NYLL
Module

mem_of_isProper_of_forall_smoothProperCurve_mem.lean

Nyll.AlgebraicCurves.Geometry.Curves.mem_of_isProper_of_forall_smoothProperCurve_mem
10 declarations indexed from sourceView source ↗

Module documentation

Membership detected by smooth proper curves in a proper scheme

Imports2
Imported by2

Declarations

abbrev

pt

L78 ↗
AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem.P2mCurveConnect.pt
Documentation

The underlying point of the scheme corresponding to a field-valued point.

Source context 9 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem
attribute [-instance] 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
  AlgebraicCurve.CurveModel.locallyOfFiniteType_gluedToBase
  AlgebraicCurve.CurveModel.isFractionRing_overlap
  AlgebraicCurve.CurveModel.isLocallyNoetherian_glued AlgebraicCurve.CurveModel.jacobsonSpace_glued
  AlgebraicCurve.CurveModel.isOpenImmersion_ι₀ AlgebraicCurve.CurveModel.isIntegral_glued
  AlgebraicCurve.CurveModel.quasiSeparated_gluedToBase AlgebraicCurve.CurveModel.compactSpace_glued
  AlgebraicCurve.CurveModel.isIntegral_adjoin_chartRing
  AlgebraicCurve.CurveModel.isFractionRing_overlap_functionField
  AlgebraicCurve.CurveModel.isProper_gluedToBase AlgebraicCurve.CurveModel.isOpenImmersion_ιU
  AlgebraicCurve.CurveModel.isOpenImmersion_f₀ AlgebraicCurve.CurveModel.isOpenImmersion_fInf
  AlgebraicCurve.CurveModel.isOpenImmersion_ιInf
  AlgebraicCurve.CurveModel.algebra_overlap_functionField
  AlgebraicCurve.CurveModel.quasiCompact_gluedToBase AlgebraicCurve.CurveModel.algebraAdjoin
  AlgebraicCurve.CurveModel.isDedekindDomain_chartRing AlgebraicCurve.CurveModel.isIntegralClosure
  AlgebraicCurve.CurveModel.finite_chartRing AlgebraicCurve.CurveModel.centre_isPrime
  AlgebraicCurve.CurveModel.isFractionRing_chartRing AlgebraicCurve.CurveModel.finiteType_chartRing
  AlgebraicCurve.CurveModel.isNoetherianRing_chartRing
  AlgebraicCurve.CurveModel.isScalarTower_base_adjoin
  AlgebraicCurve.CurveModel.isScalarTower_adjoin
  AlgebraicCurve.CurveModel.chartRing_finitePresentation
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
attribute [-simp] 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.CurveModel.ιInf_gluedToBase AlgebraicCurve.CurveModel.ι₀_gluedToBase
  AlgebraicCurve.CurveModel.coe_primeEquivChartPlaces 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
universe u
open CategoryTheory AlgebraicGeometry
namespace P2mCurveConnect
variable {k : Type u} [Field k] [IsAlgClosed k] {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of k))
Lean source · statement
abbrev pt (p : Spec (CommRingCat.of k) ⟶ Y) : Y
theorem

