Module

nonempty_pullback_iso_pullback_unit_of_isAlgEquivZero_baseChange_of_curveModel_ratFunc.lean

Nyll.JacobianVarieties.Picard.nonempty_pullback_iso_pullback_unit_of_isAlgEquivZero_baseChange_of_curveModel_ratFunc

7 declarations indexed from sourceView source ↗

Module documentation

Triviality on a rational curve after base change

Cohomology comparison and the vanishing of the rational curve genus descend the triviality of an algebraically trivial pullback.

Imports25

Imported by1

Nyll.JacobianVarieties.Picard.All

theorem

finite_and_finrank_eq

L490 ↗

JacobianVarieties.Picard.Proofs.RationalBaseChangeUnit.AlgebraicGeometry.RelPicard.LineTrivDescent.finite_and_finrank_eq

Documentation

Invertible modules have finite cohomology with dimensions independent of the cover.

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.RationalBaseChangeUnit
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
  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
attribute [-instance]
  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.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instNeg
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulInt
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAddCommGroup
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAdd
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSub
attribute [-instance]
  instModuleCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  instSMulCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulNat
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instZero
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
attribute [-instance]
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
attribute [-instance]
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
attribute [-instance]
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  SheafOfModules.isIso_ihomModelToIhom
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
attribute [-instance]
  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
attribute [-simp]
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  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
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  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
  AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
attribute [-simp]
  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
  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
attribute [-simp]
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule
  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.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.sub_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sresTop
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.add_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.IdealSheafData.coe_resLE
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.smul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_ofLE_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.ideal_range
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zero_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sres
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.comp_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.neg_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.id_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_mulRight_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.nsmul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.injEq
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zsmul_app
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
attribute [-simp]
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
attribute [-simp]
  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
  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.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicCurve hiding genus
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
namespace RelPicard
namespace LineTrivDescent
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry

Lean source · statement

theorem finite_and_finrank_eq (K : Type u) [Field K] [IsAlgClosed K] {X : Scheme.{u}}
    (x : X ⟶ Spec (CommRingCat.of K)) [IsIntegral X] [IsProper x] [SmoothOfRelativeDimension 1 x]
    (M : X.Modules) (hM : Scheme.Modules.IsInvertible M) (𝒱 𝒱' : X.TwoAffineOpenCover) :
    Module.Finite K (𝒱.sectionsOf x M).H0 ∧ Module.Finite K (𝒱.sectionsOf x M).H1 ∧
      Module.finrank K (𝒱.sectionsOf x M).H0 = Module.finrank K (𝒱'.sectionsOf x M).H0 ∧
      Module.finrank K (𝒱.sectionsOf x M).H1 = Module.finrank K (𝒱'.sectionsOf x M).H1

theorem

cover_ext

L505 ↗

JacobianVarieties.Picard.Proofs.RationalBaseChangeUnit.AlgebraicGeometry.RelPicard.LineTrivDescent.cover_ext

Documentation

A two-affine cover is determined by its two opens.

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.RationalBaseChangeUnit
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
  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
attribute [-instance]
  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.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instNeg
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulInt
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAddCommGroup
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAdd
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSub
attribute [-instance]
  instModuleCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  instSMulCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulNat
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instZero
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
attribute [-instance]
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
attribute [-instance]
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
attribute [-instance]
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  SheafOfModules.isIso_ihomModelToIhom
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
attribute [-instance]
  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
attribute [-simp]
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  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
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  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
  AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
attribute [-simp]
  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
  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
attribute [-simp]
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule
  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.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.sub_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sresTop
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.add_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.IdealSheafData.coe_resLE
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.smul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_ofLE_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.ideal_range
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zero_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sres
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.comp_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.neg_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.id_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_mulRight_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.nsmul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.injEq
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zsmul_app
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
attribute [-simp]
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
attribute [-simp]
  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
  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.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicCurve hiding genus
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
namespace RelPicard
namespace LineTrivDescent
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry

Lean source · statement

theorem cover_ext {X : Scheme.{u}} {𝒱 𝒱' : X.TwoAffineOpenCover}
    (h0 : 𝒱.U0 = 𝒱'.U0) (h1 : 𝒱.U1 = 𝒱'.U1) : 𝒱 = 𝒱'

theorem

finrank_H0_H1_congr

L510 ↗

JacobianVarieties.Picard.Proofs.RationalBaseChangeUnit.AlgebraicGeometry.RelPicard.LineTrivDescent.finrank_H0_H1_congr

Documentation

Isomorphic modules have equal cohomology dimensions.

