15.106 Weakly étale ring maps to 15.122 Determinants of…

Section15.106 Weakly étale ring maps#

Section 15.106 of the Stacks Project, Tag 092A. 24 tags here are carried by 26 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 092B, a definition

Tag 092B in the Stacks Project.

IsAbsolutelyFlat, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/AbsolutelyFlat/Basic.lean, line 85.

Stacks 092B. A commutative ring is absolutely flat if every module over it is flat.

abbrev IsAbsolutelyFlat (A : Type u) [CommRing A] : Prop

Tag 092C, a lemma

Tag 092C in the Stacks Project.

flat_of_flat_of_mul_flat, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/Flat.lean, line 431.

Stacks Tag 092C. Let A → B be a ring map such that the multiplication map μ : B ⊗_A B → B makes B a flat (B ⊗_A B)-module. Let N be a B-module. If N is flat as an A-module, then N is flat as a B-module.

theorem flat_of_flat_of_mul_flat
    (A B : Type*) [CommRing A] [CommRing B] [Algebra A B]
    (hB : Algebra.MulFlat A B)
    (N : Type*) [AddCommGroup N] [Module B N]
    [Module A N] [IsScalarTower A B N]
    (hN : Module.Flat A N) :
    Module.Flat B N

Tag 092D, a definition

Tag 092D in the Stacks Project.

HasWeakDimensionLE, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/Defs.lean, line 53.

A commutative ring has weak dimension at most d if every module over it has Tor dimension at most d.

def HasWeakDimensionLE {A : Type u} [CommRing A]
    (derivedTensor :
      DerivedCategory (ModuleCat.{u} A) ⥤
        DerivedCategory (ModuleCat.{u} A) ⥤ DerivedCategory (ModuleCat.{u} A))
    (comparison : ∀ N : ModuleCat.{u} A,
      DerivedCategory.Q ⋙
          derivedTensor.flip.obj
            ((DerivedCategory.singleFunctor (ModuleCat.{u} A) 0).obj N) ⟶
        ((tensoringRight (ModuleCat.{u} A)).obj N).mapHomologicalComplex
            (ComplexShape.up ℤ) ⋙ DerivedCategory.Q)
    (isLeftDerived : ∀ N : ModuleCat.{u} A,
      (derivedTensor.flip.obj
          ((DerivedCategory.singleFunctor (ModuleCat.{u} A) 0).obj N)).IsLeftDerivedFunctor
        (comparison N)
        (HomologicalComplex.quasiIso (ModuleCat.{u} A) (ComplexShape.up ℤ)))
    (d : ℕ) : Prop

Tag 092E, a lemma

Tag 092E in the Stacks Project.

hasWeakDimensionLE_of_weaklyEtale, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeakDimension/WeaklyEtale.lean, line 456.

Stacks Tag 092E. A weakly étale algebra over a commutative ring of weak dimension at most d also has weak dimension at most d.

theorem hasWeakDimensionLE_of_weaklyEtale
    (A : Type u) (B : Type v) [CommRing A] [CommRing B] [Algebra A B]
    (derivedTensorA :
      DerivedCategory (ModuleCat.{u} A) ⥤
        DerivedCategory (ModuleCat.{u} A) ⥤ DerivedCategory (ModuleCat.{u} A))
    (comparisonA : ∀ N : ModuleCat.{u} A,
      DerivedCategory.Q ⋙
          derivedTensorA.flip.obj
            ((DerivedCategory.singleFunctor (ModuleCat.{u} A) 0).obj N) ⟶
        ((tensoringRight (ModuleCat.{u} A)).obj N).mapHomologicalComplex
            (ComplexShape.up ℤ) ⋙ DerivedCategory.Q)
    (isLeftDerivedA : ∀ N : ModuleCat.{u} A,
      (derivedTensorA.flip.obj
          ((DerivedCategory.singleFunctor (ModuleCat.{u} A) 0).obj N)).IsLeftDerivedFunctor
        (comparisonA N)
        (HomologicalComplex.quasiIso (ModuleCat.{u} A) (ComplexShape.up ℤ)))
    (d : ℕ) (hAB : Algebra.WeaklyEtale A B)
    (hA : HasWeakDimensionLE derivedTensorA comparisonA isLeftDerivedA d) :
    ∃ (derivedTensorB :
        DerivedCategory (ModuleCat.{v} B) ⥤
          DerivedCategory (ModuleCat.{v} B) ⥤ DerivedCategory (ModuleCat.{v} B))
      (comparisonB : ∀ N : ModuleCat.{v} B,
        DerivedCategory.Q ⋙
            derivedTensorB.flip.obj
              ((DerivedCategory.singleFunctor (ModuleCat.{v} B) 0).obj N) ⟶
          ((tensoringRight (ModuleCat.{v} B)).obj N).mapHomologicalComplex
              (ComplexShape.up ℤ) ⋙ DerivedCategory.Q)
      (isLeftDerivedB : ∀ N : ModuleCat.{v} B,
        (derivedTensorB.flip.obj
            ((DerivedCategory.singleFunctor (ModuleCat.{v} B) 0).obj N)).IsLeftDerivedFunctor
          (comparisonB N)
          (HomologicalComplex.quasiIso (ModuleCat.{v} B) (ComplexShape.up ℤ))),
      HasWeakDimensionLE derivedTensorB comparisonB isLeftDerivedB d

