit [IsNoetherianRing R] hDγ hsect hZfibγ hH0one [LocallyOfFiniteType y] h𝒟U huniv hDY hUY in

Lean source · statement

theorem fibrewiseAlgEquivZero_LY_X :
    FibrewiseAlgEquivZero ((LY R c ε U hεU g e y DY hDYU).pullbackAlong
      (⟨(UY ⊓ WY).ι, rfl⟩ : SchemeHomOver ((UY ⊓ WY).ι ≫ y) y))

def

chartElem

L748 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.chartElem

Documentation

The total-presheaf element on the chosen open chart.

Source context 27 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY

Lean source · statement

def chartElem :
    (relSubPicPresheaf c ε (algEquivZeroCut c ε)).overTotal.obj (op ((UY ⊓ WY : Y.Opens) :
      Scheme.{u}))

theorem

chartElem_fst

L759 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.chartElem_fst

Documentation

The chart element has the restricted parameter structure map.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
omit [IsNoetherianRing R] hDγ hsect hZfibγ hH0one
  [LocallyOfFiniteType y] h𝒟U huniv hDY hUY in

Lean source · statement

theorem chartElem_fst : (chartElem (R := R) (c := c) (ε := ε) (U := U) (hεU := hεU)
  (g := g) (e := e) (y := y) (DY := DY) (hDYU := hDYU) (UY := UY) (WY := WY) (hWY := hWY)).1 = (UY
  ⊓ WY).ι ≫ y

def

chartMap

L764 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.chartMap

Documentation

The Yoneda morphism associated with the open-chart element.

Source context 27 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY

Lean source · statement

def chartMap :
    uliftYoneda.{u + 1}.obj ((UY ⊓ WY : Y.Opens) : Scheme.{u}) ⟶ (relSubPicPresheaf c ε
      (algEquivZeroCut c ε)).overTotal

theorem

uliftYonedaEquiv_chartMap

L773 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.uliftYonedaEquiv_chartMap

Documentation

The Yoneda equivalence recovers the chart element.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
omit [IsNoetherianRing R] hDγ hsect hZfibγ hH0one
  [LocallyOfFiniteType y] h𝒟U huniv hDY hUY in

Lean source · statement

theorem uliftYonedaEquiv_chartMap :
    uliftYonedaEquiv.{u + 1, u, u + 1} (chartMap (R := R) (c := c) (ε := ε) (U := U) (hεU := hεU)
      (g := g) (e := e) (y := y) (DY := DY) (hDYU := hDYU) (UY := UY) (WY := WY) (hWY := hWY)) =
      chartElem (R := R) (c := c) (ε := ε) (U := U) (hεU := hεU) (g := g) (e := e) (y := y) (DY :=
      DY) (hDYU := hDYU) (UY := UY) (WY := WY) (hWY := hWY)

theorem

uliftYonedaEquiv_map_comp_chartMap

L783 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.uliftYonedaEquiv_map_comp_chartMap

Documentation

Composition with the chart map is presheaf restriction.

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
omit [IsNoetherianRing R] hDγ hsect hZfibγ hH0one
  [LocallyOfFiniteType y] h𝒟U huniv hDY hUY in

Lean source · statement

theorem uliftYonedaEquiv_map_comp_chartMap {T : Scheme.{u}}
    (φ' : T ⟶ ((UY ⊓ WY : Y.Opens) : Scheme.{u})) :
    uliftYonedaEquiv.{u + 1, u, u + 1} (uliftYoneda.{u + 1}.map φ' ≫ chartMap (R := R) (c := c) (ε
      := ε) (U := U) (hεU := hεU) (g := g) (e := e) (y := y) (DY := DY) (hDYU := hDYU) (UY := UY)
      (WY := WY) (hWY := hWY)) =
      (relSubPicPresheaf c ε (algEquivZeroCut c ε)).overTotal.map φ'.op (chartElem (R := R) (c :=
        c) (ε := ε) (U := U) (hεU := hεU) (g := g) (e := e) (y := y) (DY := DY) (hDYU := hDYU) (UY
        := UY) (WY := WY) (hWY := hWY))

theorem

D_eq_pullbackAlong_lift

L813 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.D_eq_pullbackAlong_lift

Documentation

The lifted parameter recovers the twisted divisor.