isClosed_pt

L80 ↗
AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem.P2mCurveConnect.isClosed_pt
Source context 9 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem
attribute [-instance] 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
  AlgebraicCurve.CurveModel.locallyOfFiniteType_gluedToBase
  AlgebraicCurve.CurveModel.isFractionRing_overlap
  AlgebraicCurve.CurveModel.isLocallyNoetherian_glued AlgebraicCurve.CurveModel.jacobsonSpace_glued
  AlgebraicCurve.CurveModel.isOpenImmersion_ι₀ AlgebraicCurve.CurveModel.isIntegral_glued
  AlgebraicCurve.CurveModel.quasiSeparated_gluedToBase AlgebraicCurve.CurveModel.compactSpace_glued
  AlgebraicCurve.CurveModel.isIntegral_adjoin_chartRing
  AlgebraicCurve.CurveModel.isFractionRing_overlap_functionField
  AlgebraicCurve.CurveModel.isProper_gluedToBase AlgebraicCurve.CurveModel.isOpenImmersion_ιU
  AlgebraicCurve.CurveModel.isOpenImmersion_f₀ AlgebraicCurve.CurveModel.isOpenImmersion_fInf
  AlgebraicCurve.CurveModel.isOpenImmersion_ιInf
  AlgebraicCurve.CurveModel.algebra_overlap_functionField
  AlgebraicCurve.CurveModel.quasiCompact_gluedToBase AlgebraicCurve.CurveModel.algebraAdjoin
  AlgebraicCurve.CurveModel.isDedekindDomain_chartRing AlgebraicCurve.CurveModel.isIntegralClosure
  AlgebraicCurve.CurveModel.finite_chartRing AlgebraicCurve.CurveModel.centre_isPrime
  AlgebraicCurve.CurveModel.isFractionRing_chartRing AlgebraicCurve.CurveModel.finiteType_chartRing
  AlgebraicCurve.CurveModel.isNoetherianRing_chartRing
  AlgebraicCurve.CurveModel.isScalarTower_base_adjoin
  AlgebraicCurve.CurveModel.isScalarTower_adjoin
  AlgebraicCurve.CurveModel.chartRing_finitePresentation
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
attribute [-simp] 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.CurveModel.ιInf_gluedToBase AlgebraicCurve.CurveModel.ι₀_gluedToBase
  AlgebraicCurve.CurveModel.coe_primeEquivChartPlaces 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
universe u
open CategoryTheory AlgebraicGeometry
namespace P2mCurveConnect
variable {k : Type u} [Field k] [IsAlgClosed k] {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of k))
Lean source · statement
theorem isClosed_pt [LocallyOfFiniteType y] (p : Spec (CommRingCat.of k) ⟶ Y) (hp : p ≫ y = 𝟙 _) :
    IsClosed ({pt p} : Set Y)
def

CurveStable

L85 ↗
AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem.P2mCurveConnect.CurveStable
Documentation

Stability of a set of points along images of smooth proper integral curves.

Source context 9 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem
attribute [-instance] 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
  AlgebraicCurve.CurveModel.locallyOfFiniteType_gluedToBase
  AlgebraicCurve.CurveModel.isFractionRing_overlap
  AlgebraicCurve.CurveModel.isLocallyNoetherian_glued AlgebraicCurve.CurveModel.jacobsonSpace_glued
  AlgebraicCurve.CurveModel.isOpenImmersion_ι₀ AlgebraicCurve.CurveModel.isIntegral_glued
  AlgebraicCurve.CurveModel.quasiSeparated_gluedToBase AlgebraicCurve.CurveModel.compactSpace_glued
  AlgebraicCurve.CurveModel.isIntegral_adjoin_chartRing
  AlgebraicCurve.CurveModel.isFractionRing_overlap_functionField
  AlgebraicCurve.CurveModel.isProper_gluedToBase AlgebraicCurve.CurveModel.isOpenImmersion_ιU
  AlgebraicCurve.CurveModel.isOpenImmersion_f₀ AlgebraicCurve.CurveModel.isOpenImmersion_fInf
  AlgebraicCurve.CurveModel.isOpenImmersion_ιInf
  AlgebraicCurve.CurveModel.algebra_overlap_functionField
  AlgebraicCurve.CurveModel.quasiCompact_gluedToBase AlgebraicCurve.CurveModel.algebraAdjoin
  AlgebraicCurve.CurveModel.isDedekindDomain_chartRing AlgebraicCurve.CurveModel.isIntegralClosure
  AlgebraicCurve.CurveModel.finite_chartRing AlgebraicCurve.CurveModel.centre_isPrime
  AlgebraicCurve.CurveModel.isFractionRing_chartRing AlgebraicCurve.CurveModel.finiteType_chartRing
  AlgebraicCurve.CurveModel.isNoetherianRing_chartRing
  AlgebraicCurve.CurveModel.isScalarTower_base_adjoin
  AlgebraicCurve.CurveModel.isScalarTower_adjoin
  AlgebraicCurve.CurveModel.chartRing_finitePresentation
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
attribute [-simp] 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.CurveModel.ιInf_gluedToBase AlgebraicCurve.CurveModel.ι₀_gluedToBase
  AlgebraicCurve.CurveModel.coe_primeEquivChartPlaces 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
universe u
open CategoryTheory AlgebraicGeometry
namespace P2mCurveConnect
variable {k : Type u} [Field k] [IsAlgClosed k] {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of k))
Lean source · statement
def CurveStable (S : Set (Spec (CommRingCat.of k) ⟶ Y)) : Prop
def

algHomK

L96 ↗
AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem.P2mCurveConnect.algHomK
Documentation

The base-field homomorphism to the ring of sections of the selected affine open.

Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem
attribute [-instance] 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
  AlgebraicCurve.CurveModel.locallyOfFiniteType_gluedToBase
  AlgebraicCurve.CurveModel.isFractionRing_overlap
  AlgebraicCurve.CurveModel.isLocallyNoetherian_glued AlgebraicCurve.CurveModel.jacobsonSpace_glued
  AlgebraicCurve.CurveModel.isOpenImmersion_ι₀ AlgebraicCurve.CurveModel.isIntegral_glued
  AlgebraicCurve.CurveModel.quasiSeparated_gluedToBase AlgebraicCurve.CurveModel.compactSpace_glued
  AlgebraicCurve.CurveModel.isIntegral_adjoin_chartRing
  AlgebraicCurve.CurveModel.isFractionRing_overlap_functionField
  AlgebraicCurve.CurveModel.isProper_gluedToBase AlgebraicCurve.CurveModel.isOpenImmersion_ιU
  AlgebraicCurve.CurveModel.isOpenImmersion_f₀ AlgebraicCurve.CurveModel.isOpenImmersion_fInf
  AlgebraicCurve.CurveModel.isOpenImmersion_ιInf
  AlgebraicCurve.CurveModel.algebra_overlap_functionField
  AlgebraicCurve.CurveModel.quasiCompact_gluedToBase AlgebraicCurve.CurveModel.algebraAdjoin
  AlgebraicCurve.CurveModel.isDedekindDomain_chartRing AlgebraicCurve.CurveModel.isIntegralClosure
  AlgebraicCurve.CurveModel.finite_chartRing AlgebraicCurve.CurveModel.centre_isPrime
  AlgebraicCurve.CurveModel.isFractionRing_chartRing AlgebraicCurve.CurveModel.finiteType_chartRing
  AlgebraicCurve.CurveModel.isNoetherianRing_chartRing
  AlgebraicCurve.CurveModel.isScalarTower_base_adjoin
  AlgebraicCurve.CurveModel.isScalarTower_adjoin
  AlgebraicCurve.CurveModel.chartRing_finitePresentation
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
attribute [-simp] 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.CurveModel.ιInf_gluedToBase AlgebraicCurve.CurveModel.ι₀_gluedToBase
  AlgebraicCurve.CurveModel.coe_primeEquivChartPlaces 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
universe u
open CategoryTheory AlgebraicGeometry
namespace P2mCurveConnect
variable {k : Type u} [Field k] [IsAlgClosed k] {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of k))
section Link
variable {U : Y.Opens} (hU : IsAffineOpen U)
omit [IsAlgClosed k] in
Lean source · statement
def algHomK : CommRingCat.of k ⟶ Γ(Y, U)
theorem