Tag 092F, a lemma

Tag 092F in the Stacks Project.

absolutelyFlat_tfae, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/AbsolutelyFlat/Basic.lean, line 205.

For a commutative ring A, the following are equivalent: A has weak dimension at most zero; A is absolutely flat; A is reduced and every prime ideal is maximal; and the localization of A at every prime ideal is a field.

Weak dimension is HasWeakDimensionLE, which carries the derived tensor product, its comparison with termwise tensor product, and the universal property certifying that the comparison exhibits the derived tensor product as a total left derived functor. Those data are part of the statement, exactly as in the source and as in weakDimensionLE_one_tfae (Stacks 092S).

theorem absolutelyFlat_tfae (A : Type u) [CommRing A]
    (derivedTensor :
      DerivedCategory (ModuleCat.{u} A) ⥤
        DerivedCategory (ModuleCat.{u} A) ⥤ DerivedCategory (ModuleCat.{u} A))
    (comparison : ∀ N : ModuleCat.{u} A,
      DerivedCategory.Q ⋙
          derivedTensor.flip.obj
            ((DerivedCategory.singleFunctor (ModuleCat.{u} A) 0).obj N) ⟶
        ((tensoringRight (ModuleCat.{u} A)).obj N).mapHomologicalComplex
            (ComplexShape.up ℤ) ⋙ DerivedCategory.Q)
    (isLeftDerived : ∀ N : ModuleCat.{u} A,
      (derivedTensor.flip.obj
          ((DerivedCategory.singleFunctor (ModuleCat.{u} A) 0).obj N)).IsLeftDerivedFunctor
        (comparison N)
        (HomologicalComplex.quasiIso (ModuleCat.{u} A) (ComplexShape.up ℤ))) :
    [ HasWeakDimensionLE derivedTensor comparison isLeftDerived 0,
      (∀ (M : Type u) [AddCommGroup M] [Module A M], Module.Flat A M),
      IsReduced A ∧ ∀ p : Ideal A, p.IsPrime → p.IsMaximal,
      ∀ (p : Ideal A) [p.IsPrime], IsField (Localization.AtPrime p) ].TFAE

Tag 092G, a lemma

Tag 092G in the Stacks Project.

product_of_fields_absolutely_flat, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/AbsolutelyFlat/ProductOfFields.lean, line 162.

Stacks Tag 092G. A product of fields is an absolutely flat ring: every module over ∏ i, K i is flat.

theorem product_of_fields_absolutely_flat {I : Type*} (K : I → Type*) [∀ i, Field (K i)]
    (M : Type*) [AddCommGroup M] [Module (∀ i, K i) M] : Module.Flat (∀ i, K i) M

Tag 092H, a lemma

Tag 092H in the Stacks Project.

mulFlat_baseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/Basic.lean, line 113. The attribute's comment: “(1)”.

If the multiplication map μ_{B/A} : B ⊗[A] B → B is flat, then the multiplication map μ_{B'/A'} : B' ⊗[A'] B' → B' of the base change B' = A' ⊗[A] B is flat.

Under the canonical A'-algebra isomorphism e : (A' ⊗[A] B) ⊗[A'] (A' ⊗[A] B) ≃ₐ[A'] A' ⊗[A] (B ⊗[A] B) one has μ_{B'/A'} = (id_{A'} ⊗ μ_{B/A}) ∘ e, and id_{A'} ⊗ μ_{B/A} is flat by RingHom.Flat.tensorProductMap.

theorem mulFlat_baseChange (A : Type u₁) (B : Type u₂) (A' : Type u₃)
    [CommRing A] [CommRing B] [CommRing A']
    [Algebra A B] [Algebra A A']
    (h : Algebra.MulFlat A B) :
    Algebra.MulFlat A' (A' ⊗[A] B)

weaklyEtale_baseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/Basic.lean, line 123. The attribute's comment: “(2)”.

