NYLL
Module

CurveModel_exists_representsRelSubPic_abelJacobi_of_isAlgClosed.lean

Nyll.JacobianVarieties.Picard.CurveModel_exists_representsRelSubPic_abelJacobi_of_isAlgClosed
3 declarations indexed from sourceView source ↗

Module documentation

The Abel--Jacobi representation attached to a curve model

Combine closed-field Picard representability with the curve model point-class comparison.

Imports20
Imported by7

Declarations

theorem

essFiniteType_of_curveModel

L433 ↗
JacobianVarieties.Picard.Proofs.CurveModelAbelJacobi.P2mJacK.essFiniteType_of_curveModel
Documentation

The function field of the curve model is essentially of finite type.

Source context 40 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.CurveModelAbelJacobi
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-instance]
  instIsLocallyInjectiveFunUliftYonedaGluedToSheaf
attribute [-instance]
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.instIsLocallySurjectiveUliftYonedaGluedToSheaf
attribute [-instance]
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.instIsOpenImmersionToGlued
attribute [-instance]
  instIsIsoSheafZariskiTopologyTypeUliftYonedaGluedToSheaf
attribute [-instance]
  AlgebraicGeometry.RelPicard.isOpenImmersion_toGlued
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion
  AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.FGSubalgebra.instIsDirectedLe
attribute [-instance]
  instQuasiSeparatedSpaceCarrierCarrierCommRingCatObjOppositeSchemeSpecDiagram
attribute [-instance]
  AlgebraicGeometry.FGSubalgebra.instIsCofilteredOpposite
  AlgebraicGeometry.FGSubalgebra.instIsAffineObjOppositeSchemeSpecDiagram
  AlgebraicGeometry.FGSubalgebra.instNonempty
  AlgebraicGeometry.FGSubalgebra.instNonemptySubtypeLeSubalgebraValFG
attribute [-instance]
  instCompactSpaceCarrierCarrierCommRingCatObjOppositeSchemeSpecDiagram
attribute [-instance]
  AlgebraicGeometry.FGSubalgebra.instIsDirectedSubtypeLeSubalgebraValFG
  AlgebraicGeometry.FGSubalgebra.instIsAffineHomMapOppositeSchemeSpecDiagram
  AlgebraicGeometry.FGSubalgebra.instIsFiltered
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK
  AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  SheafOfModules.isIso_ihomModelToIhom
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
attribute [-instance]
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
attribute [-instance]
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
  AlgebraicGeometry.FGSubalgebra.tensorStage_directedSystem
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  CategoryTheory.Functor.OverTotal.ofFibre_fst
  CategoryTheory.Functor.overTotal_map_fst
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.glueData_J
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.uliftYoneda_toGlued_uliftYonedaGluedToSheaf
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.glueData_t'
attribute [-simp]
  uliftYoneda_toGlued_uliftYonedaGluedToSheaf_assoc
attribute [-simp]
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.uliftYonedaGluedToSheaf_app_toGlued
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.glueData_openCover_map
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.uliftYonedaGluedToSheaf_app_comp
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.glueData_V
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.glueData_t
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.glueData_U
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.glueData_f
  AlgebraicGeometry.RelPicard.designationOfRepresentableBy_P
  AlgebraicGeometry.RelPicard.designationOfRepresentableBy_toBase
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk
  AlgebraicCurve.lSpaceOn_univ
  AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single
  AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber
  AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward
  AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom
  AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring
attribute [-simp]
  AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
  AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong
  AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong
  AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def
  AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring
  AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one
  AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul
  AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one
  AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul
  AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero
  AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero
  AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec
attribute [-simp]
  AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec
  AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  AlgebraicGeometry.FGSubalgebra.cocone_ι_app_apply
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq
  AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply
  AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule
  AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule
  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
attribute [-simp]
  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
  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
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
  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.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.IsUniversal.lift_comp
  AlgebraicGeometry.RelEffCartierDiv.functor_map_fst
  AlgebraicGeometry.RelEffCartierDiv.IsUniversal.homEquiv_apply
  AlgebraicGeometry.RelEffCartierDiv.IsUniversal.lift_pullbackAlong
  AlgebraicGeometry.RelEffCartierDiv.IsUniversal.homEquiv_symm_apply
  AlgebraicGeometry.RelEffCartierDiv.IsUniversal.lift_comp_assoc
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  CoherentBaseChange.FibreH0Family.mk.sizeOf_spec
  CoherentBaseChange.FibreH0Family.mk.injEq
  AlgebraicGeometry.Scheme.Modules.ProjPresentation.mk.injEq
  AlgebraicGeometry.Scheme.Modules.ProjPresentation.mk.sizeOf_spec
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
open _root_.AlgebraicCurve
open scoped CategoryTheory.MonObj
universe u v
namespace P2mJacK
variable {K : Type u} [Field K] {F : Type v} [Field F] [Algebra K F]
Lean source · statement
theorem essFiniteType_of_curveModel (M : CurveModel K F) : Algebra.EssFiniteType K F
theorem

