Module

Exists_openChart_relSubPicPresheaf_algEquivZeroCut_of_relEffCartierDiv.lean

Nyll.JacobianVarieties.Picard.Exists_openChart_relSubPicPresheaf_algEquivZeroCut_of_relEffCartierDiv

41 declarations indexed from sourceView source ↗

Module documentation

An open Picard chart from relative effective Cartier divisors

Use a universal divisor and an auxiliary twist to construct a chart for fibrewise algebraically trivial rigidified line bundles. The vanishing of fibre H¹ identifies the chart locus, and uniqueness of divisors proves its pullback universal property.

Imports41

Imported by7

theorem

pullbackAlong_congr_hom

L365 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.pullbackAlong_congr_hom

Source context 40 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve

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

L371 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.pullbackAlong_I_eq_mul

Source context 40 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve

Lean source · statement

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

theorem

OverTotal_snd_eq_of_eq

L379 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.OverTotal_snd_eq_of_eq

Source context 40 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve

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 rw [h]; exact Category.id_comp b.1)
      : 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

L387 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.RigidifiedLineBundle_pullbackAlong_L_eq

Source context 40 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve

Lean source · statement

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

theorem

rigSection_snd'

L402 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.rigSection_snd'

Source context 43 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
omit [IsSeparated c] [SmoothOfRelativeDimension 1 c] in

Lean source · statement

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

theorem

isInvertible_sectionIdeal

L406 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.isInvertible_sectionIdeal

Source context 42 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)

Lean source · statement

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

theorem

isInvertible_sectionTwist

L413 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.isInvertible_sectionTwist

Source context 42 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)

Lean source · statement

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

abbrev

Mγ

L419 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.Mγ

Documentation

The rigidified line bundle tensored with the section twist and auxiliary divisor ideal.

Source context 43 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
omit [IsSeparated c] [SmoothOfRelativeDimension 1 c] in

Lean source · statement

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

theorem

isInvertible_Mγ

L423 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.isInvertible_Mγ

Source context 42 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)

Lean source · statement

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

def

fibreIso

L437 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.fibreIso

Documentation

Identify the two fibres associated to a morphism over the base.

Source context 44 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  {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

L443 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.fibreIso_hom_fibreAt

Source context 45 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  {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 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

L450 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.fibreIso_hom_fst_baseChangeSnd

Source context 45 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  {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 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

L466 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.nonempty_Mγ_iso_pullback

Source context 44 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  {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

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

theorem

nonempty_fibreModule_iso

L493 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.nonempty_fibreModule_iso

Source context 44 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  {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

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

theorem

fibre_transport_to_t

L507 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.fibre_transport_to_t

Source context 44 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  {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

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

theorem

fibre_transport_to_t'

L519 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.fibre_transport_to_t'

Source context 44 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  {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

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

theorem

forall_subsingleton_H1_pullbackAlong

L537 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.forall_subsingleton_H1_pullbackAlong

Source context 44 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  {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

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

theorem

forall_subsingleton_H1_of_pullbackAlong

L548 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.forall_subsingleton_H1_of_pullbackAlong

Source context 44 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
section Transport
variable {e : ℕ} (n : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  {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

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

theorem

forall_subsingleton_H1_of_iso

L562 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.forall_subsingleton_H1_of_iso

Source context 43 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
omit [IsSeparated c] [SmoothOfRelativeDimension 1 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

def

LY

L582 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.LY

Documentation

Rigidify the twist module of the universal effective divisor.

Source context 43 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable (g e : ℕ) {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R))
  (DY : RelEffCartierDiv c (g + e) y)

Lean source · statement

def LY : RigidifiedLineBundle c ε y

theorem

LY_L

L586 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.LY_L

Source context 43 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Separated
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
    [IsSeparated c] [SmoothOfRelativeDimension 1 c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable (g e : ℕ) {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R))
  (DY : RelEffCartierDiv c (g + e) y)

Lean source · statement

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

theorem

fibrewiseAlgEquivZero_LY

L612 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.fibrewiseAlgEquivZero_LY

Source context 43 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)

Lean source · statement

theorem fibrewiseAlgEquivZero_LY : FibrewiseAlgEquivZero (LY R c ε g e y DY)

def

chartElem

L623 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.chartElem

Documentation

The algebraically trivial Picard class on the open chart locus.

Source context 44 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)

Lean source · statement

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

theorem

chartElem_fst

L629 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.chartElem_fst

Source context 44 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)

Lean source · statement

theorem chartElem_fst : (chartElem R c ε g e y DY UY).1 = UY.ι ≫ y

def

chartMap

L632 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.chartMap

Documentation

The Yoneda morphism determined by the universal Picard class on the chart.