If A → B is weakly étale, then A' → B' is weakly étale, where B' = A' ⊗[A] B is the base change. This is Mathlib's base-change instance for Algebra.WeaklyEtale.

theorem weaklyEtale_baseChange (A : Type u₁) (B : Type u₂) (A' : Type u₃)
    [CommRing A] [CommRing B] [CommRing A']
    [Algebra A B] [Algebra A A']
    (h : Algebra.WeaklyEtale A B) :
    Algebra.WeaklyEtale A' (A' ⊗[A] B)

Tag 092I, a lemma

Tag 092I in the Stacks Project.

absolutelyFlat_and_isReduced_of_flat_lmul', a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/AbsolutelyFlat/Basic.lean, line 247.

If the multiplication map B ⊗[A] B → B is flat, then an absolutely flat base ring makes B absolutely flat. Under the same hypothesis, if A is reduced and A → B is weakly étale, then B is reduced.

theorem absolutelyFlat_and_isReduced_of_flat_lmul'
    (A : Type u) (B : Type v) [CommRing A] [CommRing B] [Algebra A B]
    (hμ : (Algebra.TensorProduct.lmul' A (S := B)).Flat) :
    ((∀ (M : Type u) [AddCommGroup M] [Module A M], Module.Flat A M) →
      ∀ (N : Type v) [AddCommGroup N] [Module B N], Module.Flat B N) ∧
    (IsReduced A → Algebra.WeaklyEtale A B → IsReduced B)

Tag 092J, a lemma

Tag 092J in the Stacks Project.

mulFlat_comp, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/Basic.lean, line 138. The attribute's comment: “(1)”.

If the multiplication maps μ_{B/A} and μ_{C/B} are both flat, then the multiplication map μ_{C/A} is flat.

After reducing to a single universe with Algebra.MulFlat.ulift_iff, this is the factorisation μ_{C/A} = μ_{C/B} ∘ (C ⊗_A C → C ⊗_B C) together with RingHom.Flat.mapOfCompatibleSMul and RingHom.Flat.comp.

theorem mulFlat_comp (A : Type u₁) (B : Type u₂) (C : Type u₃) [CommRing A] [CommRing B]
    [CommRing C] [Algebra A B] [Algebra B C] [Algebra A C] [IsScalarTower A B C]
    (hAB : Algebra.MulFlat A B) (hBC : Algebra.MulFlat B C) : Algebra.MulFlat A C

weaklyEtale_comp, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/Basic.lean, line 146. The attribute's comment: “(2)”.

If A → B and B → C are both weakly étale, then the composite A → C is weakly étale. This is Mathlib's Algebra.WeaklyEtale.trans.

theorem weaklyEtale_comp (A B C : Type*) [CommRing A] [CommRing B] [CommRing C]
    [Algebra A B] [Algebra B C] [Algebra A C] [IsScalarTower A B C]
    (hAB : Algebra.WeaklyEtale A B) (hBC : Algebra.WeaklyEtale B C) :
    Algebra.WeaklyEtale A C

Tag 092K, a lemma

Tag 092K in the Stacks Project.

flat_lmul'_and_weaklyEtale_of_faithfullyFlat, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/Descent.lean, line 98.

Let A → B → C be compatible maps of commutative rings and suppose that C is faithfully flat over B. Then flatness of the multiplication map C ⊗[A] C → C descends to the multiplication map B ⊗[A] B → B; consequently, if A → C is weakly étale, then A → B is weakly étale.

theorem flat_lmul'_and_weaklyEtale_of_faithfullyFlat
    (A : Type u) (B : Type v) (C : Type w)
    [CommRing A] [CommRing B] [CommRing C]
    [Algebra A B] [Algebra B C] [Algebra A C] [IsScalarTower A B C]
    [Module.FaithfullyFlat B C] :
    ((Algebra.TensorProduct.lmul' A (S := C)).Flat →
      (Algebra.TensorProduct.lmul' A (S := B)).Flat) ∧
    (Algebra.WeaklyEtale A C → Algebra.WeaklyEtale A B)

Tag 092L, a lemma

Tag 092L in the Stacks Project.

weaklyEtale_of_comp, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/Descent.lean, line 145.