single_sub_single_mem_degZero

L440 ↗
JacobianVarieties.Picard.Proofs.CurveModelAbelJacobi.P2mJacK.single_sub_single_mem_degZero
Documentation

The difference of two closed-field places has degree zero.

Source context 40 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.CurveModelAbelJacobi
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-instance]
  instIsLocallyInjectiveFunUliftYonedaGluedToSheaf
attribute [-instance]
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.instIsLocallySurjectiveUliftYonedaGluedToSheaf
attribute [-instance]
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.instIsOpenImmersionToGlued
attribute [-instance]
  instIsIsoSheafZariskiTopologyTypeUliftYonedaGluedToSheaf
attribute [-instance]
  AlgebraicGeometry.RelPicard.isOpenImmersion_toGlued
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
  AlgebraicCurve.Place.instIsLocalHomRestrictInclusion
  AlgebraicCurve.Place.instIsPrimeCenter
  AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
  AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
  AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
attribute [-instance]
  AlgebraicGeometry.RelEffCartierDiv.subsingleton_of_degree_zero
  AlgebraicGeometry.FGSubalgebra.instIsDirectedLe
attribute [-instance]
  instQuasiSeparatedSpaceCarrierCarrierCommRingCatObjOppositeSchemeSpecDiagram
attribute [-instance]
  AlgebraicGeometry.FGSubalgebra.instIsCofilteredOpposite
  AlgebraicGeometry.FGSubalgebra.instIsAffineObjOppositeSchemeSpecDiagram
  AlgebraicGeometry.FGSubalgebra.instNonempty
  AlgebraicGeometry.FGSubalgebra.instNonemptySubtypeLeSubalgebraValFG
attribute [-instance]
  instCompactSpaceCarrierCarrierCommRingCatObjOppositeSchemeSpecDiagram
attribute [-instance]
  AlgebraicGeometry.FGSubalgebra.instIsDirectedSubtypeLeSubalgebraValFG
  AlgebraicGeometry.FGSubalgebra.instIsAffineHomMapOppositeSchemeSpecDiagram
  AlgebraicGeometry.FGSubalgebra.instIsFiltered
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  AlgebraicCurve.Place.kw_ffgc_continuousSMul_adicCompletionComap
  AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersCompletionCompletion
attribute [-instance]
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
attribute [-instance]
  AlgebraicCurve.instHasCanonicalLocalResidueK
  AlgebraicCurve.Place.instAlgebra_restrictResidueField
  AlgebraicCurve.Place.instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  AlgebraicCurve.Place.instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  SheafOfModules.isIso_ihomModelToIhom
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
attribute [-instance]
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg
  ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR
attribute [-instance]
  ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype
  ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq
  ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup
  ProjSpaceCech.Twist.cochain.instModule
  AlgebraicGeometry.FGSubalgebra.tensorStage_directedSystem
  AlgebraicGeometry.RelEffCartierDiv.isClosedImmersion_subschemeι_resProdMap
  AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_resProdMap
attribute [-simp]
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  CategoryTheory.Functor.OverTotal.ofFibre_fst
  CategoryTheory.Functor.overTotal_map_fst
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.glueData_J
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.uliftYoneda_toGlued_uliftYonedaGluedToSheaf
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.glueData_t'
attribute [-simp]
  uliftYoneda_toGlued_uliftYonedaGluedToSheaf_assoc
attribute [-simp]
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.uliftYonedaGluedToSheaf_app_toGlued
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.glueData_openCover_map
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.uliftYonedaGluedToSheaf_app_comp
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.glueData_V
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.glueData_t
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.glueData_U
  AlgebraicGeometry.Scheme.LocalRepresentabilityULift.glueData_f
  AlgebraicGeometry.RelPicard.designationOfRepresentableBy_P
  AlgebraicGeometry.RelPicard.designationOfRepresentableBy_toBase
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicCurve.coe_cechH0Equiv_apply
