Module

PrimeComparison.lean

Nyll.ClassFieldTheory.LocalClassFieldTheory.Finite.LocalReciprocity.FixedFieldIntrinsicReciprocity.PrimeComparison

77 declarations indexed from sourceView source ↗

Module documentation

Intrinsic fixed-field prime comparison

This module compares intrinsic local Artin maps with the ambient fixed-field norm-residue symbol.

Imports3

Imported by3

instance

intrinsicPrimeComparison_separableClosureAlgebra

L70 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_separableClosureAlgebra

Source context 15 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal

Lean source · statement

instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F)

instance

intrinsicPrimeComparison_absoluteFinite

L74 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_absoluteFinite

Source context 16 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra

Lean source · statement

instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field))

instance

intrinsicPrimeComparison_fixedFieldFiniteDimensional

L83 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_fixedFieldFiniteDimensional

Source context 17 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite

Lean source · statement

instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F

instance

intrinsicPrimeComparison_fixedFieldNormed

L88 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_fixedFieldNormed

Source context 18 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite

Lean source · statement

instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F

instance

intrinsicPrimeComparison_fixedFieldValuative

L92 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_fixedFieldValuative

Source context 19 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F

Lean source · statement

instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F

instance

intrinsicPrimeComparison_fixedFieldLocal

L96 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_fixedFieldLocal

Source context 20 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F

Lean source · statement

instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F

instance

intrinsicPrimeComparison_extensionFiniteDimensional

L100 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_extensionFiniteDimensional

Source context 21 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F

Lean source · statement

instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E

instance

intrinsicPrimeComparison_extensionGalois

L105 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_extensionGalois

Source context 22 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite

Lean source · statement

instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E

instance

intrinsicPrimeComparison_extensionNormal

L110 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_extensionNormal

Source context 23 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal

Lean source · statement

instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal

instance

intrinsicPrimeComparison_extensionFinite

L115 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_extensionFinite

Source context 24 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal

Lean source · statement

instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below)

instance

intrinsicPrimeComparison_residueExtensionFinite

L122 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_residueExtensionFinite

Source context 25 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite

Lean source · statement

instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below)

def

intrinsicFixedFieldLocalArtinMonoidHom

L134 ↗

LocalClassFieldTheory.intrinsicFixedFieldLocalArtinMonoidHom

Documentation

The local Artin homomorphism of the actual finite fixed-field extension, using the canonical local structure constructed from the original field.

Source context 26 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e

Lean source · statement

def intrinsicFixedFieldLocalArtinMonoidHom :
    Additive Fˣ →+ Additive (Abelianization Gal(E/F))

def

intrinsicFixedFieldConcreteSymbolValue

L140 ↗

LocalClassFieldTheory.intrinsicFixedFieldConcreteSymbolValue

Documentation

The concrete norm-residue symbol of the intrinsic finite extension, evaluated at a unit of the finite fixed field.

Source context 26 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e

Lean source · statement

def intrinsicFixedFieldConcreteSymbolValue
    (x : Fˣ) : Abelianization Gal(E/F)

def

intrinsicFixedFieldConcretePrimeComparison

L148 ↗

LocalClassFieldTheory.intrinsicFixedFieldConcretePrimeComparison

Source context 26 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e

Lean source · statement

def intrinsicFixedFieldConcretePrimeComparison
    (x : Fˣ) (z : Abelianization Gal(E/F)) : Prop

def

intrinsicFixedFieldAmbientPrimeComparison

L153 ↗

LocalClassFieldTheory.intrinsicFixedFieldAmbientPrimeComparison

Source context 26 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e

Lean source · statement

def intrinsicFixedFieldAmbientPrimeComparison
    (_e : intrinsicFixedFieldSeparableClosureEquiv K H)
    (x : Fˣ) (z : Abelianization Gal(E/F)) : Prop

def

intrinsicFixedFieldSourceFrobeniusAbelianization

L166 ↗

LocalClassFieldTheory.intrinsicFixedFieldSourceFrobeniusAbelianization

Documentation

Map an intrinsic extension-quotient class to the abelianization of the intrinsic finite-extension Galois group.

Source context 26 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e

Lean source · statement

def intrinsicFixedFieldSourceFrobeniusAbelianization
    (q : (EI).extensionQuotient) :
    Abelianization Gal(E/F)

def

intrinsicFixedFieldAmbientFrobeniusAbelianization

L173 ↗

LocalClassFieldTheory.intrinsicFixedFieldAmbientFrobeniusAbelianization

Documentation

Restrict a transported ambient Frobenius element and map the resulting class to the abelianization of the intrinsic finite-extension Galois group.