If A → B and A → C are weakly étale, then a compatible algebra map B → C is weakly étale.

theorem weaklyEtale_of_comp (A : Type u) (B : Type v) (C : Type w)
    [CommRing A] [CommRing B] [CommRing C]
    [Algebra A B] [Algebra B C] [Algebra A C] [IsScalarTower A B C]
    (hAB : Algebra.WeaklyEtale A B) (hAC : Algebra.WeaklyEtale A C) :
    Algebra.WeaklyEtale B C

Tag 092M, a lemma

Tag 092M in the Stacks Project.

formallyUnramified_of_flat_lmul, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/FormallyUnramified.lean, line 49.

If the multiplication map B ⊗[A] B → B is flat, then B is formally unramified over A.

theorem formallyUnramified_of_flat_lmul (A : Type u) (B : Type v)
    [CommRing A] [CommRing B] [Algebra A B]
    (h : (Algebra.TensorProduct.lmul' A (S := B)).Flat) :
    Algebra.FormallyUnramified A B

Tag 0CKP, a lemma

Tag 0CKP in the Stacks Project.

unramified_of_finiteType_and_etale_of_finitePresentation, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/FormallyUnramified.lean, line 66.

If the multiplication map B ⊗[A] B → B is flat, finite type implies unramified and finite presentation together with flatness implies étale. In particular, every weakly étale algebra of finite presentation is étale.

theorem unramified_of_finiteType_and_etale_of_finitePresentation
    (A : Type u) (B : Type v) [CommRing A] [CommRing B] [Algebra A B] :
    ((Algebra.TensorProduct.lmul' A (S := B)).Flat →
      (Algebra.FiniteType A B → Algebra.Unramified A B) ∧
      (Algebra.FinitePresentation A B → Module.Flat A B → Algebra.Etale A B)) ∧
    (Algebra.WeaklyEtale A B →
      Algebra.FinitePresentation A B → Algebra.Etale A B)

Tag 092N, a lemma

Tag 092N in the Stacks Project.

weaklyEtale_of_isLocalization_etale_filteredColimit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/Descent.lean, line 410.

A localization and an étale algebra over a commutative ring are weakly étale, and an arbitrary filtered colimit of weakly étale algebras is weakly étale.

theorem weaklyEtale_of_isLocalization_etale_filteredColimit
    (A : Type u) [CommRing A] :
    (∀ (B : Type v) [CommRing B] [Algebra A B] (M : Submonoid A)
      [IsLocalization M B], Algebra.WeaklyEtale A B) ∧
    (∀ (B : Type v) [CommRing B] [Algebra A B] [Algebra.Etale A B],
      Algebra.WeaklyEtale A B) ∧
    (∀ (J : Type w) [SmallCategory J] [IsFiltered J] [Small.{max u (w + 1)} J]
      (F : J ⥤ CommAlgCat.{w} A) (c : Cocone F), IsColimit c →
      (∀ j, Algebra.WeaklyEtale A (F.obj j)) → Algebra.WeaklyEtale A c.pt)

Tag 092P, a lemma

Tag 092P in the Stacks Project.

isSeparable_of_flat_lmul, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/Field.lean, line 88.

Let L/K be a field extension. If the multiplication map L ⊗[K] L → L is flat, then L/K is algebraic and separable.

theorem isSeparable_of_flat_lmul (K : Type u) (L : Type v)
    [Field K] [Field L] [Algebra K L]
    (h : (Algebra.TensorProduct.lmul' K (S := L)).Flat) :
    Algebra.IsSeparable K L

Tag 092Q, a lemma

Tag 092Q in the Stacks Project.

flat_lmul_weaklyEtale_filteredColimitEtale_tfae, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/Field.lean, line 362.

Let B be an algebra over a field K. The following are equivalent: the multiplication map B ⊗[K] B → B is flat; B is weakly étale over K; and B is a filtered colimit of étale K-algebras. Under these equivalent conditions, every finitely generated K-subalgebra of B is étale over K.

theorem flat_lmul_weaklyEtale_filteredColimitEtale_tfae
    (K : Type u) (B : Type v) [Field K] [CommRing B] [Algebra K B] :
    [ (Algebra.TensorProduct.lmul' K (S := B)).Flat,
      Algebra.WeaklyEtale K B,
      ∃ (J : Type v) (_ : SmallCategory J) (_ : IsFiltered J)
        (F : J ⥤ CommAlgCat.{v} K) (c : Cocone F) (_ : IsColimit c),
        Nonempty (c.pt ≅ CommAlgCat.of K B) ∧ ∀ j, Algebra.Etale K (F.obj j) ].TFAE ∧
      ((Algebra.TensorProduct.lmul' K (S := B)).Flat →
        ∀ A : Subalgebra K B, A.FG → Algebra.Etale K A)

Tag 092R, a lemma

Tag 092R in the Stacks Project.

isSeparable_and_isAlgebraic_residueField_of_weaklyEtale, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/ResidueField.lean, line 86.

