Module

exists_openChart_openImmersion_relSubPicPresheaf_algEquivZeroCut_of_polarisation_of_fibrewise_zeroScheme.lean

Nyll.JacobianVarieties.Picard.exists_openChart_openImmersion_relSubPicPresheaf_algEquivZeroCut_of_polarisation_of_fibrewise_zeroScheme

71 declarations indexed from sourceView source ↗

Module documentation

Open Picard charts from a polarisation

The vanishing of first cohomology and the support of zero schemes cut out an open parameter space. Universal effective divisors give its chart and its lifting property.

Imports49

Imported by2

theorem

coe_support_zeroSchemeIdeal_pullbackSection

L304 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.FinTransport.coe_support_zeroSchemeIdeal_pullbackSection

Source context 21 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace FinTransport
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · statement

theorem coe_support_zeroSchemeIdeal_pullbackSection {X X' : Scheme.{u}} (F : X' ⟶ X) {M : X.Modules}
    (hM : Scheme.Modules.IsInvertible M) (s : 𝟙_ X.Modules ⟶ M) :
    ((Scheme.Modules.zeroSchemeIdeal (Scheme.Modules.pullbackSection F s)).support : Set X') =
      F.base ⁻¹' ((Scheme.Modules.zeroSchemeIdeal s).support : Set X)

def

pullbackEquivOfIso

L312 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.FinTransport.pullbackEquivOfIso

Documentation

An isomorphism of schemes induces an equivalence of their module categories.

Source context 21 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace FinTransport
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · statement

def pullbackEquivOfIso {X X' : Scheme.{u}} (e : X' ≅ X) : X.Modules ≌ X'.Modules

instance

full_pullback_of_isIso

L319 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.FinTransport.full_pullback_of_isIso

Source context 21 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace FinTransport
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · statement

instance full_pullback_of_isIso {X X' : Scheme.{u}} (F : X' ⟶ X) [IsIso F] :
    (Scheme.Modules.pullback F).Full

instance

faithful_pullback_of_isIso

L323 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.FinTransport.faithful_pullback_of_isIso

Source context 21 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace FinTransport
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · statement

instance faithful_pullback_of_isIso {X X' : Scheme.{u}} (F : X' ⟶ X) [IsIso F] :
    (Scheme.Modules.pullback F).Faithful

theorem

pullbackSection_ne_zero_of_isIso

L327 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.FinTransport.pullbackSection_ne_zero_of_isIso

Source context 21 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace FinTransport
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · statement

theorem pullbackSection_ne_zero_of_isIso {X X' : Scheme.{u}} (F : X' ⟶ X) [IsIso F] {M : X.Modules}
    {s : 𝟙_ X.Modules ⟶ M} (hs : s ≠ 0) : Scheme.Modules.pullbackSection F s ≠ 0

theorem

exists_pullbackSection_eq_of_isIso

L343 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.FinTransport.exists_pullbackSection_eq_of_isIso

Source context 21 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace FinTransport
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · statement

theorem exists_pullbackSection_eq_of_isIso {X X' : Scheme.{u}} (F : X' ⟶ X) [IsIso F]
    {M : X.Modules}
    (τ : 𝟙_ X'.Modules ⟶ (Scheme.Modules.pullback F).obj M) :
    ∃ s : 𝟙_ X.Modules ⟶ M, Scheme.Modules.pullbackSection F s = τ

theorem

forall_support_subset_iff_of_isIso

L352 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.FinTransport.forall_support_subset_iff_of_isIso

Source context 21 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace FinTransport
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · statement

theorem forall_support_subset_iff_of_isIso {Y Y' : Scheme.{u}} (F : Y' ⟶ Y) [IsIso F]
    {P : Y.Modules}
    (hP : Scheme.Modules.IsInvertible P) {P' : Y'.Modules} (j : P' ≅ (Scheme.Modules.pullback
      F).obj P)
    (V : Set Y) :
    (∀ σ' : 𝟙_ Y'.Modules ⟶ P', σ' ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ').support : Set Y') ⊆ F.base ⁻¹' V) ↔
      (∀ σ : 𝟙_ Y.Modules ⟶ P, σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set Y) ⊆ V)

abbrev

FinerAt

L385 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.FinTransport.FinerAt

Documentation

Every nonzero section on the fibre has its zero scheme in the specified open.