Source context 31 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t] (L :
  RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (W : T.Opens)
  (hWiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (W : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s (Mγ R c ε (g + e) Dγ t
          L))).H1)
  (D : RelEffCartierDiv c (g + e) (W.ι ≫ t)) (D₀ : RelEffCartierDiv c g (W.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t)
    t)).L))
  (hD₀U : D₀.SupportedIn U)
include hL hWiff hD hDL hD₀U
omit [IsNoetherianRing R] hDγ hsect hZfibγ hH0one [LocallyOfFiniteType y] h𝒟U hUY hWY
  [LocallyOfFiniteType t] hL hWiff hDL in
omit [SmoothOfRelativeDimension 1 (U.ι ≫ c)] hεU [IsProper c] hDYU in

Lean source · statement

theorem D_eq_pullbackAlong_lift : D = DY.pullbackAlong (hulift R c U g y 𝒟 huniv D₀ hD₀U)
  (hulift_comp R c U g y 𝒟 huniv D₀ hD₀U)

theorem

nonempty_LY_pullbackAlong_lift_iso

L824 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.nonempty_LY_pullbackAlong_lift_iso

Documentation

The lifted universal bundle is isomorphic to the restricted input bundle.

Source context 30 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t] (L :
  RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (W : T.Opens)
  (hWiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (W : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s (Mγ R c ε (g + e) Dγ t
          L))).H1)
  (D : RelEffCartierDiv c (g + e) (W.ι ≫ t)) (D₀ : RelEffCartierDiv c g (W.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t)
    t)).L))
  (hD₀U : D₀.SupportedIn U)
include hL hWiff hD hDL hD₀U
omit [IsNoetherianRing R] hDγ hsect hZfibγ hH0one [LocallyOfFiniteType y] h𝒟U hUY hWY
  [LocallyOfFiniteType t] hL hWiff in

Lean source · statement

theorem nonempty_LY_pullbackAlong_lift_iso :
    Nonempty (((LY R c ε U hεU g e y DY hDYU).pullbackAlong (⟨hulift R c U g y 𝒟 huniv D₀ hD₀U,
      hulift_comp R c U g y 𝒟 huniv D₀ hD₀U⟩ : SchemeHomOver (W.ι ≫ t) y)).L ≅
      (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L)

theorem

subsingleton_H1_W

L841 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.subsingleton_H1_W

Documentation

First cohomology vanishes on the input open locus.

Source context 31 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t] (L :
  RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (W : T.Opens)
  (hWiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (W : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s (Mγ R c ε (g + e) Dγ t
          L))).H1)
  (D : RelEffCartierDiv c (g + e) (W.ι ≫ t)) (D₀ : RelEffCartierDiv c g (W.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t)
    t)).L))
  (hD₀U : D₀.SupportedIn U)
include hL hWiff hD hDL hD₀U
omit [IsNoetherianRing R] hsect hZfibγ hH0one [LocallyOfFiniteType y] h𝒟U huniv hDY hUY hWY
  [LocallyOfFiniteType t] hL hD hDL hD₀U in
omit hDYU in

Lean source · statement

theorem subsingleton_H1_W (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (W : Scheme.{u}))
    (𝒲 : (pullback (pullback.snd c (W.ι ≫ t)) s).TwoAffineOpenCover) :
    Subsingleton (𝒲.sectionsOf (fibreAt c (W.ι ≫ t) s) (fibreModule c (W.ι ≫ t) s
      (Mγ R c ε (g + e) Dγ (W.ι ≫ t) (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t)
        t))))).H1

theorem

subsingleton_H1_W'

L855 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.subsingleton_H1_W'

Documentation

The lifted universal chart module has vanishing first cohomology.

Source context 30 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t] (L :
  RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (W : T.Opens)
  (hWiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (W : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s (Mγ R c ε (g + e) Dγ t
          L))).H1)
  (D : RelEffCartierDiv c (g + e) (W.ι ≫ t)) (D₀ : RelEffCartierDiv c g (W.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t)
    t)).L))
  (hD₀U : D₀.SupportedIn U)
include hL hWiff hD hDL hD₀U
omit [IsNoetherianRing R] hsect hZfibγ hH0one [LocallyOfFiniteType y]
  h𝒟U hUY hWY [LocallyOfFiniteType t] hL in

Lean source · statement

theorem subsingleton_H1_W' (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (W : Scheme.{u}))
    (𝒲 : (pullback (pullback.snd c (W.ι ≫ t)) s).TwoAffineOpenCover) :
    Subsingleton (𝒲.sectionsOf (fibreAt c (W.ι ≫ t) s) (fibreModule c (W.ι ≫ t) s
      (Mγ R c ε (g + e) Dγ (W.ι ≫ t)
        ((LY R c ε U hεU g e y DY hDYU).pullbackAlong (⟨hulift R c U g y 𝒟 huniv D₀ hD₀U,
          hulift_comp R c U g y 𝒟 huniv D₀ hD₀U⟩ : SchemeHomOver (W.ι ≫ t) y))))).H1

theorem

range_lift_subset_UY

L872 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.range_lift_subset_UY

Documentation

The classifying map lands in the cohomology-vanishing open.