Source context 26 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e

Lean source · statement

def intrinsicFixedFieldAmbientFrobeniusAbelianization
    (σ :
      (localResidueDatum F).FrobeniusElements
        RF (EI).field (EI).below) :
    Abelianization Gal(E/F)

structure

IntrinsicFixedFieldPrimeComparisonData

L185 ↗

LocalClassFieldTheory.IntrinsicFixedFieldPrimeComparisonData

Source context 26 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e

Lean source · full declaration

structure IntrinsicFixedFieldPrimeComparisonData
    (z : Abelianization Gal(E/F)) where
  xPrime : Fˣ
  concrete :
    intrinsicFixedFieldConcretePrimeComparison
      K H J hJH e xPrime z
  ambient :
    intrinsicFixedFieldAmbientPrimeComparison
      K H J hJH e xPrime z

abbrev

intrinsicFixedFieldFrobeniusSourceClosedField

L197 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusSourceClosedField

Documentation

The intrinsic closed subgroup fixed by the Frobenius element associated to the finite fixed-field extension.

Source context 26 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e

Lean source · statement

abbrev intrinsicFixedFieldFrobeniusSourceClosedField
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e)

theorem

intrinsicFixedFieldFrobeniusSourceBelow

L204 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusSourceBelow

Documentation

The intrinsic Frobenius-fixed closed subgroup lies below the intrinsic absolute-base subgroup.

Source context 26 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e

Lean source · statement

theorem intrinsicFixedFieldFrobeniusSourceBelow
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ).toSubgroup ≤
      (intrinsicAbstractBase F).toSubgroup

abbrev

intrinsicFixedFieldFrobeniusSourceField

L214 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusSourceField

Documentation

The intrinsic abstract fixed field cut out by the intrinsic Frobenius-fixed closed subgroup.

Source context 26 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e

Lean source · statement

abbrev intrinsicFixedFieldFrobeniusSourceField
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e)

abbrev

intrinsicFixedFieldFrobeniusAmbientClosedField

L222 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusAmbientClosedField

Documentation

The ambient Frobenius-fixed closed subgroup obtained after transporting the intrinsic Frobenius element.

Source context 26 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e

Lean source · statement

abbrev intrinsicFixedFieldFrobeniusAmbientClosedField
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e)

theorem

intrinsicFixedFieldFrobeniusAmbientBelow

L231 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusAmbientBelow

Documentation

The transported ambient Frobenius-fixed subgroup lies below the ambient finite fixed subgroup.

Source context 26 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e

Lean source · statement

theorem intrinsicFixedFieldFrobeniusAmbientBelow
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ).toSubgroup ≤ H.field.toSubgroup

abbrev

intrinsicFixedFieldFrobeniusAmbientField

L242 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusAmbientField

Documentation

The ambient relative fixed field cut out by the transported Frobenius-fixed subgroup.

Source context 26 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e

Lean source · statement

abbrev intrinsicFixedFieldFrobeniusAmbientField
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e)

instance

intrinsicPrimeComparison_frobeniusSourceAlgebra

L248 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusSourceAlgebra

Source context 26 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e

Lean source · statement

instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)

instance

intrinsicPrimeComparison_frobeniusSourceFiniteDimensional

L256 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusSourceFiniteDimensional

Source context 27 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra

Lean source · statement

instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ))

instance

intrinsicPrimeComparison_frobeniusSourceNormed

L287 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusSourceNormed

Source context 28 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)

Lean source · statement

instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)

instance

intrinsicPrimeComparison_frobeniusSourceValuative

L296 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusSourceValuative

Source context 29 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)

Lean source · statement

instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)

instance

intrinsicPrimeComparison_frobeniusSourceLocal

L305 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusSourceLocal

Source context 30 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)

Lean source · statement

instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)

instance

intrinsicPrimeComparison_frobeniusSourceValuationExtension

L314 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusSourceValuationExtension

Source context 31 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)

Lean source · statement

instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ))

instance

intrinsicPrimeComparison_frobeniusAmbientAlgebraK

L325 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusAmbientAlgebraK

Source context 32 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)

Lean source · statement

instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)

instance

intrinsicPrimeComparison_frobeniusAmbientAlgebraF

L334 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusAmbientAlgebraF

Source context 33 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra

Lean source · statement

instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)

instance

intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK

L342 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK

Source context 34 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra

Lean source · statement

instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ))

instance

intrinsicPrimeComparison_frobeniusAmbientNormed

L373 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusAmbientNormed

Source context 35 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))