If A → B is weakly étale and a prime q of B lies over a prime p of A, then the canonical residue field extension κ(p) → κ(q) is separable algebraic.

theorem isSeparable_and_isAlgebraic_residueField_of_weaklyEtale
    (A : Type u) (B : Type v) [CommRing A] [CommRing B] [Algebra A B]
    (h : Algebra.WeaklyEtale A B) (p : Ideal A) [p.IsPrime]
    (q : Ideal B) [q.IsPrime] (hpq : p = q.comap (algebraMap A B)) :
    letI := (Ideal.ResidueField.map p q (algebraMap A B) hpq).toAlgebra
    Algebra.IsSeparable p.ResidueField q.ResidueField ∧
      Algebra.IsAlgebraic p.ResidueField q.ResidueField

Tag 092S, a lemma

Tag 092S in the Stacks Project.

weakDimensionLE_one_tfae, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeakDimension/DimensionOne.lean, line 377.

Stacks Tag 092S. For a commutative ring, weak dimension at most one, flatness of all ideals, flatness of all finitely generated ideals, flatness of submodules of flat modules, and the valuation-ring property for all localizations at prime ideals are equivalent.

theorem weakDimensionLE_one_tfae
    (A : Type u) [CommRing A]
    (derivedTensor :
      DerivedCategory (ModuleCat.{u} A) ⥤
        DerivedCategory (ModuleCat.{u} A) ⥤ DerivedCategory (ModuleCat.{u} A))
    (comparison : ∀ N : ModuleCat.{u} A,
      DerivedCategory.Q ⋙
          derivedTensor.flip.obj
            ((DerivedCategory.singleFunctor (ModuleCat.{u} A) 0).obj N) ⟶
        ((tensoringRight (ModuleCat.{u} A)).obj N).mapHomologicalComplex
            (ComplexShape.up ℤ) ⋙ DerivedCategory.Q)
    (isLeftDerived : ∀ N : ModuleCat.{u} A,
      (derivedTensor.flip.obj
          ((DerivedCategory.singleFunctor (ModuleCat.{u} A) 0).obj N)).IsLeftDerivedFunctor
        (comparison N)
        (HomologicalComplex.quasiIso (ModuleCat.{u} A) (ComplexShape.up ℤ))) :
    List.TFAE
      [ HasWeakDimensionLE derivedTensor comparison isLeftDerived 1,
        ∀ I : Ideal A, Module.Flat A I,
        ∀ I : Ideal A, I.FG → Module.Flat A I,
        ∀ (M : Type u) [AddCommGroup M] [Module A M],
          Module.Flat A M → ∀ N : Submodule A M, Module.Flat A N,
        ∀ (p : Ideal A) [p.IsPrime],
          ∃ _ : IsDomain (Localization.AtPrime p),
            ValuationRing (Localization.AtPrime p) ]

Tag 092T, a lemma

Tag 092T in the Stacks Project.

weakDimensionLE_one_and_isLocalization_pi_fractionRing, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/WeakDimension.lean, line 208.

A product of valuation rings has weak dimension at most one, and the product of their fraction fields is the localization at the componentwise non-zero-divisors.

theorem weakDimensionLE_one_and_isLocalization_pi_fractionRing
    (J : Type v) (A : J → Type u) (K : J → Type w)
    [∀ j, CommRing (A j)] [∀ j, IsDomain (A j)] [∀ j, ValuationRing (A j)]
    [∀ j, Field (K j)] [∀ j, Algebra (A j) (K j)] [∀ j, IsFractionRing (A j) (K j)] :
    HasWeakDimensionLEOne (∀ j, A j) ∧
      IsLocalization
        (.pi Set.univ (fun j ↦ nonZeroDivisors (A j)))
        (∀ j, K j)

Tag 092U, a lemma

Tag 092U in the Stacks Project.

exists_isPullback_with_weakDimensionLE_one, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/WeakDimension.lean, line 276.

