AlgebraicGeometry.RelPicard.exists_relEffCartierDiv_lineBundle_iso_of_forall_fibre
Source context 28 commands
Surrounding namespaces, section variables, and local commands at this declaration.
section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.Exists_relEffCartierDiv_lineBundle_iso_of_forall_fibre
attribute [-instance]
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 SheafOfModules.isIso_ihomModelToIhom
AlgebraicGeometry.RelEffCartierDiv.isIso_subschemeIota_snd_of_degree_one
AlgebraicGeometry.isIso_ker_graphOver_subschemeIota_snd
AlgebraicGeometry.isClosedImmersion_graphOver AlgebraicCurve.CurveModel.isProper
AlgebraicCurve.CurveModel.isIntegral AlgebraicCurve.CurveModel.smooth
AlgebraicCurve.Place.instIsScalarTowerSubtypeMemValuationSubringToValuationSubring
AlgebraicCurve.Divisor.instDistribMulActionAlgEquiv AlgebraicCurve.Place.instSMulAlgEquiv
AlgebraicCurve.Place.instIsPrincipalIdealRingSubtypeMemValuationSubringToValuationSubring
AlgebraicCurve.Place.instIsDiscreteValuationRingSubtypeMemValuationSubringToValuationSubring
AlgebraicCurve.Pic0.instDistribMulActionAlgEquiv
AlgebraicCurve.Place.instAlgebraSubtypeMemValuationSubringToValuationSubring
AlgebraicCurve.Place.instMulActionAlgEquiv AlgebraicCurve.Pic0.instSMulAlgEquiv
AlgebraicGeometry.OModulePresheaf.isScalarTower
AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
AlgebraicGeometry.OModulePresheaf.addCommGroup
AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
AlgebraicGeometry.OModulePresheaf.moduleSections
AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
attribute [-instance]
AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
AlgebraicGeometry.OModulePresheaf.Leray.biC_module
AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
attribute [-instance]
ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
AlgebraicCurve.IsCurveOver.instNontrivialKaehler AlgebraicCurve.IsCurveOver.instFreeKaehler
AlgebraicCurve.IsCurveOver.toHasPrincipalDivisors AlgebraicCurve.IsCurveOver.instFiniteResidue
AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
AlgebraicCurve.Place.instIsLocalHomRestrictInclusion AlgebraicCurve.Place.instIsPrimeCenter
AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
attribute [-instance]
instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-instance]
AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
AlgebraicCurve.Pic0.instModuleZModTorsion
AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
AlgebraicCurve.SemilinearAut.instSMulTorsion
AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
AlgebraicCurve.Place.kw_ffgc_finiteDimensional_adicCompletion
AlgebraicCurve.Place.kw_ffgc_isScalarTower_integersIntegersCompletion
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
attribute [-simp]
AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app TwoChartCech.Mumford.dK_apply
TwoChartCech.Mumford.ι0_apply TwoChartCech.Mumford.ι1_apply TwoChartCech.KerCoprod.dK_apply
TwoChartCech.KerCoprod.ι1_apply TwoChartCech.KerCoprod.ι0_apply
AlgebraicGeometry.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.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
attribute [-simp]
AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
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.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
AlgebraicCurve.CurveModel.mk.injEq AlgebraicCurve.CurveModel.mk.sizeOf_spec
AlgebraicCurve.Place.mk.injEq AlgebraicCurve.Divisor.degree_single
AlgebraicCurve.Divisor.smul_single AlgebraicCurve.Place.smul_toValuationSubring
AlgebraicCurve.Place.heightOneSpectrum_asIdeal AlgebraicCurve.Place.coe_algebraMap
AlgebraicCurve.Place.ord_one AlgebraicCurve.Place.coe_smulRingEquiv_apply
AlgebraicCurve.Pic0.coe_degZeroSMulHom AlgebraicCurve.Place.deg_smul AlgebraicCurve.Pic0.mk_zero
AlgebraicCurve.Place.mk.sizeOf_spec AlgebraicCurve.Divisor.degree_smul AlgebraicCurve.Pic0.mk_add
AlgebraicCurve.Place.ord_zero AlgebraicCurve.Place.ofHeightOneSpectrum_toValuationSubring
AlgebraicCurve.mulAdele_apply AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd
AlgebraicCurve.weilSmul_one AlgebraicCurve.diagonalHom_apply
attribute [-simp]
AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe AlgebraicCurve.mulAdele_one
AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
AlgebraicGeometry.OModulePresheaf.mk.injEq AlgebraicGeometry.OModulePresheaf.prod_obj
AlgebraicGeometry.OModulePresheaf.restrOpen_obj
AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
AlgebraicGeometry.OModulePresheaf.Hom.id_app
AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
AlgebraicGeometry.OModulePresheaf.AffHom.id_app
AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U
attribute [-simp]
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 AlgebraicCurve.coe_cechH0Equiv_apply
AlgebraicCurve.cechH1ToH1_mk AlgebraicCurve.lSpaceOn_univ AlgebraicCurve.lSpaceOn_empty
AlgebraicCurve.Place.congrEquiv_symm_apply
AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
AlgebraicCurve.RationalFunctionField.deg_placeOfPoint AlgebraicCurve.Divisor.mapRestrict_single
AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
attribute [-simp]
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 AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
attribute [-simp]
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
AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec AlgebraicCurve.Place.poleSubmodule_one
AlgebraicCurve.Place.mem_poleSubmodule AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_neg
AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_zero
AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_sub
AlgebraicGeometry.Scheme.Modules.pullbackLocalSection_add
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.AlgebraicGeometry.SmoothProperCurveLean source · statement
theorem _root_.AlgebraicGeometry.RelPicard.exists_relEffCartierDiv_lineBundle_iso_of_forall_fibre
(R : Type u) [CommRing R] [IsNoetherianRing R] {C : Scheme.{u}}
(c : C ⟶ Spec (CommRingCat.of R))
[IsProper c] [SmoothOfRelativeDimension 1 c] [GeometricallyIntegral c]
(ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
(h𝔉 : ∀ m₀ : ℕ, ∃ 𝔉 : SmoothProperCurve.FiniteMapData c ε, m₀ ≤ 𝔉.m)
(g : ℕ)
(hg : ∀ (k : Type u) [Field k] [IsAlgClosed k]
(x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
(𝒲 : (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).TwoAffineOpenCover),
Module.finrank k (𝒲.sectionsOf (fibreAt c (𝟙 _) x)
(𝟙_ (pullback (pullback.snd c (𝟙 (Spec (CommRingCat.of R)))) x).Modules)).H1 = g)
{T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R)) [LocallyOfFiniteType t]
(M : (pullback c t).Modules) (hM : Scheme.Modules.IsInvertible M)
(hfib : ∀ (k : Type u) [Field k] (s : Spec (CommRingCat.of k) ⟶ T)
(𝒲 : (pullback (pullback.snd c t) s).TwoAffineOpenCover),
Subsingleton (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s M)).H1 ∧
Module.finrank k (𝒲.sectionsOf (fibreAt c t s) (fibreModule c t s M)).H0 = 1) :
∃ (D : RelEffCartierDiv c g t) (N : T.Modules), Scheme.Modules.IsInvertible N ∧
Nonempty (D.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c t)).obj N) ∧
∀ (d' : ℕ) (D' : RelEffCartierDiv c d' t) (N' : T.Modules), Scheme.Modules.IsInvertible N' →
Nonempty (D'.lineBundle ≅ M ⊗ (Scheme.Modules.pullback (pullback.snd c t)).obj N')
→ D'.I = D.I