Source context 44 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)

Lean source · statement

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

theorem

uliftYonedaEquiv_chartMap

L637 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.uliftYonedaEquiv_chartMap

Source context 44 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)

Lean source · statement

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

theorem

uliftYonedaEquiv_uliftYoneda_map_comp

L642 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.uliftYonedaEquiv_uliftYoneda_map_comp

Source context 45 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)
omit [SmoothOfRelativeDimension 1 c] [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c] in

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)

theorem

uliftYonedaEquiv_map_comp_chartMap

L649 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.uliftYonedaEquiv_map_comp_chartMap

Source context 44 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)

Lean source · statement

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

def

pointElem

L657 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.pointElem

Documentation

View a fibrewise algebraically trivial rigidified bundle as a total presheaf element.

Source context 45 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)
omit [SmoothOfRelativeDimension 1 c] [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c] in

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

L663 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.pointElem_fst

Source context 45 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)
omit [SmoothOfRelativeDimension 1 c] [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c] in

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

D_eq_pullbackAlong_lift

L683 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.D_eq_pullbackAlong_lift

Source context 47 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
    (L : RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (U : T.Opens)
  (hUiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (U : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s)
            (fibreModule c t s (Mγ R c ε (g + e) Dγ t L))).H1)
  (D : RelEffCartierDiv c (g + e) (U.ι ≫ t)) (D₀ : RelEffCartierDiv c g (U.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (U.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong
      (⟨U.ι, rfl⟩ : SchemeHomOver (U.ι ≫ t) t)).L))
omit [SmoothOfRelativeDimension 1 c] [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
    [LocallyOfFiniteType t] in
include hDY hD in

Lean source · statement

theorem D_eq_pullbackAlong_lift : D = DY.pullbackAlong (hU.lift D₀) (hU.lift_comp D₀)

theorem

nonempty_LY_pullbackAlong_lift_iso

L691 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.nonempty_LY_pullbackAlong_lift_iso

Source context 47 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
    (L : RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (U : T.Opens)
  (hUiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (U : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s)
            (fibreModule c t s (Mγ R c ε (g + e) Dγ t L))).H1)
  (D : RelEffCartierDiv c (g + e) (U.ι ≫ t)) (D₀ : RelEffCartierDiv c g (U.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (U.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong
      (⟨U.ι, rfl⟩ : SchemeHomOver (U.ι ≫ t) t)).L))
omit [IsNoetherianRing R] [GeometricallyIntegral c] [LocallyOfFiniteType t] in
include hDY hD hDL in

Lean source · statement

theorem nonempty_LY_pullbackAlong_lift_iso :
    Nonempty (((LY R c ε g e y DY).pullbackAlong
        (⟨hU.lift D₀, hU.lift_comp D₀⟩ : SchemeHomOver (U.ι ≫ t) y)).L ≅
      (L.pullbackAlong (⟨U.ι, rfl⟩ : SchemeHomOver (U.ι ≫ t) t)).L)

theorem

subsingleton_H1_U

L702 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.subsingleton_H1_U

Source context 47 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
    (L : RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (U : T.Opens)
  (hUiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (U : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s)
            (fibreModule c t s (Mγ R c ε (g + e) Dγ t L))).H1)
  (D : RelEffCartierDiv c (g + e) (U.ι ≫ t)) (D₀ : RelEffCartierDiv c g (U.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (U.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong
      (⟨U.ι, rfl⟩ : SchemeHomOver (U.ι ≫ t) t)).L))
omit [IsNoetherianRing R] [GeometricallyIntegral c] [LocallyOfFiniteType t] in
include hUiff in

Lean source · statement

theorem subsingleton_H1_U (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (U : Scheme.{u}))
    (𝒲 : (pullback (pullback.snd c (U.ι ≫ t)) s).TwoAffineOpenCover) :
    Subsingleton (𝒲.sectionsOf (fibreAt c (U.ι ≫ t) s) (fibreModule c (U.ι ≫ t) s
      (Mγ R c ε (g + e) Dγ (U.ι ≫ t)
          (L.pullbackAlong (⟨U.ι, rfl⟩ : SchemeHomOver (U.ι ≫ t) t))))).H1

theorem

subsingleton_H1_U'

L716 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.subsingleton_H1_U'

Source context 47 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
    (L : RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (U : T.Opens)
  (hUiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (U : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s)
            (fibreModule c t s (Mγ R c ε (g + e) Dγ t L))).H1)
  (D : RelEffCartierDiv c (g + e) (U.ι ≫ t)) (D₀ : RelEffCartierDiv c g (U.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (U.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong
      (⟨U.ι, rfl⟩ : SchemeHomOver (U.ι ≫ t) t)).L))
omit [IsNoetherianRing R] [GeometricallyIntegral c] [LocallyOfFiniteType t] in
include hDY hUiff hD hDL in

