NYLL
Module

Place_diffCoeff_smul_D_of_ord_ne_zero.lean

Nyll.AlgebraicCurves.Places.Place_diffCoeff_smul_D_of_ord_ne_zero
9 declarations indexed from sourceView source ↗

Module documentation

Place diff Coeff smul D of ord ne zero

Imports7
Imported by6

Declarations

theorem

isSeparable_adjoin

L60 ↗
AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero.AlgebraicCurve.FF2R3.isSeparable_adjoin
Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero
attribute [-instance]
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
attribute [-simp] AlgebraicCurve.Place.placeOfPrime_toValuationSubring
  AlgebraicCurve.Place.mem_fiberOver AlgebraicCurve.Place.fiberEquiv_symm_apply
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  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
noncomputable section
open KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace FF2R3
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [CharZero K] (x : F)
  [Algebra.IsAlgebraic (IntermediateField.adjoin K ({x} : Set F)) F]
include x in
Lean source · statement
theorem isSeparable_adjoin (v : Place K F) {t : F} (ht : v.ord t ≠ 0) :
    Algebra.IsSeparable (IntermediateField.adjoin K ({t} : Set F)) F
theorem

D_ne_zero_of_ord_ne_zero

L67 ↗
AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero.AlgebraicCurve.FF2R3.D_ne_zero_of_ord_ne_zero
Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero
attribute [-instance]
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
attribute [-simp] AlgebraicCurve.Place.placeOfPrime_toValuationSubring
  AlgebraicCurve.Place.mem_fiberOver AlgebraicCurve.Place.fiberEquiv_symm_apply
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  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
noncomputable section
open KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace FF2R3
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [CharZero K] (x : F)
  [Algebra.IsAlgebraic (IntermediateField.adjoin K ({x} : Set F)) F]
include x in
Lean source · statement
theorem D_ne_zero_of_ord_ne_zero (v : Place K F) {t : F} (ht : v.ord t ≠ 0) : D K F t ≠ 0
theorem

exists_eq_smul_D_of_ord_ne_zero

L71 ↗
AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero.AlgebraicCurve.FF2R3.exists_eq_smul_D_of_ord_ne_zero
Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero
attribute [-instance]
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
attribute [-simp] AlgebraicCurve.Place.placeOfPrime_toValuationSubring
  AlgebraicCurve.Place.mem_fiberOver AlgebraicCurve.Place.fiberEquiv_symm_apply
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  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
noncomputable section
open KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace FF2R3
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [CharZero K] (x : F)
  [Algebra.IsAlgebraic (IntermediateField.adjoin K ({x} : Set F)) F]
include x in
Lean source · statement
theorem exists_eq_smul_D_of_ord_ne_zero (v : Place K F) {t : F} (ht : v.ord t ≠ 0) (ω : Ω[F⁄K]) :
    ∃ g : F, ω = g • D K F t
theorem

diffCoeff_smul_D_of_ord_ne_zero

L80 ↗
AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero.AlgebraicCurve.FF2R3.diffCoeff_smul_D_of_ord_ne_zero
Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero
attribute [-instance]
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
attribute [-simp] AlgebraicCurve.Place.placeOfPrime_toValuationSubring
  AlgebraicCurve.Place.mem_fiberOver AlgebraicCurve.Place.fiberEquiv_symm_apply
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  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
noncomputable section
open KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace FF2R3
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [CharZero K] (x : F)
  [Algebra.IsAlgebraic (IntermediateField.adjoin K ({x} : Set F)) F]
include x in
Lean source · statement
theorem diffCoeff_smul_D_of_ord_ne_zero (v : Place K F) {t : F} (ht : v.ord t ≠ 0) (ω : Ω[F⁄K]) :
    Place.diffCoeff t ω • D K F t = ω
theorem