Source context 21 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace FinTransport
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · statement

abbrev FinerAt {R : Type u} [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (U : C.Opens)
    {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (M : (pullback c t).Modules)
    {k : Type u} [Field k] (s : Spec (CommRingCat.of k) ⟶ T) : Prop

theorem

finerAt_transport

L393 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.FinTransport.finerAt_transport

Source context 21 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
namespace AlgebraicGeometry
namespace RelPicard
namespace FinTransport
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · statement

theorem finerAt_transport {R : Type u} [CommRing R] {C : Scheme.{u}}
    (c : C ⟶ Spec (CommRingCat.of R))
    (U : C.Opens) {T T' : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (t' : T' ⟶ Spec
      (CommRingCat.of R))
    (ψ : SchemeHomOver t' t) (M : (pullback c t).Modules) (hM : Scheme.Modules.IsInvertible M)
    (M' : (pullback c t').Modules) (i : M' ≅ (Scheme.Modules.pullback (baseChangeSnd c ψ)).obj M)
    {k : Type u} [Field k] (x' : Spec (CommRingCat.of k) ⟶ T') :
    FinerAt c U t' M' x' ↔ FinerAt c U t M (x' ≫ ψ.1)

theorem

pullbackAlong_congr_hom

L509 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.pullbackAlong_congr_hom

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open 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

L515 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.pullbackAlong_I_eq_mul

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open 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')
    (F : RelEffCartierDiv f l g') (hD : D.I = D₀.I * F.I) {T : Scheme.{u}} {g : T ⟶ S} (φ : T ⟶ T')
    (hφ : φ ≫ g' = g) :
    (D.pullbackAlong φ hφ).I = (D₀.pullbackAlong φ hφ).I * (F.pullbackAlong φ hφ).I

theorem

OverTotal_snd_eq_of_eq

L523 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.OverTotal_snd_eq_of_eq

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open 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

L531 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.RigidifiedLineBundle_pullbackAlong_L_eq

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open 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

abbrev

FinerAt

L539 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.FinerAt

Documentation

Every nonzero section on the fibre has its zero scheme in the specified open.

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · statement

abbrev FinerAt {R : Type u} [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    (U : C.Opens)
    {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (M : (pullback c t).Modules)
    {k : Type u} [Field k] (s : Spec (CommRingCat.of k) ⟶ T) : Prop

theorem

finerAt_congr_hom

L547 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.finerAt_congr_hom

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · statement

theorem finerAt_congr_hom {R : Type u} [CommRing R] {C : Scheme.{u}}
    (c : C ⟶ Spec (CommRingCat.of R)) (U : C.Opens)
    {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (M : (pullback c t).Modules)
    {k : Type u} [Field k] {s s' : Spec (CommRingCat.of k) ⟶ T} (h : s = s') :
    FinerAt c U t M s ↔ FinerAt c U t M s'

theorem

finerAt_congr_iso

L554 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.finerAt_congr_iso

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra

Lean source · statement

theorem finerAt_congr_iso {R : Type u} [CommRing R] {C : Scheme.{u}}
    (c : C ⟶ Spec (CommRingCat.of R)) (U : C.Opens)
    {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) {M M' : (pullback c t).Modules} (i : M ≅ M')
    {k : Type u} [Field k] (s : Spec (CommRingCat.of k) ⟶ T) :
    FinerAt c U t M s ↔ FinerAt c U t M' s

theorem

rigSection_snd'

L583 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.rigSection_snd'

Source context 26 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
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_EI

L588 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.isInvertible_EI

Source context 26 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
include hEU in

Lean source · statement

theorem isInvertible_EI {V : Scheme.{u}} (u : V ⟶ Spec (CommRingCat.of R)) :
    (E.pullbackAlong u (Category.comp_id u)).I.IsInvertible

abbrev

Mγ

L594 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.Mγ

Documentation

Twist the bundle by the polarising divisor and the ideal of the auxiliary divisor.

Source context 25 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)

Lean source · statement

abbrev Mγ {e : 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

L600 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.isInvertible_Dγ

Source context 25 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)

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

L607 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.isInvertible_Mγ

Source context 26 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
include hEU in

Lean source · statement

theorem isInvertible_Mγ {e : 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 E Dγ u LV)

def

fibreIso

L623 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.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.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
section Transport
variable {e : 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

L629 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.fibreIso_hom_fibreAt

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
section Transport
variable {e : 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

L636 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.fibreIso_hom_fst_baseChangeSnd

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
section Transport
variable {e : 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

L653 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.nonempty_Mγ_iso_pullback

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
section Transport
variable {e : 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 hEU hDγ in

Lean source · statement

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

theorem

nonempty_fibreModule_iso

L682 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.nonempty_fibreModule_iso