  AlgebraicCurve.cechH1ToH1_mk
  AlgebraicCurve.lSpaceOn_univ
  AlgebraicCurve.lSpaceOn_empty
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single
  AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber
  AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward
  AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom
  AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring
attribute [-simp]
  AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
  AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong
  AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong
  AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def
  AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring
  AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one
  AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul
  AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one
  AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul
  AlgebraicCurve.SemilinearAut.toRingAut_mul
attribute [-simp]
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero
  AlgebraicCurve.Place.evalAt_one
  AlgebraicCurve.Place.differentialCoeff_zero
  AlgebraicCurve.Place.differentialCoeff_dCoord
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec
attribute [-simp]
  AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq
  AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec
  AlgebraicCurve.PoleDivisorPackage.mk.injEq
  AlgebraicGeometry.prodKerGraph_one
  AlgebraicGeometry.fibrePowOver.proj_comp
  AlgebraicGeometry.prodKerGraph_zero
  AlgebraicGeometry.RelEffCartierDiv.empty_I
  AlgebraicGeometry.fibrePowOver.proj_comp_assoc
  AlgebraicCurve.SymmetricPowerPackage.mk.sizeOf_spec
  AlgebraicCurve.SymmetricPowerPackage.mk.injEq
  AlgebraicGeometry.FGSubalgebra.cocone_ι_app_apply
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq
  AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply
  AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule
  AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one
  AlgebraicCurve.Place.mem_poleSubmodule
  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
attribute [-simp]
  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
  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
attribute [-simp]
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec
  DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq
  AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq
  ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq
  ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap
  ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
  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.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelEffCartierDiv.IsUniversal.lift_comp
  AlgebraicGeometry.RelEffCartierDiv.functor_map_fst
  AlgebraicGeometry.RelEffCartierDiv.IsUniversal.homEquiv_apply
  AlgebraicGeometry.RelEffCartierDiv.IsUniversal.lift_pullbackAlong
  AlgebraicGeometry.RelEffCartierDiv.IsUniversal.homEquiv_symm_apply
  AlgebraicGeometry.RelEffCartierDiv.IsUniversal.lift_comp_assoc
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top
  AlgebraicGeometry.RelEffCartierDiv.mem_supportedIn_iff
  AlgebraicGeometry.RelEffCartierDiv.supportedIn_top_eq
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_extendAlong
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_I
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd
  AlgebraicGeometry.RelEffCartierDiv.restrictAlong_I
  AlgebraicGeometry.RelEffCartierDiv.extendAlong_restrictAlong
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_snd_assoc
  AlgebraicGeometry.RelEffCartierDiv.resProdMap_fst
  CoherentBaseChange.FibreH0Family.mk.sizeOf_spec
  CoherentBaseChange.FibreH0Family.mk.injEq
  AlgebraicGeometry.Scheme.Modules.ProjPresentation.mk.injEq
  AlgebraicGeometry.Scheme.Modules.ProjPresentation.mk.sizeOf_spec
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
open _root_.AlgebraicCurve
open scoped CategoryTheory.MonObj
universe u v
namespace P2mJacK
variable {K : Type u} [Field K] {F : Type v} [Field F] [Algebra K F]
Lean source · statement
theorem single_sub_single_mem_degZero [IsAlgClosed K] [IsCurveOver K F] (v w : Place K F) :
    Finsupp.single v (1 : ℤ) - Finsupp.single w 1 ∈ Divisor.degZero (K := K) (F := F)
theorem