Lean source · statement

theorem subsingleton_H1_U' (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ (U : Scheme.{u}))
    (𝒲 : (pullback (pullback.snd c (U.ι ≫ t)) s).TwoAffineOpenCover) :
    Subsingleton (𝒲.sectionsOf (fibreAt c (U.ι ≫ t) s) (fibreModule c (U.ι ≫ t) s
      (Mγ R c ε (g + e) Dγ (U.ι ≫ t)
        ((LY R c ε g e y DY).pullbackAlong
            (⟨hU.lift D₀, hU.lift_comp D₀⟩ : SchemeHomOver (U.ι ≫ t) y))))).H1

theorem

range_lift_subset

L728 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.range_lift_subset

Source context 47 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
    (L : RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (U : T.Opens)
  (hUiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (U : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s)
            (fibreModule c t s (Mγ R c ε (g + e) Dγ t L))).H1)
  (D : RelEffCartierDiv c (g + e) (U.ι ≫ t)) (D₀ : RelEffCartierDiv c g (U.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (U.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong
      (⟨U.ι, rfl⟩ : SchemeHomOver (U.ι ≫ t) t)).L))
omit [IsNoetherianRing R] [GeometricallyIntegral c] [LocallyOfFiniteType t] in
include hDY hUY hUiff hD hDL in

Lean source · statement

theorem range_lift_subset : Set.range ⇑(hU.lift D₀) ⊆ (UY : Set Y)

def

φ

L744 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.φ

Documentation

Factor the universal divisor lift through the open chart locus.

Source context 46 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
    (L : RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (U : T.Opens)
  (hUiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (U : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s)
            (fibreModule c t s (Mγ R c ε (g + e) Dγ t L))).H1)
  (D : RelEffCartierDiv c (g + e) (U.ι ≫ t)) (D₀ : RelEffCartierDiv c g (U.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (U.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong
      (⟨U.ι, rfl⟩ : SchemeHomOver (U.ι ≫ t) t)).L))
omit [IsNoetherianRing R] [GeometricallyIntegral c] [LocallyOfFiniteType t] in

Lean source · statement

def φ : (U : Scheme.{u}) ⟶ (UY : Scheme.{u})

theorem

φ_ι

L751 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.φ_ι

Source context 46 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
    (L : RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (U : T.Opens)
  (hUiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (U : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s)
            (fibreModule c t s (Mγ R c ε (g + e) Dγ t L))).H1)
  (D : RelEffCartierDiv c (g + e) (U.ι ≫ t)) (D₀ : RelEffCartierDiv c g (U.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (U.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong
      (⟨U.ι, rfl⟩ : SchemeHomOver (U.ι ≫ t) t)).L))
omit [IsNoetherianRing R] [GeometricallyIntegral c] [LocallyOfFiniteType t] in

Lean source · statement

theorem φ_ι : φ R c ε g e Dγ y 𝒟 hU DY hDY UY hUY t L U hUiff D D₀ hD hDL ≫ UY.ι = hU.lift D₀

theorem

clause_i

L757 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.clause_i

Source context 47 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
    (L : RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (U : T.Opens)
  (hUiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (U : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s)
            (fibreModule c t s (Mγ R c ε (g + e) Dγ t L))).H1)
  (D : RelEffCartierDiv c (g + e) (U.ι ≫ t)) (D₀ : RelEffCartierDiv c g (U.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (U.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong
      (⟨U.ι, rfl⟩ : SchemeHomOver (U.ι ≫ t) t)).L))
omit [SmoothOfRelativeDimension 1 c] [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
    [LocallyOfFiniteType t] in
include hUiff 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 ⊆ (U : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s)
            (fibreModule c t s (Mγ R c ε (g + e) Dγ t L'))).H1

theorem

clause_ii

L768 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.clause_ii

Source context 46 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
    (L : RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (U : T.Opens)
  (hUiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (U : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s)
            (fibreModule c t s (Mγ R c ε (g + e) Dγ t L))).H1)
  (D : RelEffCartierDiv c (g + e) (U.ι ≫ t)) (D₀ : RelEffCartierDiv c g (U.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (U.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong
      (⟨U.ι, rfl⟩ : SchemeHomOver (U.ι ≫ t) t)).L))
omit [LocallyOfFiniteType t] in