Lean source · statement

instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)

instance

intrinsicPrimeComparison_frobeniusAmbientValuative

L382 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusAmbientValuative

Source context 36 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)

Lean source · statement

instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)

instance

intrinsicPrimeComparison_frobeniusAmbientLocal

L391 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusAmbientLocal

Source context 37 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)

Lean source · statement

instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)

instance

intrinsicPrimeComparison_frobeniusAmbientValuationExtension

L400 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusAmbientValuationExtension

Source context 38 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)

Lean source · statement

instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ))

instance

intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF

L411 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF

Source context 39 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)

Lean source · statement

instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ))

def

intrinsicFixedFieldFrobeniusAmbientEmbedding

L443 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusAmbientEmbedding

Documentation

The canonical algebra embedding of the ambient Frobenius fixed field into the ambient separable closure.

Source context 40 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))

Lean source · statement

def intrinsicFixedFieldFrobeniusAmbientEmbedding
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ →ₐ[K]
      SeparableClosure K

instance

intrinsicPrimeComparison_frobeniusAmbientSeparable

L452 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusAmbientSeparable

Source context 40 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))

Lean source · statement

instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)

def

intrinsicFixedFieldFrobeniusAlgEquiv

L466 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusAlgEquiv

Documentation

The canonical algebra equivalence from the intrinsic Frobenius fixed field to the corresponding ambient Frobenius fixed field.

Source context 41 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance

Lean source · statement

def intrinsicFixedFieldFrobeniusAlgEquiv
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e)

def

intrinsicFixedFieldFrobeniusRingEquiv

L473 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusRingEquiv

Documentation

The underlying ring equivalence of the canonical equivalence between the intrinsic and ambient Frobenius fixed fields.

Source context 41 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance

Lean source · statement

def intrinsicFixedFieldFrobeniusRingEquiv
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ ≃+*
      intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ

theorem

intrinsicFixedFieldFrobenius_valuationSubring_mem

L484 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobenius_valuationSubring_mem

Documentation

The canonical intrinsic-to-ambient Frobenius fixed-field equivalence identifies their valuation subrings.

Source context 41 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance

Lean source · statement

theorem intrinsicFixedFieldFrobenius_valuationSubring_mem
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e)
    (x :
      intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :
    x ∈
        (ValuativeRel.valuation
          (intrinsicFixedFieldFrobeniusSourceField
            K H J hJH e σ)).valuationSubring ↔
      intrinsicFixedFieldFrobeniusRingEquiv
          K H J hJH e σ x ∈
        (ValuativeRel.valuation
          (intrinsicFixedFieldFrobeniusAmbientField
            K H J hJH e σ)).valuationSubring

def

intrinsicFixedFieldFrobeniusSourcePrimeUnit

L530 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusSourcePrimeUnit

Source context 41 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance

Lean source · statement

def intrinsicFixedFieldFrobeniusSourcePrimeUnit
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)ˣ

theorem

intrinsicFixedFieldFrobeniusSourcePrimeUnit_valuation

L545 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusSourcePrimeUnit_valuation

Source context 41 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance

Lean source · statement

theorem intrinsicFixedFieldFrobeniusSourcePrimeUnit_valuation
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField.valuationMap
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        (Additive.ofMul
          (intrinsicFixedFieldFrobeniusSourcePrimeUnit
            K H J hJH e σ)) =
      1

def

intrinsicFixedFieldFrobeniusAmbientPrimeUnit

L565 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusAmbientPrimeUnit

Source context 41 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance

Lean source · statement

def intrinsicFixedFieldFrobeniusAmbientPrimeUnit
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)ˣ

theorem

intrinsicFixedFieldFrobeniusAmbientPrimeUnit_valuation

L575 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusAmbientPrimeUnit_valuation

Source context 41 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance

Lean source · statement

theorem intrinsicFixedFieldFrobeniusAmbientPrimeUnit_valuation
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField.valuationMap
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        (Additive.ofMul
          (intrinsicFixedFieldFrobeniusAmbientPrimeUnit
            K H J hJH e σ)) =
      1

def

intrinsicFixedFieldFrobeniusPrimeNorm

L595 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusPrimeNorm

Source context 41 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance

Lean source · statement

def intrinsicFixedFieldFrobeniusPrimeNorm
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) : Fˣ

instance

intrinsicPrimeComparison_frobeniusSourceQuotientFinite

L603 ↗

LocalClassFieldTheory.intrinsicPrimeComparison_frobeniusSourceQuotientFinite

Source context 41 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance

Lean source · statement

instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ))

def

intrinsicFixedFieldFrobeniusSourceAbstractField

L627 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusSourceAbstractField

Documentation

Package the intrinsic Frobenius-fixed subgroup as a finite abstract field inside the intrinsic absolute Galois group.