exists_representsRelSubPic_abelJacobi_of_isAlgClosed

L466 ↗
AlgebraicCurve.CurveModel.exists_representsRelSubPic_abelJacobi_of_isAlgClosed
Documentation

The curve model has an Abel--Jacobi map and a pointwise Picard-class comparison.

Source context 13 commands

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

section ProofImplementation
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
open _root_.AlgebraicCurve
open scoped CategoryTheory.MonObj
namespace AlgebraicCurve.CurveModel
universe u v
open JacobianVarieties.Picard.Proofs.CurveModelAbelJacobi.P2mJacK in
Lean source · statement
theorem exists_representsRelSubPic_abelJacobi_of_isAlgClosed
    (K : Type u) [Field K] [IsAlgClosed K] (F : Type v) [Field F] [Algebra K F] [IsCurveOver K F]
    (M : CurveModel K F)
    (s : {q : Spec (CommRingCat.of K) ⟶ M.C // q ≫ M.toBase = 𝟙 _}) :
    ∃ (D : RelativePic0Designation K M.toBase)
      (h : RepresentsRelSubPic M.toBase s (algEquivZeroCut M.toBase s) D)
      (aj : SchemeHomOver M.toBase D.toBase),
      Smooth D.toBase ∧ IsProper D.toBase ∧ GeometricallyConnected D.toBase ∧
      s.1 ≫ aj.1 = D.zeroSection ∧
      (∀ (K' : Type u) [Field K'] (t : Spec (CommRingCat.of K') ⟶ Spec (CommRingCat.of K))
          (x : SchemeHomOver t M.toBase),
        Nonempty ((h.poincare.pullbackAlong
            ⟨x.1 ≫ aj.1, (Category.assoc _ _ _).trans ((congrArg (x.1 ≫ ·) aj.2).trans x.2)⟩).L ≅
          (RelEffCartierDiv.ofPoint M.toBase x.1 x.2).lineBundle ⊗
            (RelEffCartierDiv.ofPoint M.toBase (t ≫ s.1)
              ((Category.assoc _ _ _).trans ((congrArg (t ≫ ·) s.2).trans (Category.comp_id
                t)))).idealModule)) ∧
      ∃ pts : Pic0 K F ≃ SchemeHomOver (𝟙 (Spec (CommRingCat.of K))) D.toBase,
        (∀ x y : Pic0 K F, pts (x + y) =
          (RepresentsRelSubPic.relativeGroupLaw (P := algEquivZeroGroupCut M.toBase s) h).mul _
            (pts x) (pts y)) ∧
        ∀ x : {q : Spec (CommRingCat.of K) ⟶ M.C // q ≫ M.toBase = 𝟙 _},
          ∃ Dv : Divisor.degZero (K := K) (F := F),
            (Dv : Divisor K F) =
              Finsupp.single (M.pointEquivPlace x) 1 - Finsupp.single (M.pointEquivPlace s) 1 ∧
            (pts (Pic0.mk Dv)).1 = x.1 ≫ aj.1

Statements and documentation are read from the Lean source at the revision shown. Source entries are not a measure of mathematical completeness.