Source context 30 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t] (L :
  RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (W : T.Opens)
  (hWiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (W : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s (Mγ R c ε (g + e) Dγ t
          L))).H1)
  (D : RelEffCartierDiv c (g + e) (W.ι ≫ t)) (D₀ : RelEffCartierDiv c g (W.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t)
    t)).L))
  (hD₀U : D₀.SupportedIn U)
include hL hWiff hD hDL hD₀U
omit [IsNoetherianRing R] hsect hZfibγ hH0one [LocallyOfFiniteType y] h𝒟U [LocallyOfFiniteType t]
  hWY hL in

Lean source · statement

theorem range_lift_subset_UY : Set.range ⇑(hulift R c U g y 𝒟 huniv D₀ hD₀U) ⊆ (UY : Set Y)

theorem

range_lift_subset_WY

L890 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.range_lift_subset_WY

Documentation

The classifying map lands in the algebraic-equivalence open.

Source context 30 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t] (L :
  RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (W : T.Opens)
  (hWiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (W : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s (Mγ R c ε (g + e) Dγ t
          L))).H1)
  (D : RelEffCartierDiv c (g + e) (W.ι ≫ t)) (D₀ : RelEffCartierDiv c g (W.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t)
    t)).L))
  (hD₀U : D₀.SupportedIn U)
include hL hWiff hD hDL hD₀U
omit [IsNoetherianRing R] hDγ hsect hZfibγ hH0one
  [LocallyOfFiniteType y] h𝒟U hUY [LocallyOfFiniteType t] hWiff in

Lean source · statement

theorem range_lift_subset_WY : Set.range ⇑(hulift R c U g y 𝒟 huniv D₀ hD₀U) ⊆ (WY : Set Y)

theorem

range_lift_subset

L954 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.range_lift_subset

Documentation

The classifying map lands in the chosen chart intersection.

Source context 30 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t] (L :
  RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (W : T.Opens)
  (hWiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (W : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s (Mγ R c ε (g + e) Dγ t
          L))).H1)
  (D : RelEffCartierDiv c (g + e) (W.ι ≫ t)) (D₀ : RelEffCartierDiv c g (W.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t)
    t)).L))
  (hD₀U : D₀.SupportedIn U)
include hL hWiff hD hDL hD₀U
omit [IsNoetherianRing R] hsect hZfibγ hH0one [LocallyOfFiniteType y] h𝒟U [LocallyOfFiniteType t] in

Lean source · statement

theorem range_lift_subset : Set.range ⇑(hulift R c U g y 𝒟 huniv D₀ hD₀U) ⊆ ((UY ⊓ WY : Y.Opens) :
  Set Y)

def

φ

L965 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.φ

Documentation

The factorization of the classifying map through the open chart.

Source context 29 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t] (L :
  RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (W : T.Opens)
  (hWiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (W : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s (Mγ R c ε (g + e) Dγ t
          L))).H1)
  (D : RelEffCartierDiv c (g + e) (W.ι ≫ t)) (D₀ : RelEffCartierDiv c g (W.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t)
    t)).L))
  (hD₀U : D₀.SupportedIn U)
include hL hWiff hD hDL hD₀U

Lean source · statement

def φ : (W : Scheme.{u}) ⟶ ((UY ⊓ WY : Y.Opens) : Scheme.{u})

theorem

φ_ι

L977 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.φ_ι

Documentation

The open-chart lift composes to the original classifying map.

Source context 30 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t] (L :
  RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (W : T.Opens)
  (hWiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (W : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s (Mγ R c ε (g + e) Dγ t
          L))).H1)
  (D : RelEffCartierDiv c (g + e) (W.ι ≫ t)) (D₀ : RelEffCartierDiv c g (W.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t)
    t)).L))
  (hD₀U : D₀.SupportedIn U)
include hL hWiff hD hDL hD₀U
omit [IsNoetherianRing R] hsect hZfibγ hH0one [LocallyOfFiniteType y] h𝒟U [LocallyOfFiniteType t] in

Lean source · statement

theorem φ_ι : φ (R := R) (c := c) (ε := ε) (U := U) (hεU := hεU) (g := g) (e := e) (Dγ := Dγ) (hDγ
  := hDγ) (y := y) (𝒟 := 𝒟) (huniv := huniv) (DY := DY) (hDY := hDY) (hDYU := hDYU) (UY := UY) (hUY
  := hUY) (WY := WY) (hWY := hWY) (t := t) (L := L) (hL := hL) (W := W) (hWiff := hWiff) (D := D)
  (D₀ := D₀) (hD := hD) (hDL := hDL) (hD₀U := hD₀U) ≫ (UY ⊓ WY).ι = hulift R c U g y 𝒟 huniv D₀
  hD₀U

theorem

clause_i

L988 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.clause_i

Documentation

The input open is characterized by first-cohomology vanishing.