spec_map_algHomK

L98 ↗
AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem.P2mCurveConnect.spec_map_algHomK
Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem
attribute [-instance] 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
  AlgebraicCurve.CurveModel.locallyOfFiniteType_gluedToBase
  AlgebraicCurve.CurveModel.isFractionRing_overlap
  AlgebraicCurve.CurveModel.isLocallyNoetherian_glued AlgebraicCurve.CurveModel.jacobsonSpace_glued
  AlgebraicCurve.CurveModel.isOpenImmersion_ι₀ AlgebraicCurve.CurveModel.isIntegral_glued
  AlgebraicCurve.CurveModel.quasiSeparated_gluedToBase AlgebraicCurve.CurveModel.compactSpace_glued
  AlgebraicCurve.CurveModel.isIntegral_adjoin_chartRing
  AlgebraicCurve.CurveModel.isFractionRing_overlap_functionField
  AlgebraicCurve.CurveModel.isProper_gluedToBase AlgebraicCurve.CurveModel.isOpenImmersion_ιU
  AlgebraicCurve.CurveModel.isOpenImmersion_f₀ AlgebraicCurve.CurveModel.isOpenImmersion_fInf
  AlgebraicCurve.CurveModel.isOpenImmersion_ιInf
  AlgebraicCurve.CurveModel.algebra_overlap_functionField
  AlgebraicCurve.CurveModel.quasiCompact_gluedToBase AlgebraicCurve.CurveModel.algebraAdjoin
  AlgebraicCurve.CurveModel.isDedekindDomain_chartRing AlgebraicCurve.CurveModel.isIntegralClosure
  AlgebraicCurve.CurveModel.finite_chartRing AlgebraicCurve.CurveModel.centre_isPrime
  AlgebraicCurve.CurveModel.isFractionRing_chartRing AlgebraicCurve.CurveModel.finiteType_chartRing
  AlgebraicCurve.CurveModel.isNoetherianRing_chartRing
  AlgebraicCurve.CurveModel.isScalarTower_base_adjoin
  AlgebraicCurve.CurveModel.isScalarTower_adjoin
  AlgebraicCurve.CurveModel.chartRing_finitePresentation
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
attribute [-simp] 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.CurveModel.ιInf_gluedToBase AlgebraicCurve.CurveModel.ι₀_gluedToBase
  AlgebraicCurve.CurveModel.coe_primeEquivChartPlaces 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
universe u
open CategoryTheory AlgebraicGeometry
namespace P2mCurveConnect
variable {k : Type u} [Field k] [IsAlgClosed k] {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of k))
section Link
variable {U : Y.Opens} (hU : IsAffineOpen U)
omit [IsAlgClosed k] in
Lean source · statement
theorem spec_map_algHomK : Spec.map (algHomK y hU) = hU.fromSpec ≫ y
def

algK

L101 ↗
AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem.P2mCurveConnect.algK
Documentation

The base-field algebra structure on the sections of the selected affine open.

Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem
attribute [-instance] 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
  AlgebraicCurve.CurveModel.locallyOfFiniteType_gluedToBase
  AlgebraicCurve.CurveModel.isFractionRing_overlap
  AlgebraicCurve.CurveModel.isLocallyNoetherian_glued AlgebraicCurve.CurveModel.jacobsonSpace_glued
  AlgebraicCurve.CurveModel.isOpenImmersion_ι₀ AlgebraicCurve.CurveModel.isIntegral_glued
  AlgebraicCurve.CurveModel.quasiSeparated_gluedToBase AlgebraicCurve.CurveModel.compactSpace_glued
  AlgebraicCurve.CurveModel.isIntegral_adjoin_chartRing
  AlgebraicCurve.CurveModel.isFractionRing_overlap_functionField
  AlgebraicCurve.CurveModel.isProper_gluedToBase AlgebraicCurve.CurveModel.isOpenImmersion_ιU
  AlgebraicCurve.CurveModel.isOpenImmersion_f₀ AlgebraicCurve.CurveModel.isOpenImmersion_fInf
  AlgebraicCurve.CurveModel.isOpenImmersion_ιInf
  AlgebraicCurve.CurveModel.algebra_overlap_functionField
  AlgebraicCurve.CurveModel.quasiCompact_gluedToBase AlgebraicCurve.CurveModel.algebraAdjoin
  AlgebraicCurve.CurveModel.isDedekindDomain_chartRing AlgebraicCurve.CurveModel.isIntegralClosure
  AlgebraicCurve.CurveModel.finite_chartRing AlgebraicCurve.CurveModel.centre_isPrime
  AlgebraicCurve.CurveModel.isFractionRing_chartRing AlgebraicCurve.CurveModel.finiteType_chartRing
  AlgebraicCurve.CurveModel.isNoetherianRing_chartRing
  AlgebraicCurve.CurveModel.isScalarTower_base_adjoin
  AlgebraicCurve.CurveModel.isScalarTower_adjoin
  AlgebraicCurve.CurveModel.chartRing_finitePresentation
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
attribute [-simp] 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.CurveModel.ιInf_gluedToBase AlgebraicCurve.CurveModel.ι₀_gluedToBase
  AlgebraicCurve.CurveModel.coe_primeEquivChartPlaces 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
universe u
open CategoryTheory AlgebraicGeometry
namespace P2mCurveConnect
variable {k : Type u} [Field k] [IsAlgClosed k] {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of k))
section Link
variable {U : Y.Opens} (hU : IsAffineOpen U)
omit [IsAlgClosed k] in
Lean source · statement
def algK : Algebra k Γ(Y, U)
theorem