A normal domain and its fraction field fit into a cartesian square whose lower-left ring has weak dimension at most one and whose lower map is flat, injective, and a ring epimorphism.

theorem exists_isPullback_with_weakDimensionLE_one
    (A : Type u) [CommRing A] [IsDomain A] [IsIntegrallyClosed A]
    (K : Type v) [Field K] [Algebra A K] [IsFractionRing A K] :
    ∃ (V L : Type (max u v)) (_ : CommRing V) (_ : CommRing L)
      (_ : Algebra A V) (_ : Algebra K L) (_ : Algebra V L)
      (_ : Module.Flat V L) (_ : Algebra.IsEpi V L),
      HasWeakDimensionLEOne V ∧
        Function.Injective (algebraMap V L) ∧
        IsPullback
          (CommRingCat.ofHom (algebraMap A K).ulift)
          (CommRingCat.ofHom
            ((algebraMap A V).comp ULift.ringEquiv.toRingHom))
          (CommRingCat.ofHom
            ((algebraMap K L).comp ULift.ringEquiv.toRingHom))
          (CommRingCat.ofHom (algebraMap V L))

Tag 092V, a lemma

Tag 092V in the Stacks Project.

isIntegrallyClosedIn_of_isEpi_of_flat_of_weakDimensionLE_one, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/WeakDimension.lean, line 242.

A flat injective epimorphism of commutative rings whose source has weak dimension at most one makes the source integrally closed in the target.

theorem isIntegrallyClosedIn_of_isEpi_of_flat_of_weakDimensionLE_one
    {A : Type u} {B : Type v} [CommRing A] [CommRing B] [Algebra A B]
    [Module.Flat A B] [Algebra.IsEpi A B]
    (hA : HasWeakDimensionLEOne A)
    (hinjective : Function.Injective (algebraMap A B)) :
    IsIntegrallyClosedIn A B

Tag 092W, a lemma

Tag 092W in the Stacks Project.

isIntegrallyClosedIn_tensorProduct_fractionField_of_weaklyEtale, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/WeakDimension.lean, line 740.

A weakly étale algebra over a normal domain is integrally closed in its scalar extension to any fraction field of the domain.

theorem isIntegrallyClosedIn_tensorProduct_fractionField_of_weaklyEtale
    (A : Type u) (B : Type v) (K : Type w)
    [CommRing A] [CommRing B] [Field K] [Algebra A B] [Algebra A K]
    [IsDomain A] [IsIntegrallyClosed A] [IsFractionRing A K]
    (hAB : Algebra.WeaklyEtale A B) :
    IsIntegrallyClosedIn B (B ⊗[A] K)

Tag 092X, a lemma

Tag 092X in the Stacks Project.

RingHom.exists_henselianLocalRing_of_isIntegral_of_isDomain, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Henselian/IntegralDomain.lean, line 262.

An integral homomorphism from a henselian local ring to a domain makes the target a henselian local ring and is a local homomorphism. If the source residue field is separably closed, then the target residue field is separably closed and its extension along the induced residue-field homomorphism is purely inseparable.

theorem RingHom.exists_henselianLocalRing_of_isIntegral_of_isDomain
    {A : Type u} {B : Type v} [CommRing A] [CommRing B]
    [HenselianLocalRing A] [IsDomain B] (f : A →+* B) (hf : f.IsIntegral) :
    ∃ (hB : HenselianLocalRing B) (hlocal : IsLocalHom f),
      letI : IsLocalRing B := hB.toIsLocalRing
      letI : IsLocalHom f := hlocal
      IsSepClosed (ResidueField A) →
        IsSepClosed (ResidueField B) ∧ (ResidueField.map f).IsPurelyInseparable

Tag 092Z, a theorem

Tag 092Z in the Stacks Project.

bijective_algebraMap_of_weaklyEtale_of_isLocalHom, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/StrictlyHenselian.lean, line 326.

A local weakly étale algebra over a strictly henselian local ring has bijective structure homomorphism.

theorem bijective_algebraMap_of_weaklyEtale_of_isLocalHom
    (A : Type u) (B : Type v) [CommRing A] [HenselianLocalRing A]
    [IsSepClosed (ResidueField A)] [CommRing B] [IsLocalRing B] [Algebra A B]
    [IsLocalHom (algebraMap A B)] (hAB : Algebra.WeaklyEtale A B) :
    Function.Bijective (algebraMap A B)

Sources

• Section 15.106, Tag 092A
• AxiomMath/Stacks at 5ed82d14

LeanIsAbsolutelyFlat · flat_of_flat_of_mul_flat · HasWeakDimensionLE · hasWeakDimensionLE_of_weaklyEtale · absolutelyFlat_tfae · product_of_fields_absolutely_flat · mulFlat_baseChange · weaklyEtale_baseChange

Section15.107 Weakly étale algebras over fields#

Section 15.107 of the Stacks Project, Tag 0CKQ. 4 tags here are carried by 4 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0CKR, a lemma

Tag 0CKR in the Stacks Project.

structure_of_weaklyEtale_over_field, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/Field.lean, line 573. References: EGA IV (18.3.3), “partial”; SGA 1 IX (1.8), “decomposition direction”; SGA 4½ Arcata (2.1.2), “partial”; SGA 4½ Arcata (2.4.2), “more general”.

