AlgebraicGeometry.SmoothProperCurve.exists_curveModel_iso_pullback_of_isAlgClosed
Source context 8 commands
Surrounding namespaces, section variables, and local commands at this declaration.
section ProofImplementation
namespace AlgebraicCurves.Geometry.Curves.Proofs
namespace SmoothProperCurve_exists_curveModel_iso_pullback_of_isAlgClosed
attribute [-instance]
AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
AlgebraicCurve.Place.instIsScalarTowerResidueFieldRestrictPushforward
AlgebraicCurve.Place.instAlgebraResidueFieldRestrictPushforward
AlgebraicCurve.Place.instIsLocalHomRestrictInclusion
AlgebraicCurve.Place.instIsPrimeCenter
AlgebraicCurve.Place.instIsFractionRingIntegralClosureAt
instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
AlgebraicCurve.Place.instIsDedekindDomainIntegralClosureAt
AlgebraicCurve.Place.instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
attribute [-simp]
AlgebraicCurve.Divisor.mapRestrict_single
AlgebraicCurve.Divisor.pushforward_single
AlgebraicCurve.Place.coe_restrictInclusion
AlgebraicCurve.Place.mem_fiber
AlgebraicCurve.Place.restrict_toValuationSubring
AlgebraicCurve.Divisor.degree_pushforward
AlgebraicCurve.Place.restrictResidueMap_residue
AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom
AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
AlgebraicCurve.Place.placeOfPrime_toValuationSubring
AlgebraicCurve.Place.mem_fiberOver
AlgebraicCurve.Place.fiberEquiv_symm_apply
AlgebraicCurve.Place.fiberEquiv_apply
AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
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
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.mulAdele_apply
AlgebraicCurve.residuePairing_apply_coe
AlgebraicCurve.mem_adeleBdd
AlgebraicCurve.weilSmul_one
AlgebraicCurve.diagonalHom_apply
AlgebraicCurve.weilSmul_apply
AlgebraicCurve.adeleSpaceMul_coe
AlgebraicCurve.mulAdele_one
AlgebraicCurve.Place.differentialCoeff_zero
AlgebraicCurve.Place.differentialCoeff_dCoord
AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec
AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
AlgebraicCurve.TranscendenceTower.mk.injEq
AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec
AlgebraicCurve.PoleDivisorPackage.mk.injEq
universe u v
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry AlgebraicGeometry.SmoothProperCurve
NeronModelInfra AlgebraicCurveLean source · statement
theorem _root_.AlgebraicGeometry.SmoothProperCurve.exists_curveModel_iso_pullback_of_isAlgClosed
(R : Type u) [CommRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of R)) [IsProper c]
[SmoothOfRelativeDimension 1 c] [GeometricallyIntegral c]
(k : Type u) [Field k] [IsAlgClosed k] (s : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R)) :
∃ (L : Type u) (_ : Field L) (_ : Algebra k L) (_ : IsCurveOver k L) (_ :
Algebra.EssFiniteType k L)
(M : CurveModel k L) (e : M.C ≅ pullback c s), e.hom ≫ pullback.snd c s = M.toBase