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
section Transport
variable {e : 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 hEU hDγ in

Lean source · statement

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

theorem

fibre_transport_to_t

L697 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.fibre_transport_to_t

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
section Transport
variable {e : 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 hEU 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 E Dγ t
        L))).H1 ≃ₗ[k]
        (𝒲.sectionsOf (fibreAt c t' s') (fibreModule c t' s' (Mγ R c ε E Dγ t' (L.pullbackAlong
          ψ)))).H1)

theorem

fibre_transport_to_t'

L710 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.fibre_transport_to_t'

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
section Transport
variable {e : 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 hEU 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 E 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 E Dγ t
          L))).H1)

theorem

forall_subsingleton_H1_pullbackAlong

L729 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.forall_subsingleton_H1_pullbackAlong

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
section Transport
variable {e : 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 hEU 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 E
        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 E Dγ t'
      (L.pullbackAlong ψ)))).H1

theorem

forall_subsingleton_H1_of_pullbackAlong

L741 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.forall_subsingleton_H1_of_pullbackAlong

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
section Transport
variable {e : 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 hEU 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 E 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 E Dγ
      t L))).H1

theorem

finerAt_transport

L753 ↗

AlgebraicGeometry.RelPicard.ChartE.finerAt_transport

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
section Transport
variable {e : 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] [SmoothOfRelativeDimension 1 (U.ι ≫ c)] in

Lean source · statement

theorem _root_.AlgebraicGeometry.RelPicard.ChartE.finerAt_transport
    (M : (pullback c t).Modules) (hM : Scheme.Modules.IsInvertible M) (M' : (pullback c t').Modules)
    (i : M' ≅ (Scheme.Modules.pullback (baseChangeSnd c ψ)).obj M) :
    FinerAt c U t' M' s' ↔ FinerAt c U t M (s' ≫ ψ.1)

theorem

finerAt_Mγ_pullbackAlong_iff

L760 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.finerAt_Mγ_pullbackAlong_iff

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
section Transport
variable {e : 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 hEU hDγ in

Lean source · statement

theorem finerAt_Mγ_pullbackAlong_iff :
    FinerAt c U t' (Mγ R c ε E Dγ t' (L.pullbackAlong ψ)) s' ↔ FinerAt c U t (Mγ R c ε E Dγ t L)
      (s' ≫ ψ.1)

theorem

forall_subsingleton_H1_of_iso

L775 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.forall_subsingleton_H1_of_iso

Source context 28 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
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

L796 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.uliftYonedaEquiv_uliftYoneda_map_comp

Source context 30 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
section Generic
omit [IsSeparated c] [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

L804 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.pointElem

Documentation

The relative Picard class of a fibrewise algebraically trivial rigidified bundle.

Source context 30 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
section Generic
omit [IsSeparated c] [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

L810 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.pointElem_fst

Source context 30 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
section Generic
omit [IsSeparated c] [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

theorem

forall_subsingleton_H1_congr_hom

L814 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.forall_subsingleton_H1_congr_hom

Source context 30 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
section Generic
omit [IsSeparated c] [SmoothOfRelativeDimension 1 (U.ι ≫ c)]

Lean source · statement

theorem forall_subsingleton_H1_congr_hom {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R))
    (F : (pullback c t).Modules)
    {k : Type u} [Field k] {s s' : Spec (CommRingCat.of k) ⟶ T} (h : s = s') :
    (∀ 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

exists_finerOpen

L890 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.exists_finerOpen

Source context 35 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
section ProperOpen
variable [IsNoetherianRing R] [IsProper c] [Flat c]
omit [IsSeparated c] in
include R c ε U E hEU 𝒱 Dγ hDγ hH0one in

Lean source · statement

theorem exists_finerOpen {V : Scheme.{u}} (v : V ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType v]
    (LV : RigidifiedLineBundle c ε v) (hLV : FibrewiseAlgEquivZero LV) :
    ∃ W' : V.Opens, ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ V),
      Set.range ⇑s ⊆ (W' : Set V) ↔
        (∀ 𝒲 : (pullback (pullback.snd c v) s).TwoAffineOpenCover,
          Subsingleton (𝒲.sectionsOf (fibreAt c v s) (fibreModule c v s (Mγ R c ε E Dγ v LV))).H1) ∧
        FinerAt c U v (Mγ R c ε E Dγ v LV) s

def

LY

L957 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.LY

Documentation

The rigidified bundle on the parameter scheme defined by the universal divisor.