A weakly étale algebra B over a field K is reduced and integral. Every finitely generated K-subalgebra is a finite product of finite separable extensions of K. Moreover, B is a field or the zero ring exactly when it has no nontrivial idempotents; if it is a field, then it is a separable algebraic extension. Subalgebras and quotient algebras of B are weakly étale over K, as is the tensor product of B with any weakly étale K-algebra.

theorem structure_of_weaklyEtale_over_field
    (K : Type u) (B : Type v) [Field K] [CommRing B] [Algebra K B]
    (hB : Algebra.WeaklyEtale K B) :
    IsReduced B ∧
      Algebra.IsIntegral K B ∧
      (∀ A : Subalgebra K B, A.FG →
        ∃ (ι : Type v) (_ : Finite ι) (L : ι → Type v) (_ : ∀ i, Field (L i))
          (_ : ∀ i, Algebra K (L i)) (_ : A ≃ₐ[K] ∀ i, L i),
          ∀ i, Module.Finite K (L i) ∧ Algebra.IsSeparable K (L i)) ∧
      (((Subsingleton B ∨ IsField B) ↔
          ∀ e : B, IsIdempotentElem e → e = 0 ∨ e = 1) ∧
        (IsField B → Algebra.IsSeparable K B ∧ Algebra.IsAlgebraic K B)) ∧
      ((∀ S : Subalgebra K B, Algebra.WeaklyEtale K S) ∧
        (∀ (Q : Type w) [CommRing Q] [Algebra K Q],
          (∃ f : B →ₐ[K] Q, Function.Surjective f) → Algebra.WeaklyEtale K Q)) ∧
      (∀ (B' : Type w) [CommRing B'] [Algebra K B'],
        Algebra.WeaklyEtale K B' → Algebra.WeaklyEtale K (B ⊗[K] B'))

Tag 0CKS, a lemma

Tag 0CKS in the Stacks Project.

exists_greatest_weaklyEtale_subalgebra, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/MaximalSubalgebra.lean, line 136.

An algebra over a field has a greatest weakly étale subalgebra.

theorem exists_greatest_weaklyEtale_subalgebra
    (K : Type u) (A : Type v) [Field K] [CommRing A] [Algebra K A] :
    ∃ Bmax : Subalgebra K A,
      Algebra.WeaklyEtale K Bmax ∧
        ∀ B : Subalgebra K A, Algebra.WeaklyEtale K B → B ≤ Bmax

Tag 0CKT, a lemma

Tag 0CKT in the Stacks Project.

exists_maximalWeaklyEtaleSubalgebraFunctor, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/MaximalSubalgebra.lean, line 722.

The maximal weakly étale subalgebra over a field is functorial and commutes with filtered colimits, reduction, and finite products. It pulls back along injective algebra maps, and over an algebra without nontrivial idempotents it is either the zero ring or a separable algebraic field extension.

theorem exists_maximalWeaklyEtaleSubalgebraFunctor
    (K : Type u) [Field K] :
    ∃ (Bmax : CommAlgCat.{v} K ⥤ CommAlgCat.{v} K)
      (ι : Bmax ⟶ 𝟭 (CommAlgCat.{v} K)),
      (∀ A, Function.Injective (ι.app A)) ∧
      (∀ A, Algebra.WeaklyEtale K (Bmax.obj A)) ∧
      (∀ (A : CommAlgCat.{v} K) (S : Subalgebra K A),
        Algebra.WeaklyEtale K S →
          ∃ f : CommAlgCat.of K S ⟶ Bmax.obj A,
            f ≫ ι.app A = CommAlgCat.ofHom S.val) ∧
      (∀ (A' A : CommAlgCat.{v} K) (f : A' ⟶ A), Function.Injective f →
        ∀ x : A',
          (∃ b : Bmax.obj A', ι.app A' b = x) ↔
            ∃ b : Bmax.obj A, ι.app A b = f x) ∧
      PreservesFilteredColimits Bmax ∧
      (∀ A : CommAlgCat.{v} K,
        IsIso (Bmax.map (CommAlgCat.ofHom (Ideal.Quotient.mkₐ K (nilradical A))))) ∧
      (∀ (J : Type v) [Finite J], PreservesLimitsOfShape (Discrete J) Bmax) ∧
      (∀ A : CommAlgCat.{v} K,
        (∀ e : A, IsIdempotentElem e → e = 0 ∨ e = 1) →
          Subsingleton A ∨
            (IsField (Bmax.obj A) ∧ Algebra.IsSeparable K (Bmax.obj A) ∧
              Algebra.IsAlgebraic K (Bmax.obj A)))

Tag 0CKU, a lemma

Tag 0CKU in the Stacks Project.

maximalWeaklyEtaleSubalgebra_baseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/WeaklyEtale/MaximalSubalgebra.lean, line 1086.