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

def intrinsicFixedFieldFrobeniusSourceAbstractField
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    FiniteAbstractField Gal(SeparableClosure F/F)

def

intrinsicFixedFieldFrobeniusSourcePrimeElement

L649 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusSourcePrimeElement

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

def intrinsicFixedFieldFrobeniusSourcePrimeElement
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ambientFixedAddSubgroup
      (intrinsicAbsoluteUnits F)
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)

theorem

intrinsicFixedFieldFrobeniusSourcePrimeElement_isPrime

L663 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusSourcePrimeElement_isPrime

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

theorem intrinsicFixedFieldFrobeniusSourcePrimeElement_isPrime
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    (localHenselianValuation F).IsPrimeElement
      (intrinsicFixedFieldFrobeniusSourceAbstractField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusSourcePrimeElement
        K H J hJH e σ)

theorem

intrinsicFixedFieldFrobeniusPrimeNorm_relativeNorm

L679 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusPrimeNorm_relativeNorm

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

theorem intrinsicFixedFieldFrobeniusPrimeNorm_relativeNorm
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    relativeNorm
        (intrinsicAbsoluteUnits F)
        (intrinsicAbstractBase F)
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourcePrimeElement
          K H J hJH e σ) =
      baseUnitsEquivGaloisAmbientFixed F (SeparableClosure F)
        (Additive.ofMul
          (intrinsicFixedFieldFrobeniusPrimeNorm
            K H J hJH e σ))

def

intrinsicFixedFieldFrobeniusAmbientAbstractField

L705 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusAmbientAbstractField

Documentation

Package the transported ambient Frobenius-fixed subgroup as a finite abstract field inside the ambient absolute Galois group.

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

def intrinsicFixedFieldFrobeniusAmbientAbstractField
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    FiniteAbstractField Gal(SeparableClosure K/K)

def

intrinsicFixedFieldFrobeniusAmbientPrimeElement

L715 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusAmbientPrimeElement

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

def intrinsicFixedFieldFrobeniusAmbientPrimeElement
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ambientFixedAddSubgroup
      (galoisAmbientUnitsRep K (SeparableClosure K))
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)

theorem

intrinsicFixedFieldFrobeniusAmbientPrimeElement_isPrime

L731 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusAmbientPrimeElement_isPrime

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

theorem intrinsicFixedFieldFrobeniusAmbientPrimeElement_isPrime
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    (localHenselianValuation K).IsPrimeElement
      (intrinsicFixedFieldFrobeniusAmbientAbstractField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientPrimeElement
        K H J hJH e σ)

theorem

intrinsicFixedFieldFrobeniusPrimeNorm_concrete

L747 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusPrimeNorm_concrete

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

theorem intrinsicFixedFieldFrobeniusPrimeNorm_concrete
    (q : (EI).extensionQuotient)
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e)
    (hσ :
      (localResidueDatum F).frobeniusRestriction
        RF (EI).field (EI).below σ = q) :
    intrinsicFixedFieldConcretePrimeComparison
      K H J hJH e
      (intrinsicFixedFieldFrobeniusPrimeNorm
        K H J hJH e σ)
      (intrinsicFixedFieldSourceFrobeniusAbelianization
        K H J hJH e q)

theorem

intrinsicFixedFieldFrobeniusPrimeNorm_ambient

L782 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusPrimeNorm_ambient

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

theorem intrinsicFixedFieldFrobeniusPrimeNorm_ambient
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    intrinsicFixedFieldAmbientPrimeComparison
      K H J hJH e
      (intrinsicFixedFieldFrobeniusPrimeNorm
        K H J hJH e σ)
      (intrinsicFixedFieldAmbientFrobeniusAbelianization
        K H J hJH e σ)

def

intrinsicFixedFieldPrimeRepresentative

L838 ↗

LocalClassFieldTheory.intrinsicFixedFieldPrimeRepresentative

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

def intrinsicFixedFieldPrimeRepresentative
    (z : Abelianization Gal(E/F)) :
    (EI).extensionQuotient

theorem

intrinsicFixedFieldPrimeRepresentative_abelianization

L847 ↗

LocalClassFieldTheory.intrinsicFixedFieldPrimeRepresentative_abelianization

Source context 43 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ
omit [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K] in