finiteType_algK

L103 ↗
AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem.P2mCurveConnect.finiteType_algK
Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem
attribute [-instance] 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
  AlgebraicCurve.CurveModel.locallyOfFiniteType_gluedToBase
  AlgebraicCurve.CurveModel.isFractionRing_overlap
  AlgebraicCurve.CurveModel.isLocallyNoetherian_glued AlgebraicCurve.CurveModel.jacobsonSpace_glued
  AlgebraicCurve.CurveModel.isOpenImmersion_ι₀ AlgebraicCurve.CurveModel.isIntegral_glued
  AlgebraicCurve.CurveModel.quasiSeparated_gluedToBase AlgebraicCurve.CurveModel.compactSpace_glued
  AlgebraicCurve.CurveModel.isIntegral_adjoin_chartRing
  AlgebraicCurve.CurveModel.isFractionRing_overlap_functionField
  AlgebraicCurve.CurveModel.isProper_gluedToBase AlgebraicCurve.CurveModel.isOpenImmersion_ιU
  AlgebraicCurve.CurveModel.isOpenImmersion_f₀ AlgebraicCurve.CurveModel.isOpenImmersion_fInf
  AlgebraicCurve.CurveModel.isOpenImmersion_ιInf
  AlgebraicCurve.CurveModel.algebra_overlap_functionField
  AlgebraicCurve.CurveModel.quasiCompact_gluedToBase AlgebraicCurve.CurveModel.algebraAdjoin
  AlgebraicCurve.CurveModel.isDedekindDomain_chartRing AlgebraicCurve.CurveModel.isIntegralClosure
  AlgebraicCurve.CurveModel.finite_chartRing AlgebraicCurve.CurveModel.centre_isPrime
  AlgebraicCurve.CurveModel.isFractionRing_chartRing AlgebraicCurve.CurveModel.finiteType_chartRing
  AlgebraicCurve.CurveModel.isNoetherianRing_chartRing
  AlgebraicCurve.CurveModel.isScalarTower_base_adjoin
  AlgebraicCurve.CurveModel.isScalarTower_adjoin
  AlgebraicCurve.CurveModel.chartRing_finitePresentation
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
attribute [-simp] 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.CurveModel.ιInf_gluedToBase AlgebraicCurve.CurveModel.ι₀_gluedToBase
  AlgebraicCurve.CurveModel.coe_primeEquivChartPlaces 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
universe u
open CategoryTheory AlgebraicGeometry
namespace P2mCurveConnect
variable {k : Type u} [Field k] [IsAlgClosed k] {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of k))
section Link
variable {U : Y.Opens} (hU : IsAffineOpen U)
omit [IsAlgClosed k] in
Lean source · statement
theorem finiteType_algK [LocallyOfFiniteType y] : @Algebra.FiniteType k Γ(Y, U) _ _ (algK y hU)
theorem