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.RationalBaseChangeUnit
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
  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
attribute [-instance]
  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.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instNeg
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulInt
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAddCommGroup
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAdd
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSub
attribute [-instance]
  instModuleCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  instSMulCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulNat
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instZero
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
attribute [-instance]
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
attribute [-instance]
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
attribute [-instance]
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  SheafOfModules.isIso_ihomModelToIhom
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
attribute [-instance]
  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
attribute [-simp]
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  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
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  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
  AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
attribute [-simp]
  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
  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
attribute [-simp]
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule
  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.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.sub_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sresTop
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.add_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.IdealSheafData.coe_resLE
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.smul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_ofLE_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.ideal_range
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zero_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sres
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.comp_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.neg_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.id_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_mulRight_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.nsmul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.injEq
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zsmul_app
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
attribute [-simp]
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
attribute [-simp]
  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
  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.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicCurve hiding genus
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
namespace RelPicard
namespace LineTrivDescent
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry

Lean source · statement

theorem finrank_H0_H1_congr {k : Type u} [Field k] {X : Scheme.{u}}
    (x : X ⟶ Spec (CommRingCat.of k))
    (𝒱 : X.TwoAffineOpenCover) {M M' : X.Modules} (e : M ≅ M') :
    Module.finrank k (𝒱.sectionsOf x M).H0 = Module.finrank k (𝒱.sectionsOf x M').H0 ∧
      Module.finrank k (𝒱.sectionsOf x M).H1 = Module.finrank k (𝒱.sectionsOf x M').H1

theorem

nonempty_linearEquiv_H0_H1_congr

L523 ↗

JacobianVarieties.Picard.Proofs.RationalBaseChangeUnit.AlgebraicGeometry.RelPicard.LineTrivDescent.nonempty_linearEquiv_H0_H1_congr

Documentation

An isomorphism of modules induces linear equivalences on both cohomology groups.

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.RationalBaseChangeUnit
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
  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
attribute [-instance]
  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.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instNeg
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulInt
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAddCommGroup
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAdd
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSub
attribute [-instance]
  instModuleCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  instSMulCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulNat
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instZero
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
attribute [-instance]
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
attribute [-instance]
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
attribute [-instance]
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  SheafOfModules.isIso_ihomModelToIhom
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
attribute [-instance]
  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
attribute [-simp]
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  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
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  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
  AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
attribute [-simp]
  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
  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
attribute [-simp]
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule
  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.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.sub_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sresTop
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.add_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.IdealSheafData.coe_resLE
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.smul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_ofLE_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.ideal_range
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zero_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sres
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.comp_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.neg_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.id_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_mulRight_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.nsmul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.injEq
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zsmul_app
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
attribute [-simp]
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
attribute [-simp]
  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
  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.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicCurve hiding genus
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
namespace RelPicard
namespace LineTrivDescent
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry

Lean source · statement

theorem nonempty_linearEquiv_H0_H1_congr {k : Type u} [Field k] {X : Scheme.{u}}
    (x : X ⟶ Spec (CommRingCat.of k))
    (𝒱 : X.TwoAffineOpenCover) {M M' : X.Modules} (e : M ≅ M') :
    Nonempty ((𝒱.sectionsOf x M).H0 ≃ₗ[k] (𝒱.sectionsOf x M').H0) ∧
      Nonempty ((𝒱.sectionsOf x M).H1 ≃ₗ[k] (𝒱.sectionsOf x M').H1)

theorem

geometricallyIntegral_toBase

L536 ↗

JacobianVarieties.Picard.Proofs.RationalBaseChangeUnit.AlgebraicGeometry.RelPicard.LineTrivDescent.geometricallyIntegral_toBase

Documentation

A smooth rational curve model over an algebraically closed field is geometrically integral.