Source context 33 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
open _root_.AlgebraicGeometry.RelEffCartierDiv in

Lean source · statement

def LY : RigidifiedLineBundle c ε y

theorem

LY_L

L965 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.LY_L

Source context 32 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)

Lean source · statement

theorem LY_L : (LY R c ε U E hEU y DY hDYU).L =
    Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
      (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule)

def

hulift

L976 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.hulift

Documentation

The unique morphism classifying the supported effective divisor.

Source context 35 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
section HunivAPI
omit [IsSeparated c] [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
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

L979 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.hulift_comp

Source context 36 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
section HunivAPI
omit [IsSeparated c] [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
variable {T₁ : Scheme.{u}} {g₁ : T₁ ⟶ Spec (CommRingCat.of R)} (D₁ : RelEffCartierDiv c g g₁) (hD₁
  : D₁.SupportedIn U)
include huniv in

Lean source · statement

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

theorem

pullsBackOver_hulift

L983 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.pullsBackOver_hulift

Source context 36 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
section HunivAPI
omit [IsSeparated c] [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
variable {T₁ : Scheme.{u}} {g₁ : T₁ ⟶ Spec (CommRingCat.of R)} (D₁ : RelEffCartierDiv c g g₁) (hD₁
  : D₁.SupportedIn U)
include huniv in

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

L988 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.pullbackAlong_hulift

Source context 36 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
section HunivAPI
omit [IsSeparated c] [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
variable {T₁ : Scheme.{u}} {g₁ : T₁ ⟶ Spec (CommRingCat.of R)} (D₁ : RelEffCartierDiv c g g₁) (hD₁
  : D₁.SupportedIn U)
include R c U g y 𝒟 huniv in

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

L993 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.eq_hulift

Source context 36 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
section HunivAPI
omit [IsSeparated c] [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
variable {T₁ : Scheme.{u}} {g₁ : T₁ ⟶ Spec (CommRingCat.of R)} (D₁ : RelEffCartierDiv c g g₁) (hD₁
  : D₁.SupportedIn U)
include huniv in

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_WY

L1016 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.fibrewiseAlgEquivZero_LY_WY

Source context 36 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
include R c y WY hWY in

Lean source · statement

theorem fibrewiseAlgEquivZero_LY_WY :
    FibrewiseAlgEquivZero ((LY R c ε U E hEU y DY hDYU).pullbackAlong
      (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))

abbrev

jX

L1052 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.jX

Documentation

The composite open immersion from the refined chart to the parameter scheme.

Source context 36 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)

Lean source · statement

abbrev jX : (X : Scheme.{u}) ⟶ Y

theorem

fibrewiseAlgEquivZero_LY_X

L1055 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.fibrewiseAlgEquivZero_LY_X

Source context 37 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
include R c ε U E hEU y DY hDYU WY hWY X in

Lean source · statement

theorem fibrewiseAlgEquivZero_LY_X :
    FibrewiseAlgEquivZero ((LY R c ε U E hEU y DY hDYU).pullbackAlong
      (⟨jX WY X, rfl⟩ : SchemeHomOver (jX WY X ≫ y) y))

def

chartElem

L1065 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.chartElem

Documentation

The relative Picard class of the universal bundle restricted to the chart.

Source context 36 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)

Lean source · statement

def chartElem :
    (relSubPicPresheaf c ε (algEquivZeroCut c ε)).overTotal.obj (op (X : Scheme.{u}))

theorem

chartElem_fst

L1071 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.chartElem_fst

Source context 36 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)

Lean source · statement

theorem chartElem_fst :
    (chartElem R c ε U E hEU y DY hDYU WY hWY X).1 = jX WY X ≫ y

def

chartMap

L1075 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.chartMap

Documentation

The natural transformation represented by the restricted universal bundle.

Source context 36 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)

Lean source · statement

def chartMap :
    uliftYoneda.{u + 1}.obj (X : Scheme.{u}) ⟶ (relSubPicPresheaf c ε (algEquivZeroCut c
      ε)).overTotal

theorem

uliftYonedaEquiv_chartMap

L1081 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.uliftYonedaEquiv_chartMap

Source context 36 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)

Lean source · statement

theorem uliftYonedaEquiv_chartMap :
    uliftYonedaEquiv.{u + 1, u, u + 1} (chartMap R c ε U E hEU y DY hDYU WY hWY X) =
      chartElem R c ε U E hEU y DY hDYU WY hWY X

theorem

uliftYonedaEquiv_map_comp_chartMap

L1086 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.uliftYonedaEquiv_map_comp_chartMap