Lean source · statement

theorem intrinsicFixedFieldPrimeRepresentative_abelianization
    (z : Abelianization Gal(E/F)) :
    (qF).abelianizationCongr
        (Abelianization.of
          (intrinsicFixedFieldPrimeRepresentative
            K H J hJH e z)) =
      z

def

intrinsicFixedFieldPrimeFrobeniusLift

L867 ↗

LocalClassFieldTheory.intrinsicFixedFieldPrimeFrobeniusLift

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

def intrinsicFixedFieldPrimeFrobeniusLift
    (z : Abelianization Gal(E/F)) :
    intrinsicFixedFieldFrobeniusElements K H J hJH e

theorem

intrinsicFixedFieldPrimeFrobeniusLift_restriction

L876 ↗

LocalClassFieldTheory.intrinsicFixedFieldPrimeFrobeniusLift_restriction

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

theorem intrinsicFixedFieldPrimeFrobeniusLift_restriction
    (z : Abelianization Gal(E/F)) :
    (localResidueDatum F).frobeniusRestriction
        RF (EI).field (EI).below
        (intrinsicFixedFieldPrimeFrobeniusLift
          K H J hJH e z) =
      intrinsicFixedFieldPrimeRepresentative
        K H J hJH e z

def

intrinsicFixedFieldPrimeComparisonWitness

L890 ↗

LocalClassFieldTheory.intrinsicFixedFieldPrimeComparisonWitness

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

def intrinsicFixedFieldPrimeComparisonWitness
    (z : Abelianization Gal(E/F)) : Fˣ

theorem

intrinsicFixedFieldPrimeComparisonWitness_concrete

L897 ↗

LocalClassFieldTheory.intrinsicFixedFieldPrimeComparisonWitness_concrete

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

theorem intrinsicFixedFieldPrimeComparisonWitness_concrete
    (z : Abelianization Gal(E/F)) :
    intrinsicFixedFieldConcretePrimeComparison
      K H J hJH e
      (intrinsicFixedFieldPrimeComparisonWitness
        K H J hJH e z) z

def

intrinsicFixedFieldAmbientQuotientResult

L923 ↗

LocalClassFieldTheory.intrinsicFixedFieldAmbientQuotientResult

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

def intrinsicFixedFieldAmbientQuotientResult
    (q :
      H.field.toSubgroup ⧸
        extensionSubgroup H.field J hJH) :
    Gal(E/F)

def

intrinsicFixedFieldSourceQuotientResult

L930 ↗

LocalClassFieldTheory.intrinsicFixedFieldSourceQuotientResult

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

def intrinsicFixedFieldSourceQuotientResult
    (q : (EI).extensionQuotient) :
    Gal(E/F)

def

intrinsicFixedFieldFrobeniusAmbientRestrictionResult

L935 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusAmbientRestrictionResult

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

def
    intrinsicFixedFieldFrobeniusAmbientRestrictionResult
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Gal(E/F)

def

intrinsicFixedFieldFrobeniusSourceRestrictionResult

L945 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusSourceRestrictionResult

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

def
    intrinsicFixedFieldFrobeniusSourceRestrictionResult
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Gal(E/F)

theorem

intrinsicFixedFieldQuotientResult_mk_compatibility

L955 ↗

LocalClassFieldTheory.intrinsicFixedFieldQuotientResult_mk_compatibility

Source context 43 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ
omit [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K] in

Lean source · statement

theorem
    intrinsicFixedFieldQuotientResult_mk_compatibility
    (τ : (intrinsicAbstractBase F).toSubgroup) :
    intrinsicFixedFieldAmbientQuotientResult K H J hJH
        (QuotientGroup.mk
          (intrinsicBaseEquivAmbientFixedField K H e τ)) =
      intrinsicFixedFieldSourceQuotientResult K H J hJH e
        (QuotientGroup.mk τ)

theorem

intrinsicFixedFieldFrobeniusRestriction_mk_compatibility

L968 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusRestriction_mk_compatibility