diffCoeff_smul_of_ord_ne_zero

L84 ↗
AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero.AlgebraicCurve.FF2R3.diffCoeff_smul_of_ord_ne_zero
Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero
attribute [-instance]
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
attribute [-simp] AlgebraicCurve.Place.placeOfPrime_toValuationSubring
  AlgebraicCurve.Place.mem_fiberOver AlgebraicCurve.Place.fiberEquiv_symm_apply
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  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
noncomputable section
open KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace FF2R3
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [CharZero K] (x : F)
  [Algebra.IsAlgebraic (IntermediateField.adjoin K ({x} : Set F)) F]
include x in
Lean source · statement
theorem diffCoeff_smul_of_ord_ne_zero (v : Place K F) {t : F} (ht : v.ord t ≠ 0) (g : F) :
    Place.diffCoeff t (g • D K F t) = g
theorem

ordDiff_smul_D_uniformizer

L89 ↗
AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero.AlgebraicCurve.FF2R3.ordDiff_smul_D_uniformizer
Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero
attribute [-instance]
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
attribute [-simp] AlgebraicCurve.Place.placeOfPrime_toValuationSubring
  AlgebraicCurve.Place.mem_fiberOver AlgebraicCurve.Place.fiberEquiv_symm_apply
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  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
noncomputable section
open KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace FF2R3
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [CharZero K] (x : F)
  [Algebra.IsAlgebraic (IntermediateField.adjoin K ({x} : Set F)) F]
include x in
Lean source · statement
theorem ordDiff_smul_D_uniformizer (v : Place K F) (g : F) :
    v.ordDiff (g • D K F v.uniformizer_alt) = v.ord g
theorem

ordDiff_zero

L94 ↗
AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero.AlgebraicCurve.FF2R3.ordDiff_zero
Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero
attribute [-instance]
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
attribute [-simp] AlgebraicCurve.Place.placeOfPrime_toValuationSubring
  AlgebraicCurve.Place.mem_fiberOver AlgebraicCurve.Place.fiberEquiv_symm_apply
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  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
noncomputable section
open KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace FF2R3
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [CharZero K] (x : F)
  [Algebra.IsAlgebraic (IntermediateField.adjoin K ({x} : Set F)) F]
include x in
Lean source · statement
theorem ordDiff_zero (v : Place K F) : v.ordDiff 0 = 0
theorem

ordDiff_D_uniformizer

L97 ↗
AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero.AlgebraicCurve.FF2R3.ordDiff_D_uniformizer
Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero
attribute [-instance]
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
attribute [-simp] AlgebraicCurve.Place.placeOfPrime_toValuationSubring
  AlgebraicCurve.Place.mem_fiberOver AlgebraicCurve.Place.fiberEquiv_symm_apply
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  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
noncomputable section
open KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace FF2R3
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [CharZero K] (x : F)
  [Algebra.IsAlgebraic (IntermediateField.adjoin K ({x} : Set F)) F]
include x in
Lean source · statement
theorem ordDiff_D_uniformizer (v : Place K F) : v.ordDiff (D K F v.uniformizer_alt) = 0
theorem

diffCoeff_smul_D_of_ord_ne_zero

L104 ↗
AlgebraicCurve.Place.diffCoeff_smul_D_of_ord_ne_zero
Source context 4 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_diffCoeff_smul_D_of_ord_ne_zero
attribute [-instance]
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
attribute [-simp] AlgebraicCurve.Place.placeOfPrime_toValuationSubring
  AlgebraicCurve.Place.mem_fiberOver AlgebraicCurve.Place.fiberEquiv_symm_apply
  AlgebraicCurve.Place.fiberEquiv_apply AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  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
Lean source · statement
theorem _root_.AlgebraicCurve.Place.diffCoeff_smul_D_of_ord_ne_zero {K F : Type*} [Field K] [Field
  F] [Algebra K F] [CharZero K] (x : F)
    [Algebra.IsAlgebraic (IntermediateField.adjoin K ({x} : Set F)) F] (v : AlgebraicCurve.Place K
      F) {t : F} (ht : v.ord t ≠ 0) (ω : Ω[F⁄K]) :
    AlgebraicCurve.Place.diffCoeff t ω • KaehlerDifferential.D K F t = ω

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