Source context 36 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)

Lean source · statement

theorem uliftYonedaEquiv_map_comp_chartMap {T : Scheme.{u}} (φ' : T ⟶ (X : Scheme.{u})) :
    uliftYonedaEquiv.{u + 1, u, u + 1} (uliftYoneda.{u + 1}.map φ' ≫
        chartMap R c ε U E hEU y DY hDYU WY hWY X) =
      (relSubPicPresheaf c ε (algEquivZeroCut c ε)).overTotal.map φ'.op
        (chartElem R c ε U E hEU y DY hDYU WY hWY X)

theorem

D_eq_pullbackAlong_lift

L1109 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.D_eq_pullbackAlong_lift

Source context 39 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)
omit [IsSeparated c] [SmoothOfRelativeDimension 1 (U.ι ≫ c)] in
include R c U g Dγ y 𝒟 huniv DY hDY D₀ hD hD₀U 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)

abbrev

φ₀

L1119 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.φ₀

Documentation

The morphism over the base that classifies the supported divisor.

Source context 37 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)

Lean source · statement

abbrev φ₀ : SchemeHomOver (W.ι ≫ t) y

theorem

nonempty_LY_pullbackAlong_lift_iso

L1124 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.nonempty_LY_pullbackAlong_lift_iso

Source context 39 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)
include R c ε U E hEU g e Dγ y 𝒟 huniv DY hDY hDYU t W D D₀ hD hDL hD₀U in
open _root_.AlgebraicGeometry.RelEffCartierDiv in

Lean source · statement

theorem nonempty_LY_pullbackAlong_lift_iso :
    Nonempty (((LY R c ε U E hEU y DY hDYU).pullbackAlong (φ₀ R c U g y 𝒟 huniv t W D₀ hD₀U)).L ≅
      (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L)

theorem

nonempty_Mγ_LY_pullbackAlong_lift_iso

L1136 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.nonempty_Mγ_LY_pullbackAlong_lift_iso

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)
include R c ε U E hEU g e Dγ y 𝒟 huniv DY hDY hDYU t L W D D₀ hD hDL hD₀U in

Lean source · statement

theorem nonempty_Mγ_LY_pullbackAlong_lift_iso :
    Nonempty (Mγ R c ε E Dγ (W.ι ≫ t) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (φ₀ R c U g y 𝒟
      huniv t W D₀ hD₀U)) ≅
      Mγ R c ε E Dγ (W.ι ≫ t) (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)))

theorem

subsingleton_H1_W

L1144 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.subsingleton_H1_W

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)
include R c ε U E hEU Dγ hDγ t L W hWiff 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 E Dγ (W.ι ≫ t) (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t))))).H1

theorem

subsingleton_H1_W'

L1156 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.subsingleton_H1_W'

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)
include R c ε U E hEU g e Dγ hDγ y 𝒟 huniv DY hDY hDYU t L W hWiff D D₀ hD hDL hD₀U 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 E Dγ (W.ι ≫ t)
        ((LY R c ε U E hEU y DY hDYU).pullbackAlong (φ₀ R c U g y 𝒟 huniv t W D₀ hD₀U))))).H1

theorem

finerAt_W'

L1167 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.finerAt_W'

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)
include R c ε U E hEU g e Dγ hDγ y 𝒟 huniv DY hDY hDYU t L W hWiff D D₀ hD hDL hD₀U in

Lean source · statement

theorem finerAt_W' (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (W : Scheme.{u})) :
    FinerAt c U (W.ι ≫ t) (Mγ R c ε E Dγ (W.ι ≫ t)
      ((LY R c ε U E hEU y DY hDYU).pullbackAlong (φ₀ R c U g y 𝒟 huniv t W D₀ hD₀U))) s

theorem

H1_and_finerAt_Y

L1180 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.H1_and_finerAt_Y

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)
include R c ε U E hEU g e Dγ hDγ y 𝒟 huniv DY hDY hDYU t L W hWiff D D₀ hD hDL hD₀U in