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

theorem
    intrinsicFixedFieldFrobeniusRestriction_mk_compatibility
    (τ : (intrinsicAbstractBase F).toSubgroup) :
    intrinsicFixedFieldAmbientQuotientResult K H J hJH
        ((localResidueDatum K).extensionRestriction
          H.field J hJH
          (intrinsicFrobeniusQuotientEquivAmbientFixedField
            K H J hJH e (QuotientGroup.mk τ))) =
      intrinsicFixedFieldSourceQuotientResult K H J hJH e
        ((localResidueDatum F).extensionRestriction
          (intrinsicAbstractBase F) (EI).field (EI).below
          (QuotientGroup.mk τ))

theorem

intrinsicFixedFieldFrobeniusAmbientRestrictionResult_eq

L1033 ↗

LocalClassFieldTheory.intrinsicFixedFieldFrobeniusAmbientRestrictionResult_eq

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

theorem
    intrinsicFixedFieldFrobeniusAmbientRestrictionResult_eq
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    intrinsicFixedFieldFrobeniusAmbientRestrictionResult
        K H J hJH e σ =
      intrinsicFixedFieldFrobeniusSourceRestrictionResult
        K H J hJH e σ

theorem

intrinsicFixedFieldPrimeFrobeniusSourceRestrictionResult_eq

L1064 ↗

LocalClassFieldTheory.intrinsicFixedFieldPrimeFrobeniusSourceRestrictionResult_eq

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

theorem
    intrinsicFixedFieldPrimeFrobeniusSourceRestrictionResult_eq
    (z : Abelianization Gal(E/F)) :
    intrinsicFixedFieldFrobeniusSourceRestrictionResult
        K H J hJH e
        (intrinsicFixedFieldPrimeFrobeniusLift
          K H J hJH e z) =
      qF
        (intrinsicFixedFieldPrimeRepresentative
          K H J hJH e z)

theorem

intrinsicFixedFieldPrimeFrobeniusLift_ambientRestriction

L1081 ↗

LocalClassFieldTheory.intrinsicFixedFieldPrimeFrobeniusLift_ambientRestriction

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

theorem intrinsicFixedFieldPrimeFrobeniusLift_ambientRestriction
    (z : Abelianization Gal(E/F)) :
    intrinsicFixedFieldFrobeniusAmbientRestrictionResult
        K H J hJH e
        (intrinsicFixedFieldPrimeFrobeniusLift
          K H J hJH e z) =
      qF
        (intrinsicFixedFieldPrimeRepresentative
          K H J hJH e z)

theorem

intrinsicFixedFieldPrimeComparisonWitness_ambient

L1110 ↗

LocalClassFieldTheory.intrinsicFixedFieldPrimeComparisonWitness_ambient

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

theorem intrinsicFixedFieldPrimeComparisonWitness_ambient
    (z : Abelianization Gal(E/F)) :
    intrinsicFixedFieldAmbientPrimeComparison
      K H J hJH e
      (intrinsicFixedFieldPrimeComparisonWitness
        K H J hJH e z) z

def

intrinsicFixedFieldPrimeComparison

L1156 ↗

LocalClassFieldTheory.intrinsicFixedFieldPrimeComparison

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

def intrinsicFixedFieldPrimeComparison
    (z : Abelianization Gal(E/F)) :
    IntrinsicFixedFieldPrimeComparisonData
      K H J hJH e z

theorem

exists_intrinsicFixedFieldPrimeComparison

L1173 ↗

LocalClassFieldTheory.exists_intrinsicFixedFieldPrimeComparison

Documentation

Every abelianized Galois element of the intrinsic finite fixed-field extension is represented by a unit with both its concrete norm-residue value and its actual ambient fixed-field norm-residue value.

Source context 42 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ

Lean source · statement

theorem exists_intrinsicFixedFieldPrimeComparison
    (z : Abelianization Gal(E/F)) :
    ∃ x : Fˣ,
      concreteNormResidueSymbolOfEmbedding
          F E iFE
          (localResidueDatum F)
          (localHenselianValuation F)
          (separableClosureUnits_isClassFormation F) x = z ∧
        abstractFixedFieldNormResidueSymbol
          K (SeparableClosure K)
          (localResidueDatum K)
          (localHenselianValuation K)
          (separableClosureUnits_isClassFormation K)
          H.field J hJH (Additive.ofMul x) =
          Additive.ofMul z

theorem

exists_intrinsicFixedFieldNormClassRepresentative

L1198 ↗

LocalClassFieldTheory.exists_intrinsicFixedFieldNormClassRepresentative

Documentation

Every unit of the intrinsic finite fixed field has a norm-class-equivalent representative whose ambient fixed-field norm-residue value is the actual local Artin value of the original unit.