Source context 31 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t] (L :
  RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (W : T.Opens)
  (hWiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (W : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s (Mγ R c ε (g + e) Dγ t
          L))).H1)
  (D : RelEffCartierDiv c (g + e) (W.ι ≫ t)) (D₀ : RelEffCartierDiv c g (W.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t)
    t)).L))
  (hD₀U : D₀.SupportedIn U)
include hL hWiff hD hDL hD₀U
omit [IsNoetherianRing R] hDγ hsect hZfibγ hH0one [LocallyOfFiniteType y] h𝒟U huniv hDY hUY hWY
  [LocallyOfFiniteType t] hL hD hDL hD₀U in
omit [SmoothOfRelativeDimension 1 (U.ι ≫ c)] hεU [IsProper c] hDYU in

Lean source · statement

theorem clause_i (L' : RigidifiedLineBundle c ε t) (hL' : Nonempty (L'.L ≅ L.L))
    (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) :
    Set.range ⇑s ⊆ (W : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s (Mγ R c ε (g + e) Dγ t
          L'))).H1

theorem

clause_ii

L1000 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.clause_ii

Documentation

The open-chart factorization agrees with the input presheaf point.

Source context 30 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t] (L :
  RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (W : T.Opens)
  (hWiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (W : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s (Mγ R c ε (g + e) Dγ t
          L))).H1)
  (D : RelEffCartierDiv c (g + e) (W.ι ≫ t)) (D₀ : RelEffCartierDiv c g (W.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t)
    t)).L))
  (hD₀U : D₀.SupportedIn U)
include hL hWiff hD hDL hD₀U
omit [IsNoetherianRing R] hsect hZfibγ hH0one [LocallyOfFiniteType y] h𝒟U [LocallyOfFiniteType t] in

Lean source · statement

theorem clause_ii :
    uliftYoneda.{u + 1}.map (φ (R := R) (c := c) (ε := ε) (U := U) (hεU := hεU) (g := g) (e := e)
      (Dγ := Dγ) (hDγ := hDγ) (y := y) (𝒟 := 𝒟) (huniv := huniv) (DY := DY) (hDY := hDY) (hDYU :=
      hDYU) (UY := UY) (hUY := hUY) (WY := WY) (hWY := hWY) (t := t) (L := L) (hL := hL) (W := W)
      (hWiff := hWiff) (D := D) (D₀ := D₀) (hD := hD) (hDL := hDL) (hD₀U := hD₀U)) ≫
        chartMap (R := R) (c := c) (ε := ε) (U := U) (hεU := hεU) (g := g) (e := e) (y := y) (DY :=
          DY) (hDYU := hDYU) (UY := UY) (WY := WY) (hWY := hWY)
          =
      uliftYoneda.{u + 1}.map W.ι ≫ uliftYonedaEquiv.{u + 1, u, u + 1}.symm (pointElem R c ε t L
        hL)

theorem

clause_iii

L1037 ↗

JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard.ChartOneSS.clause_iii

Documentation

Every compatible chart morphism factors uniquely through the input open.