Lean source · statement

theorem H1_and_finerAt_Y (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (W : Scheme.{u})) :
    (∀ 𝒲 : (pullback (pullback.snd c y) (s ≫ hulift R c U g y 𝒟 huniv D₀ hD₀U)).TwoAffineOpenCover,
      Subsingleton (𝒲.sectionsOf (fibreAt c y (s ≫ hulift R c U g y 𝒟 huniv D₀ hD₀U))
        (fibreModule c y (s ≫ hulift R c U g y 𝒟 huniv D₀ hD₀U) (Mγ R c ε E Dγ y (LY R c ε U E hEU
          y DY hDYU)))).H1) ∧
    FinerAt c U y (Mγ R c ε E Dγ y (LY R c ε U E hEU y DY hDYU)) (s ≫ hulift R c U g y 𝒟 huniv D₀
      hD₀U)

theorem

range_lift_subset_WY

L1197 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.range_lift_subset_WY

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)
include R c ε U E hEU g e Dγ y 𝒟 huniv DY hDY hDYU hWY t L hL W D D₀ hD hDL hD₀U in

Lean source · statement

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

def

φ₁

L1268 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.φ₁

Documentation

Factor the classifying morphism through the algebraically trivial locus.

Source context 37 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)

Lean source · statement

def φ₁ : (W : Scheme.{u}) ⟶ (WY : Scheme.{u})

theorem

φ₁_ι

L1277 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.φ₁_ι

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)
include R c ε U E hEU g e Dγ y 𝒟 huniv DY hDY hDYU WY hWY t L hL W D D₀ hD hDL hD₀U in

Lean source · statement

theorem φ₁_ι : φ₁ R c ε U E hEU g e Dγ y 𝒟 huniv DY hDY hDYU WY hWY t L hL W D D₀ hD hDL hD₀U ≫
  WY.ι = hulift R c U g y 𝒟 huniv D₀ hD₀U

theorem

range_φ₁_subset_X

L1282 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.range_φ₁_subset_X

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)
include R c ε U E hEU g e Dγ hDγ y 𝒟 huniv DY hDY hDYU WY hWY hX t L hL W hWiff D D₀ hD hDL hD₀U in

Lean source · statement

theorem range_φ₁_subset_X : Set.range ⇑(φ₁ R c ε U E hEU g e Dγ y 𝒟 huniv DY hDY hDYU WY hWY t L hL
  W D D₀ hD hDL hD₀U) ⊆ (X : Set WY)

def

φ

L1308 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.φ

Documentation

Factor the classifying morphism through the refined open chart.

Source context 37 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)

Lean source · statement

def φ : (W : Scheme.{u}) ⟶ (X : Scheme.{u})

theorem

φ_jX

L1319 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.φ_jX

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)
include R c ε U E hEU g e Dγ hDγ y 𝒟 huniv DY hDY hDYU WY hWY X hX t L hL W hWiff D D₀ hD hDL hD₀U
  in

Lean source · statement

theorem φ_jX : φ R c ε U E hEU g e Dγ hDγ y 𝒟 huniv DY hDY hDYU WY hWY X hX t L hL W hWiff D D₀ hD
  hDL hD₀U ≫ jX WY X = hulift R c U g y 𝒟 huniv D₀ hD₀U

theorem

clause_i

L1325 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.clause_i

Source context 39 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)
omit [IsSeparated c] [SmoothOfRelativeDimension 1 (U.ι ≫ c)] in
include R c ε U E Dγ t L hWiff 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 E Dγ t L'))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L') s

theorem

clause_ii

L1340 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.clause_ii

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry
namespace RelPicard
namespace ChartE
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 AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
section Chart
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable (𝒱 : C.TwoAffineOpenCover)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
  (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
variable {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
variable
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
variable
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) (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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)
include R c ε U E hEU g e Dγ hDγ y 𝒟 huniv DY hDY hDYU WY hWY X hX t L hL W hWiff D D₀ hD hDL hD₀U
  in

Lean source · statement

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

theorem

clause_iii

L1380 ↗

JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE.clause_iii

Source context 15 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart
attribute [-instance]
  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
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
attribute [-instance]
  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
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  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
  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
attribute [-simp]
  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
  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
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U0
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.preimage_U1
attribute [-simp]
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.ofPreimage_hom
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.mk.injEq
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply
  TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
universe u
attribute [local instance] FinTransport.full_pullback_of_isIso
attribute [local instance] FinTransport.faithful_pullback_of_isIso
namespace AlgebraicGeometry.RelPicard.ChartE
open _root_.CategoryTheory _root_.CategoryTheory.Limits _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard _root_.NeronModelInfra
open _root_.AlgebraicGeometry.RelEffCartierDiv in