Lean source · statement

theorem clause_ii :
    uliftYoneda.{u + 1}.map (φ R c ε g e Dγ y 𝒟 hU DY hDY UY hUY t L U hUiff D D₀ hD hDL) ≫
        chartMap R c ε g e y DY UY =
      uliftYoneda.{u + 1}.map U.ι ≫ uliftYonedaEquiv.{u + 1, u, u + 1}.symm
        (pointElem R c ε t L hL)

theorem

clause_iii

L792 ↗

JacobianVarieties.Picard.Proofs.OpenChart.AlgebraicGeometry.RelPicard.ChartOne.clause_iii

Source context 46 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChart
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField SheafOfModules.isIso_ihomModelToIhom
  AlgebraicCurve.CurveModel.isProper AlgebraicCurve.CurveModel.isIntegral
  AlgebraicCurve.CurveModel.smooth
  AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
  AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
  AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
  AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
  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.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
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
attribute [-instance]
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx 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
  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
  AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
  AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors
attribute [-instance]
  AlgebraicCurve.IsCurveOver.instFiniteResidue
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  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.RelPicard.fst_toProdSpec AlgebraicGeometry.RelPicard.toProdSpec_fst_assoc
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_cons
  AlgebraicGeometry.RelPicard.pointsSubBasepointModule_nil
  AlgebraicGeometry.RelPicard.fst_toProdSpec_assoc AlgebraicGeometry.RelPicard.toProdSpec_fst
  AlgebraicGeometry.prodKerGraph_one AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app 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
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
  AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
  AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
  AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
  AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
  AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
  AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
  AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
  AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
  AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply
attribute [-simp]
  AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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.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
  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.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp]
  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 AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp]
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
attribute [-simp]
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace ChartOne
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurve
section Chart
variable (R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R))
  [SmoothOfRelativeDimension 1 c]
  (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
variable [IsNoetherianRing R] [IsProper c] [GeometricallyIntegral c]
  (h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
  (g e : ℕ) (Dγ : RelEffCartierDiv c e (𝟙 (Spec (CommRingCat.of R))))
  (hχ : ∀ (k : Type u) [Field k] [IsAlgClosed k]
    (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
      (Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H0 : ℤ) -
        Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
          (fibreModule c (𝟙 _) x (sectionTwist c ε (𝟙 _) (g + e) ⊗ Dγ.idealModule))).H1 = 1)
  {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of R)) (𝒟 : RelEffCartierDiv c g y)
  (hU : 𝒟.IsUniversal)
  (DY : RelEffCartierDiv c (g + e) y)
  (hDY : DY.I = 𝒟.I * (Dγ.pullbackAlong y (Category.comp_id y)).I)
variable (UY : Y.Opens)
  (hUY : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ Y),
    Set.range ⇑s ⊆ (UY : Set Y) ↔
      ∀ (𝒲 : (pullback (pullback.snd c y) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c y s)
            (fibreModule c y s (Mγ R c ε (g + e) Dγ y (LY R c ε g e y DY)))).H1)
variable {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
    (L : RigidifiedLineBundle c ε t)
  (hL : FibrewiseAlgEquivZero L) (U : T.Opens)
  (hUiff : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T),
    Set.range ⇑s ⊆ (U : Set T) ↔
      ∀ (𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
        Subsingleton (𝒲.sectionsOf (fibreAt c t s)
            (fibreModule c t s (Mγ R c ε (g + e) Dγ t L))).H1)
  (D : RelEffCartierDiv c (g + e) (U.ι ≫ t)) (D₀ : RelEffCartierDiv c g (U.ι ≫ t))
  (hD : D.I = D₀.I * (Dγ.pullbackAlong (U.ι ≫ t) (Category.comp_id _)).I)
  (hDL : Nonempty (D.twistModule c ε ≅ (L.pullbackAlong
      (⟨U.ι, rfl⟩ : SchemeHomOver (U.ι ≫ t) t)).L))
include h𝔉 hχ in

Lean source · statement

theorem clause_iii {T' : Scheme.{u}} (ψ : T' ⟶ T) (φ' : T' ⟶ (UY : Scheme.{u}))
    [LocallyOfFiniteType (ψ ≫ t)]
    (hφ' : uliftYoneda.{u + 1}.map φ' ≫ chartMap R c ε g e y DY UY =
      uliftYoneda.{u + 1}.map ψ ≫ uliftYonedaEquiv.{u + 1, u, u + 1}.symm
        (pointElem R c ε t L hL)) :
    ∃ χ : T' ⟶ (U : Scheme.{u}), χ ≫ U.ι = ψ ∧
      χ ≫ φ R c ε g e Dγ y 𝒟 hU DY hDY UY hUY t L U hUiff D D₀ hD hDL = φ'

theorem