Source context 30 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
attribute [-instance]
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U1
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.injEq
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.mk.sizeOf_spec
  AlgebraicGeometry.SmoothProperCurve.FiniteMapData.twoAffineOpenCover_U0
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
attribute [-simp]
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartOneSS
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs.ZeroSchemeOpenChart.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)] (hεU : Set.range ε.1 ⊆ (U : Set C))
section Chart
variable [IsNoetherianRing R] [IsProper c]
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ : Dγ.SupportedIn U)
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ Spec
    (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  (hsect : ∀ ⦃V : Scheme.{u}⦄ (u : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType u] (M :
    (pullback c u).Modules),
      Scheme.Modules.IsInvertible M →
      (∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V) (𝒲 : (pullback (pullback.snd c u)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H1 ∧
          Module.finrank k (𝒲.sectionsOf (fibreAt c u s) (fibreModule c u s M)).H0 = 1) →
      (∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ V)
        (σ : 𝟙_ (pullback c (x ≫ u)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x
          rfl)).obj M), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ u), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U) →
      ∃ (D₀ : RelEffCartierDiv c g u) (N : V.Modules), Scheme.Modules.IsInvertible N ∧
        Nonempty (D₀.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N) ∧
        ∀ (d' : ℕ) (D' : RelEffCartierDiv c d' u) (N' : V.Modules), Scheme.Modules.IsInvertible N'
          → D'.SupportedIn U →
          Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c u)).obj N') → D'.I
            = D₀.I)
  (hZfibγ : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (L : RigidifiedLineBundle c ε t), FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] [IsAlgClosed k] (x : Spec (CommRingCat.of k) ⟶ T)
        (σ : 𝟙_ (pullback c (x ≫ t)).Modules ⟶ (Scheme.Modules.pullback (mapOnProdOver c x rfl)).obj
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule))), σ ≠ 0 →
        ∃ Dx : RelEffCartierDiv c g (x ≫ t), Dx.I = Scheme.Modules.zeroSchemeIdeal σ ∧
          Dx.SupportedIn U)
  (hH0one : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) (L : RigidifiedLineBundle c ε t),
    FibrewiseAlgEquivZero L →
      ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T) (𝒲 : (pullback (pullback.snd c t)
        s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ (sectionTwist c ε t (g + e) ⊗ (Dγ.pullbackAlong t (Category.comp_id
            t)).idealModule)))).H0 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] (𝒟 : RelEffCartierDiv
    c g y) (h𝒟U : 𝒟.SupportedIn U)
  (huniv : ∀ ⦃T : Scheme.{u}⦄ (g' : T ⟶ Spec (CommRingCat.of R)) (D : RelEffCartierDiv c g g'),
    D.SupportedIn U →
    ∃! φ : {φ : T ⟶ Y // φ ≫ y = g'}, PullsBackOver 𝒟 φ.1 φ.2 D)
  (DY : RelEffCartierDiv c (g + e) y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id
    y)).I) (hDYU : DY.SupportedIn U)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s) (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R
          c ε U hεU g e y DY hDYU)))).H1)
  (WY : Y.Opens)
  (hWY : ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (WY : Set Y) ↔
      IsAlgEquivZero (fibreAt c y s) ((Scheme.Modules.pullback (pullback.fst (pullback.snd c y)
        s)).obj (DY.twistModule c ε)))
include hεU hDγ hsect hZfibγ hH0one h𝒟U huniv hDY hDYU hUY hWY
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t] (L :
  RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (W : T.Opens)
  (hWiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (W : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s (Mγ R c ε (g + e) Dγ t
          L))).H1)
  (D : RelEffCartierDiv c (g + e) (W.ι ≫ t)) (D₀ : RelEffCartierDiv c g (W.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t)
    t)).L))
  (hD₀U : D₀.SupportedIn U)
include hL hWiff hD hDL hD₀U
omit [LocallyOfFiniteType y] in

Lean source · statement

theorem clause_iii {T' : Scheme.{u}} (ψ : T' ⟶ T) (φ' : T' ⟶ ((UY ⊓ WY : Y.Opens) : Scheme.{u}))
    [LocallyOfFiniteType (ψ ≫ t)]
    (hφ' : uliftYoneda.{u + 1}.map φ' ≫ chartMap (R := R) (c := c) (ε := ε) (U := U) (hεU := hεU)
      (g := g) (e := e) (y := y) (DY := DY) (hDYU := hDYU) (UY := UY) (WY := WY) (hWY := hWY) =
      uliftYoneda.{u + 1}.map ψ ≫ uliftYonedaEquiv.{u + 1, u, u + 1}.symm (pointElem R c ε t L hL))
        :
    ∃ χ : T' ⟶ (W : Scheme.{u}), χ ≫ W.ι = ψ ∧
      χ ≫ φ (R := R) (c := c) (ε := ε) (U := U) (hεU := hεU) (g := g) (e := e) (Dγ := Dγ) (hDγ :=
        hDγ) (y := y) (𝒟 := 𝒟) (huniv := huniv) (DY := DY) (hDY := hDY) (hDYU := hDYU) (UY := UY)
        (hUY := hUY) (WY := WY) (hWY := hWY) (t := t) (L := L) (hL := hL) (W := W) (hWiff := hWiff)
        (D := D) (D₀ := D₀) (hD := hD) (hDL := hDL) (hD₀U := hD₀U) = φ'

theorem