Source context 43 commands

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

noncomputable section
namespace LocalClassFieldTheory
open LocalFieldTheory RamificationTheory CyclicCohomology KummerTheory
open ClassFormation
open scoped ValuativeRel
section IntrinsicFixedFieldPrimeComparison
variable
    (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K]
    [IsNonarchimedeanLocalField K]
    (H : FiniteAbstractField
      Gal(SeparableClosure K/K))
    (J : ClosedSubgroup Gal(SeparableClosure K/K))
    (hJH : J.toSubgroup ≤ H.field.toSubgroup)
    [hJnormal : (extensionSubgroup H.field J hJH).Normal]
    [hJfinite : Finite
      (H.field.toSubgroup ⧸ extensionSubgroup H.field J hJH)]
    (e : intrinsicFixedFieldSeparableClosureEquiv K H)
local notation "F" =>
  abstractFixedField K (SeparableClosure K) H.field
local notation "E" =>
  abstractRelativeFixedField K (SeparableClosure K) hJH
local notation "iFE" =>
  e.symm.toAlgHom.comp
    (IntermediateField.val
      (abstractRelativeFixedField
        K (SeparableClosure K) hJH))
local notation "EI" =>
  finiteGaloisAbstractExtensionOfEmbedding F E iFE
local notation "RF" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    (intrinsicFiniteAbstractBase F)
    (localResidueDatum F)
local notation "RH" =>
  FiniteAbstractField.toFiniteResidueAbstractField
    H (localResidueDatum K)
local notation "qF" =>
  finiteGaloisAbstractQuotientEquivGaloisGroupOfEmbedding F E iFE
local notation "qH" =>
  abstractExtensionQuotientEquivGaloisGroup
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_separableClosureAlgebra :
    Algebra F (SeparableClosure F) :=
  (separableClosure F (AlgebraicClosure F)).algebra
local instance intrinsicPrimeComparison_absoluteFinite :
    Finite
      ((baseField
        Gal(SeparableClosure K/K)).toSubgroup ⧸
        extensionSubgroup
          (baseField Gal(SeparableClosure K/K))
          H.field (le_baseField H.field)) :=
  H.finite
local instance intrinsicPrimeComparison_fixedFieldFiniteDimensional :
    FiniteDimensional K F :=
  abstractFixedField_finiteDimensional
    K (SeparableClosure K) H.field H.finite
local instance intrinsicPrimeComparison_fixedFieldNormed :
    NontriviallyNormedField F :=
  finiteExtensionSpectralNormedField K F
local instance intrinsicPrimeComparison_fixedFieldValuative :
    ValuativeRel F :=
  finiteExtensionSpectralValuativeRel K F
local instance intrinsicPrimeComparison_fixedFieldLocal :
    IsNonarchimedeanLocalField F :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K F
local instance intrinsicPrimeComparison_extensionFiniteDimensional :
    FiniteDimensional F E :=
  abstractRelativeFixedField_finiteDimensional
    K (SeparableClosure K) H.field J hJH H.finite hJfinite
local instance intrinsicPrimeComparison_extensionGalois :
    IsGalois F E :=
  abstractRelativeFixedField_isGalois
    K (SeparableClosure K) H.field J hJH hJnormal
local instance intrinsicPrimeComparison_extensionNormal :
    (extensionSubgroup
      (intrinsicAbstractBase F) (EI).field (EI).below).Normal :=
  (EI).normal
local instance intrinsicPrimeComparison_extensionFinite :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F) (EI).field (EI).below) :=
  (EI).finite
local instance intrinsicPrimeComparison_residueExtensionFinite :
    Finite
      ((RF).field.toSubgroup ⧸
        extensionSubgroup (RF).field (EI).field (EI).below) := by
  change Finite
    ((intrinsicAbstractBase F).toSubgroup ⧸
      extensionSubgroup
        (intrinsicAbstractBase F) (EI).field (EI).below)
  exact intrinsicPrimeComparison_extensionFinite K H J hJH e
local instance intrinsicPrimeComparison_frobeniusSourceAlgebra
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusSourceField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusSourceFiniteDimensional
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusSourceAlgebra
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusSourceAlgebra
      K H J hJH e σ
  let _hRFFinite :
      Finite
        ((RF).field.toSubgroup ⧸
          extensionSubgroup (RF).field (EI).field (EI).below) :=
    intrinsicPrimeComparison_residueExtensionFinite
      K H J hJH e
  exact
    @abstractFixedField_finiteDimensional
      F (SeparableClosure F) _ _ inferInstance inferInstance
      (intrinsicFixedFieldFrobeniusSourceClosedField
        K H J hJH e σ)
      ((localResidueDatum F).frobeniusFixedField_absoluteFinite
        (intrinsicFiniteAbstractBase F) (EI).field (EI).below σ)
local instance intrinsicPrimeComparison_frobeniusSourceNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusSourceField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusSourceValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation F)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusSourceField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension F