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.RationalBaseChangeUnit
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
  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
attribute [-instance]
  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.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instNeg
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulInt
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAddCommGroup
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAdd
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSub
attribute [-instance]
  instModuleCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  instSMulCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulNat
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instZero
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
attribute [-instance]
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
attribute [-instance]
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
attribute [-instance]
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  SheafOfModules.isIso_ihomModelToIhom
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
attribute [-instance]
  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
attribute [-simp]
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  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
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  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
  AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
attribute [-simp]
  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
  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
attribute [-simp]
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule
  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.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.sub_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sresTop
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.add_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.IdealSheafData.coe_resLE
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.smul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_ofLE_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.ideal_range
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zero_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sres
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.comp_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.neg_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.id_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_mulRight_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.nsmul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.injEq
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zsmul_app
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
attribute [-simp]
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
attribute [-simp]
  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
  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.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicCurve hiding genus
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
namespace RelPicard
namespace LineTrivDescent
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry

Lean source · statement

theorem geometricallyIntegral_toBase (k : Type u) [Field k] [IsAlgClosed k]
    (M : CurveModel k (RatFunc k)) :
    GeometricallyIntegral M.toBase

theorem

nonempty_pullback_iso_of_isAlgEquivZero_baseChange

L551 ↗

JacobianVarieties.Picard.Proofs.RationalBaseChangeUnit.AlgebraicGeometry.RelPicard.LineTrivDescent.nonempty_pullback_iso_of_isAlgEquivZero_baseChange

Documentation

Algebraic triviality after field extension forces triviality on the rational curve model.

Source context 38 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.RationalBaseChangeUnit
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
  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
attribute [-instance]
  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.isIso_subschemeIota_snd_of_degree_one
  AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
  AlgebraicGeometry.isClosedImmersion_graphOver
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instNeg
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulInt
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAddCommGroup
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAdd
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSub
attribute [-instance]
  instModuleCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  instSMulCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulNat
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instZero
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
attribute [-instance]
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
attribute [-instance]
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
attribute [-instance]
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  SheafOfModules.isIso_ihomModelToIhom
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
attribute [-instance]
  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
attribute [-simp]
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  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
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  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
  AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
attribute [-simp]
  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
  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
attribute [-simp]
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule
  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.graphOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp
  AlgebraicGeometry.RelEffCartierDiv.toPoint_comp_assoc
  AlgebraicGeometry.graphOver_fst
  AlgebraicGeometry.RelEffCartierDiv.ofPoint_I
  AlgebraicGeometry.graphOver_snd
  AlgebraicGeometry.graphOver_snd_assoc
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.sub_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sresTop
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.add_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.IdealSheafData.coe_resLE
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.smul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_ofLE_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.ideal_range
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zero_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sres
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.comp_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.neg_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.id_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_mulRight_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.nsmul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.injEq
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zsmul_app
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
attribute [-simp]
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
  AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
attribute [-simp]
  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
  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.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicCurve hiding genus
open _root_.NeronModelInfra
noncomputable section
namespace AlgebraicGeometry
namespace RelPicard
namespace LineTrivDescent
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry

Lean source · statement

theorem nonempty_pullback_iso_of_isAlgEquivZero_baseChange
    (k : Type u) [Field k] [IsAlgClosed k] (Ω : Type u) [Field Ω] [IsAlgClosed Ω] [Algebra k Ω]
    {X : Scheme.{u}} (x : X ⟶ Spec (CommRingCat.of k))
    (M : CurveModel k (RatFunc k)) (𝒲M : M.C.TwoAffineOpenCover) (i : M.C ⟶ X) (hi : i ≫ x =
      M.toBase)
    (L : X.Modules) (hL : Scheme.Modules.IsInvertible L)
    (hΩ : IsAlgEquivZero (pullback.snd x (Spec.map (CommRingCat.ofHom (algebraMap k Ω))))
      ((Scheme.Modules.pullback (pullback.fst x (Spec.map (CommRingCat.ofHom (algebraMap k
        Ω))))).obj L)) :
    Nonempty ((Scheme.Modules.pullback i).obj L ≅
      (Scheme.Modules.pullback i).obj (SheafOfModules.unit X.ringCatSheaf))

theorem