Lean source · statement

theorem clause_iii
    (R : Type u) [CommRing R] [IsNoetherianRing R] {C : Scheme.{u}}
    (c : C ⟶ Spec (CommRingCat.of R)) [IsProper c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    {ρ : ℕ} (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R))))
    (hEU : E.SupportedIn U)
  (g e : ℕ) (hr : g + e = ρ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R)))) (hDγ :
    Dγ.SupportedIn U)
  (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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
    {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (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 ρ y) (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
    (hDYU : DY.SupportedIn U)
  (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
        (Scheme.Modules.rigidify (rigSection c y ε) (pullback.snd c y)
          (DY.lineBundle ⊗ (E.pullbackAlong y (Category.comp_id y)).idealModule))))
  (X : (WY : Scheme.{u}).Opens)
  (hX : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (WY : Scheme.{u})),
    Set.range ⇑s ⊆ (X : Set WY) ↔
      (∀ 𝒲 : (pullback (pullback.snd c (WY.ι ≫ y)) s).TwoAffineOpenCover,
        Subsingleton (𝒲.sectionsOf (fibreAt c (WY.ι ≫ y) s) (fibreModule c (WY.ι ≫ y) s
          (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong (⟨WY.ι, rfl⟩ :
            SchemeHomOver (WY.ι ≫ y) y))))).H1) ∧
      FinerAt c U (WY.ι ≫ y) (Mγ R c ε E Dγ (WY.ι ≫ y) ((LY R c ε U E hEU y DY hDYU).pullbackAlong
        (⟨WY.ι, rfl⟩ : SchemeHomOver (WY.ι ≫ y) y))) s)
    {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 E Dγ t L))).H1) ∧
      FinerAt c U t (Mγ R c ε E Dγ t L) s)
  (D : RelEffCartierDiv c ρ (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 (Scheme.Modules.rigidify (rigSection c (W.ι ≫ t) ε) (pullback.snd c (W.ι ≫ t))
      (D.lineBundle ⊗ (E.pullbackAlong (W.ι ≫ t) (Category.comp_id _)).idealModule) ≅
    (L.pullbackAlong (⟨W.ι, rfl⟩ : SchemeHomOver (W.ι ≫ t) t)).L))
  (hD₀U : D₀.SupportedIn U)
    {T' : Scheme.{u}} (ψ : T' ⟶ T) (φ' : T' ⟶ (X : Scheme.{u}))
    [LocallyOfFiniteType (ψ ≫ t)]
    (hφ' : uliftYoneda.{u + 1}.map φ' ≫ chartMap R c ε U E hEU y DY hDYU WY hWY X =
      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 c ε U E hEU g e Dγ hDγ y 𝒟 huniv DY hDY hDYU
      WY hWY X hX t L hL W hWiff D D₀ hD hDL hD₀U = φ'

theorem

exists_openChart_of_polarisation_of_fibrewise_zeroScheme

L1607 ↗

AlgebraicGeometry.RelPicard.exists_openChart_of_polarisation_of_fibrewise_zeroScheme

Documentation

The polarisation and fibrewise zero-scheme conditions define a universal open Picard chart.

Source context 5 commands

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

open CategoryTheory CategoryTheory.Limits Opposite CategoryTheory.MonoidalCategory
open AlgebraicGeometry AlgebraicGeometry.RelPicard NeronModelInfra
namespace AlgebraicGeometry.RelPicard
universe u
open JacobianVarieties.Picard.Proofs.PolarizedZeroSchemeChart.AlgebraicGeometry.RelPicard.ChartE in