link

L109 ↗
AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem.P2mCurveConnect.link
Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem
attribute [-instance] 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
  AlgebraicCurve.CurveModel.locallyOfFiniteType_gluedToBase
  AlgebraicCurve.CurveModel.isFractionRing_overlap
  AlgebraicCurve.CurveModel.isLocallyNoetherian_glued AlgebraicCurve.CurveModel.jacobsonSpace_glued
  AlgebraicCurve.CurveModel.isOpenImmersion_ι₀ AlgebraicCurve.CurveModel.isIntegral_glued
  AlgebraicCurve.CurveModel.quasiSeparated_gluedToBase AlgebraicCurve.CurveModel.compactSpace_glued
  AlgebraicCurve.CurveModel.isIntegral_adjoin_chartRing
  AlgebraicCurve.CurveModel.isFractionRing_overlap_functionField
  AlgebraicCurve.CurveModel.isProper_gluedToBase AlgebraicCurve.CurveModel.isOpenImmersion_ιU
  AlgebraicCurve.CurveModel.isOpenImmersion_f₀ AlgebraicCurve.CurveModel.isOpenImmersion_fInf
  AlgebraicCurve.CurveModel.isOpenImmersion_ιInf
  AlgebraicCurve.CurveModel.algebra_overlap_functionField
  AlgebraicCurve.CurveModel.quasiCompact_gluedToBase AlgebraicCurve.CurveModel.algebraAdjoin
  AlgebraicCurve.CurveModel.isDedekindDomain_chartRing AlgebraicCurve.CurveModel.isIntegralClosure
  AlgebraicCurve.CurveModel.finite_chartRing AlgebraicCurve.CurveModel.centre_isPrime
  AlgebraicCurve.CurveModel.isFractionRing_chartRing AlgebraicCurve.CurveModel.finiteType_chartRing
  AlgebraicCurve.CurveModel.isNoetherianRing_chartRing
  AlgebraicCurve.CurveModel.isScalarTower_base_adjoin
  AlgebraicCurve.CurveModel.isScalarTower_adjoin
  AlgebraicCurve.CurveModel.chartRing_finitePresentation
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
attribute [-simp] 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.CurveModel.ιInf_gluedToBase AlgebraicCurve.CurveModel.ι₀_gluedToBase
  AlgebraicCurve.CurveModel.coe_primeEquivChartPlaces 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
universe u
open CategoryTheory AlgebraicGeometry
namespace P2mCurveConnect
variable {k : Type u} [Field k] [IsAlgClosed k] {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of k))
section Link
variable {U : Y.Opens} (hU : IsAffineOpen U)
include hU in
Lean source · statement
theorem link [IsProper y] [IsIntegral Y] (S : Set (Spec (CommRingCat.of k) ⟶ Y)) (hS : CurveStable
  y S)
    (p q : Spec (CommRingCat.of k) ⟶ Y) (hp : p ≫ y = 𝟙 _) (hq : q ≫ y = 𝟙 _)
    (hpU : pt p ∈ U) (hqU : pt q ∈ U) (hpS : p ∈ S) : q ∈ S
theorem

main

L206 ↗
AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem.P2mCurveConnect.main
Source context 9 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem
attribute [-instance] 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
  AlgebraicCurve.CurveModel.locallyOfFiniteType_gluedToBase
  AlgebraicCurve.CurveModel.isFractionRing_overlap
  AlgebraicCurve.CurveModel.isLocallyNoetherian_glued AlgebraicCurve.CurveModel.jacobsonSpace_glued
  AlgebraicCurve.CurveModel.isOpenImmersion_ι₀ AlgebraicCurve.CurveModel.isIntegral_glued
  AlgebraicCurve.CurveModel.quasiSeparated_gluedToBase AlgebraicCurve.CurveModel.compactSpace_glued
  AlgebraicCurve.CurveModel.isIntegral_adjoin_chartRing
  AlgebraicCurve.CurveModel.isFractionRing_overlap_functionField
  AlgebraicCurve.CurveModel.isProper_gluedToBase AlgebraicCurve.CurveModel.isOpenImmersion_ιU
  AlgebraicCurve.CurveModel.isOpenImmersion_f₀ AlgebraicCurve.CurveModel.isOpenImmersion_fInf
  AlgebraicCurve.CurveModel.isOpenImmersion_ιInf
  AlgebraicCurve.CurveModel.algebra_overlap_functionField
  AlgebraicCurve.CurveModel.quasiCompact_gluedToBase AlgebraicCurve.CurveModel.algebraAdjoin
  AlgebraicCurve.CurveModel.isDedekindDomain_chartRing AlgebraicCurve.CurveModel.isIntegralClosure
  AlgebraicCurve.CurveModel.finite_chartRing AlgebraicCurve.CurveModel.centre_isPrime
  AlgebraicCurve.CurveModel.isFractionRing_chartRing AlgebraicCurve.CurveModel.finiteType_chartRing
  AlgebraicCurve.CurveModel.isNoetherianRing_chartRing
  AlgebraicCurve.CurveModel.isScalarTower_base_adjoin
  AlgebraicCurve.CurveModel.isScalarTower_adjoin
  AlgebraicCurve.CurveModel.chartRing_finitePresentation
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
attribute [-simp] 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.CurveModel.ιInf_gluedToBase AlgebraicCurve.CurveModel.ι₀_gluedToBase
  AlgebraicCurve.CurveModel.coe_primeEquivChartPlaces 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
universe u
open CategoryTheory AlgebraicGeometry
namespace P2mCurveConnect
variable {k : Type u} [Field k] [IsAlgClosed k] {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of k))
Lean source · statement
theorem main [IsProper y] [IsIntegral Y]
    (S : Set (Spec (CommRingCat.of k) ⟶ Y))
    (y₀ : Spec (CommRingCat.of k) ⟶ Y) (hy₀ : y₀ ≫ y = 𝟙 _) (h₀ : y₀ ∈ S)
    (hS : CurveStable y S)
    (y₁ : Spec (CommRingCat.of k) ⟶ Y) (hy₁ : y₁ ≫ y = 𝟙 _) : y₁ ∈ S
theorem