    (intrinsicFixedFieldFrobeniusSourceField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (abstractFixedField K (SeparableClosure K)
    (intrinsicFixedFieldFrobeniusAmbientClosedField
      K H J hJH e σ)).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientAlgebraF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  (intrinsicFixedFieldFrobeniusAmbientField
    K H J hJH e σ).algebra
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalK
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule K
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraK
          K H J hJH e σ)) := by
  let : Algebra K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraK
      K H J hJH e σ
  change FiniteDimensional K
    (abstractFixedField K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ))
  exact
    abstractFixedField_finiteDimensional
      K (SeparableClosure K)
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      ((localResidueDatum K).frobeniusFixedField_absoluteFinite
        H J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientNormed
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    NontriviallyNormedField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralNormedField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuative
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    ValuativeRel
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralValuativeRel K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientLocal
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    IsNonarchimedeanLocalField
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  finiteExtensionSpectralIsNonarchimedeanLocalField K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientValuationExtension
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Valuation.HasExtension
      (ValuativeRel.valuation K)
      (ValuativeRel.valuation
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)) :=
  finiteExtensionSpectralValuation_hasExtension K
    (intrinsicFixedFieldFrobeniusAmbientField
      K H J hJH e σ)
local instance intrinsicPrimeComparison_frobeniusAmbientFiniteDimensionalF
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    @FiniteDimensional F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ)
      _ _
      (@Algebra.toModule F
        (intrinsicFixedFieldFrobeniusAmbientField
          K H J hJH e σ)
        _ _
        (intrinsicPrimeComparison_frobeniusAmbientAlgebraF
          K H J hJH e σ)) := by
  let : Algebra F
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
    intrinsicPrimeComparison_frobeniusAmbientAlgebraF
      K H J hJH e σ
  exact
    abstractRelativeFixedField_finiteDimensional
      K (SeparableClosure K) H.field
      (intrinsicFixedFieldFrobeniusAmbientClosedField
        K H J hJH e σ)
      (intrinsicFixedFieldFrobeniusAmbientBelow
        K H J hJH e σ)
      H.finite
      ((localResidueDatum K).frobeniusFixedField_finite
        RH J hJH
        (intrinsicFrobeniusElementToAmbientFixedField
          K H J hJH e σ))
local instance intrinsicPrimeComparison_frobeniusAmbientSeparable
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Algebra.IsSeparable K
      (intrinsicFixedFieldFrobeniusAmbientField
        K H J hJH e σ) :=
  by
    change Algebra.IsSeparable K
      (abstractFixedField K (SeparableClosure K)
        (intrinsicFixedFieldFrobeniusAmbientClosedField
          K H J hJH e σ))
    infer_instance
local instance intrinsicPrimeComparison_frobeniusSourceQuotientFinite
    (σ : intrinsicFixedFieldFrobeniusElements K H J hJH e) :
    Finite
      ((intrinsicAbstractBase F).toSubgroup ⧸
        extensionSubgroup
          (intrinsicAbstractBase F)
          (intrinsicFixedFieldFrobeniusSourceClosedField
            K H J hJH e σ)
          (intrinsicFixedFieldFrobeniusSourceBelow
            K H J hJH e σ)) := by
  change Finite
    ((RF).field.toSubgroup ⧸
      extensionSubgroup
        (RF).field
        (intrinsicFixedFieldFrobeniusSourceClosedField
          K H J hJH e σ)
        (intrinsicFixedFieldFrobeniusSourceBelow
          K H J hJH e σ))
  exact
    (localResidueDatum F).frobeniusFixedField_finite
      RF (EI).field (EI).below σ
include e in

Lean source · statement

theorem exists_intrinsicFixedFieldNormClassRepresentative
    (a : Fˣ) :
    ∃ x : Fˣ,
      normClass F E a = normClass F E x ∧
        abstractFixedFieldNormResidueSymbol
          K (SeparableClosure K)
          (localResidueDatum K)
          (localHenselianValuation K)
          (separableClosureUnits_isClassFormation K)
          H.field J hJH (Additive.ofMul x) =
          intrinsicFixedFieldLocalArtinMonoidHom
            K H J hJH (Additive.ofMul a)

Module documentation

Intrinsic fixed-field prime comparison

Declarations