Lean source · statement

theorem exists_openChart_of_polarisation_of_fibrewise_zeroScheme
    (R : Type u) [CommRing R] [IsNoetherianRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of
      R))
    [IsProper c] [Flat c]
    (𝒱 : C.TwoAffineOpenCover)
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (U : C.Opens) [SmoothOfRelativeDimension 1 (U.ι ≫ c)]
    (g e ρ : ℕ) (hr : g + e = ρ)
    (E : RelEffCartierDiv c ρ (𝟙 (Spec (CommRingCat.of R)))) (hEU : E.SupportedIn U)
    (Y : Scheme.{u}) (y : Y ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType y] [CompactSpace Y]
    (Duniv : RelEffCartierDiv c g y) (hDunivU : Duniv.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 Duniv φ.1 φ.2 D)
    (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 (E.lineBundle ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (E.lineBundle ⊗ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))), σ ≠ 0 →
        ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆ ((pullback.fst c
          (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t))) →
        ∃ 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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H1 →
        Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)))).H0 = 1)
    (havoid : ∀ ⦃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 ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule))),
        σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (x ≫ t))) ⊆
          ((pullback.fst c (x ≫ t)) ⁻¹ᵁ U : Set ↥(pullback c (x ≫ t)))) →
      ∀ (D₀ : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N →
        Nonempty (D₀.lineBundle ≅
          (L.L ⊗ ((E.pullbackAlong t (Category.comp_id t)).lineBundle ⊗ (Dγ.pullbackAlong t
            (Category.comp_id t)).idealModule)) ⊗
            (Scheme.Modules.pullback (pullback.snd c t)).obj N) →
        D₀.SupportedIn U)
    (hcut : ∀ ⦃T : Scheme.{u}⦄ (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
      (D : RelEffCartierDiv c ρ t) (D₀ : RelEffCartierDiv c g t), D₀.SupportedIn U →
      D.I = D₀.I * (Dγ.pullbackAlong t (Category.comp_id t)).I →
      ∃ W : T.Opens, ∀ (k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ T),
        Set.range ⇑s ⊆ (W : Set T) ↔
          IsAlgEquivZero (fibreAt c t s)
            ((Scheme.Modules.pullback (pullback.fst (pullback.snd c t) s)).obj
              (Scheme.Modules.rigidify (RelPicard.rigSection c t ε) (pullback.snd c t)
              (D.lineBundle ⊗ (E.pullbackAlong t (Category.comp_id t)).idealModule)))) :
    ∃ (X : Scheme.{u}) (f : uliftYoneda.{u + 1}.obj X ⟶ (relSubPicPresheaf c ε (algEquivZeroCut c
      ε)).overTotal),
      LocallyOfFiniteType (uliftYonedaEquiv f).1 ∧ CompactSpace X ∧
      (∃ j : X ⟶ Y, IsOpenImmersion j ∧ j ≫ y = (uliftYonedaEquiv f).1) ∧
      ∀ ⦃T : Scheme.{u}⦄ (x : uliftYoneda.{u + 1}.obj T ⟶ (relSubPicPresheaf c ε (algEquivZeroCut c
        ε)).overTotal),
        LocallyOfFiniteType (uliftYonedaEquiv x).1 →
        ∃ (U' : T.Opens) (φ : (↑U' : Scheme.{u}) ⟶ X),
          (∀ (L : RigidifiedLineBundle c ε (uliftYonedaEquiv x).1), Quotient.mk _ L =
            (uliftYonedaEquiv x).2.1 →
            ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
              Set.range ⇑s ⊆ (U' : Set T) ↔
                (∀ (𝒲 : (pullback (pullback.snd c (uliftYonedaEquiv x).1) s).TwoAffineOpenCover),
                  Subsingleton (𝒲.sectionsOf (fibreAt c (uliftYonedaEquiv x).1 s) (fibreModule c
                    (uliftYonedaEquiv x).1 s
                    (L.L ⊗ ((E.pullbackAlong (uliftYonedaEquiv x).1 (Category.comp_id
                      _)).lineBundle ⊗
                      (Dγ.pullbackAlong (uliftYonedaEquiv x).1 (Category.comp_id
                        _)).idealModule)))).H1) ∧
                (∀ σ : 𝟙_ (pullback c (s ≫ (uliftYonedaEquiv x).1)).Modules ⟶
                    (Scheme.Modules.pullback (mapOnProdOver c s rfl)).obj
                      (L.L ⊗ ((E.pullbackAlong (uliftYonedaEquiv x).1 (Category.comp_id
                        _)).lineBundle ⊗
                        (Dγ.pullbackAlong (uliftYonedaEquiv x).1 (Category.comp_id
                          _)).idealModule)),
                  σ ≠ 0 → ((Scheme.Modules.zeroSchemeIdeal σ).support : Set ↥(pullback c (s ≫
                    (uliftYonedaEquiv x).1))) ⊆
                    ((pullback.fst c (s ≫ (uliftYonedaEquiv x).1)) ⁻¹ᵁ U : Set ↥(pullback c (s ≫
                      (uliftYonedaEquiv x).1))))) ∧
          uliftYoneda.{u + 1}.map φ ≫ f = uliftYoneda.{u + 1}.map U'.ι ≫ x ∧
          ∀ ⦃T' : Scheme.{u}⦄ (ψ : T' ⟶ T) (φ' : T' ⟶ X),
            LocallyOfFiniteType (ψ ≫ (uliftYonedaEquiv x).1) →
            uliftYoneda.{u + 1}.map φ' ≫ f = uliftYoneda.{u + 1}.map ψ ≫ x →
            ∃ χ : T' ⟶ ↑U', χ ≫ U'.ι = ψ ∧ χ ≫ φ = φ'

Module documentation

Open Picard charts from a polarisation

Declarations