mem_of_isProper_of_forall_smoothProperCurve_mem

L229 ↗
AlgebraicGeometry.mem_of_isProper_of_forall_smoothProperCurve_mem
Source context 8 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Curves.Proofs.mem_of_isProper_of_forall_smoothProperCurve_mem
attribute [-instance] 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
  AlgebraicCurve.CurveModel.locallyOfFiniteType_gluedToBase
  AlgebraicCurve.CurveModel.isFractionRing_overlap
  AlgebraicCurve.CurveModel.isLocallyNoetherian_glued AlgebraicCurve.CurveModel.jacobsonSpace_glued
  AlgebraicCurve.CurveModel.isOpenImmersion_ι₀ AlgebraicCurve.CurveModel.isIntegral_glued
  AlgebraicCurve.CurveModel.quasiSeparated_gluedToBase AlgebraicCurve.CurveModel.compactSpace_glued
  AlgebraicCurve.CurveModel.isIntegral_adjoin_chartRing
  AlgebraicCurve.CurveModel.isFractionRing_overlap_functionField
  AlgebraicCurve.CurveModel.isProper_gluedToBase AlgebraicCurve.CurveModel.isOpenImmersion_ιU
  AlgebraicCurve.CurveModel.isOpenImmersion_f₀ AlgebraicCurve.CurveModel.isOpenImmersion_fInf
  AlgebraicCurve.CurveModel.isOpenImmersion_ιInf
  AlgebraicCurve.CurveModel.algebra_overlap_functionField
  AlgebraicCurve.CurveModel.quasiCompact_gluedToBase AlgebraicCurve.CurveModel.algebraAdjoin
  AlgebraicCurve.CurveModel.isDedekindDomain_chartRing AlgebraicCurve.CurveModel.isIntegralClosure
  AlgebraicCurve.CurveModel.finite_chartRing AlgebraicCurve.CurveModel.centre_isPrime
  AlgebraicCurve.CurveModel.isFractionRing_chartRing AlgebraicCurve.CurveModel.finiteType_chartRing
  AlgebraicCurve.CurveModel.isNoetherianRing_chartRing
  AlgebraicCurve.CurveModel.isScalarTower_base_adjoin
  AlgebraicCurve.CurveModel.isScalarTower_adjoin
  AlgebraicCurve.CurveModel.chartRing_finitePresentation
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
attribute [-simp] 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.CurveModel.ιInf_gluedToBase AlgebraicCurve.CurveModel.ι₀_gluedToBase
  AlgebraicCurve.CurveModel.coe_primeEquivChartPlaces 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
universe u
open CategoryTheory AlgebraicGeometry
open P2mCurveConnect in
Lean source · statement
theorem _root_.AlgebraicGeometry.mem_of_isProper_of_forall_smoothProperCurve_mem
    {k : Type u} [Field k] [IsAlgClosed k] {Y : Scheme.{u}} (y : Y ⟶ Spec (CommRingCat.of k))
    [IsProper y] [IsIntegral Y]
    (S : Set (Spec (CommRingCat.of k) ⟶ Y))
    (y₀ : Spec (CommRingCat.of k) ⟶ Y) (hy₀ : y₀ ≫ y = 𝟙 _) (h₀ : y₀ ∈ S)
    (hS : ∀ (C : Scheme.{u}) (c : C ⟶ Spec (CommRingCat.of k)) [IsProper c]
      [SmoothOfRelativeDimension 1 c] [IsIntegral C] (ψ : C ⟶ Y), ψ ≫ y = c →
      ∀ p q : Spec (CommRingCat.of k) ⟶ C, p ≫ c = 𝟙 _ → q ≫ c = 𝟙 _ → p ≫ ψ ∈ S → q ≫ ψ ∈ S)
    (y₁ : Spec (CommRingCat.of k) ⟶ Y) (hy₁ : y₁ ≫ y = 𝟙 _) : y₁ ∈ S

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