Extending the base field carries the maximal weakly étale subalgebra to the maximal weakly étale subalgebra of the scalar extension.

theorem maximalWeaklyEtaleSubalgebra_baseChange
    (K : Type u) (L : Type v) (A : Type w)
    [Field K] [Field L] [CommRing A] [Algebra K L] [Algebra K A]
    (B : Subalgebra K A) (hB : Algebra.WeaklyEtale K B)
    (hBmax : ∀ C : Subalgebra K A, Algebra.WeaklyEtale K C → C ≤ B) :
    let f : L ⊗[K] B →ₐ[K] L ⊗[K] A :=
      Algebra.TensorProduct.map (AlgHom.id K L) B.val
    Function.Injective f ∧
      Algebra.WeaklyEtale L (L ⊗[K] B) ∧
      ∀ C : Subalgebra L (L ⊗[K] A), Algebra.WeaklyEtale L C →
        ∀ x : C, ∃ y : L ⊗[K] B, f y = x

Sources

• Section 15.107, Tag 0CKQ
• AxiomMath/Stacks at 5ed82d14

Leanstructure_of_weaklyEtale_over_field · exists_greatest_weaklyEtale_subalgebra · exists_maximalWeaklyEtaleSubalgebraFunctor · maximalWeaklyEtaleSubalgebra_baseChange

Section15.108 Local irreducibility#

Section 15.108 of the Stacks Project, Tag 06DT. 8 tags here are carried by 10 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0BPZ, a definition

Tag 0BPZ in the Stacks Project.

IsUnibranch, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Defs.lean, line 62. The attribute's comment: “(1)”. References: EGA 0 (23.2.1), “def. of unibranch”; EGA III (4.3.6), “unibranch local ring”.

A local ring is unibranch if its reduction is a domain and the integral closure of its reduction in its field of fractions is local.

def IsUnibranch (A : Type u) [CommRing A] [IsLocalRing A] : Prop

IsGeometricallyUnibranch, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Defs.lean, line 70. The attribute's comment: “(2)”. References: EGA 0 (23.2.1), “def. of geom. unibranch”.

A local ring is geometrically unibranch if it is unibranch and the residue field of the normalization of its reduction is purely inseparable over its residue field.

noncomputable def IsGeometricallyUnibranch
    (A : Type u) [CommRing A] [IsLocalRing A] : Prop

Tag 0C24, a lemma

Tag 0C24 in the Stacks Project.

bijective_maximalIdeals_and_minimalPrimes_of_henselization, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Henselization/BranchCorrespondence.lean, line 598.

Let A' be the integral closure of a local ring A in the total ring of fractions of its reduction, and let Aʰ be a henselization of A. Contraction from A' ⊗[A] Aʰ is bijective on maximal ideals toward A' and on minimal primes toward Aʰ. Each minimal prime of the tensor product lies in a unique maximal ideal, and each maximal ideal contains a unique minimal prime.

theorem bijective_maximalIdeals_and_minimalPrimes_of_henselization
    (A : Type u) (Ah : Type v) [CommRing A] [IsLocalRing A] [CommRing Ah] [IsLocalRing Ah]
    [Algebra A Ah]
    [IsLocalRing.IsHenselization A (IsLocalRing.ResidueField A) Ah]
    (hmin : (minimalPrimes A).Finite) :
    let A' := integralClosure A (FractionRing (A ⧸ nilradical A))
    let T := TensorProduct A A' Ah
    Set.BijOn
        (Ideal.comap (Algebra.TensorProduct.includeLeftRingHom : A' →+* T))
        {I : Ideal T | I.IsMaximal} {I : Ideal A' | I.IsMaximal} ∧
      Set.BijOn
        (Ideal.comap
          (Algebra.TensorProduct.includeRight : Ah →ₐ[A] T).toRingHom)
        (minimalPrimes T) (minimalPrimes Ah) ∧
      (∀ q ∈ minimalPrimes T, ∃! m : Ideal T, m.IsMaximal ∧ q ≤ m) ∧
      (∀ m : Ideal T, m.IsMaximal → ∃! q : Ideal T, q ∈ minimalPrimes T ∧ q ≤ m)

Tag 0BQ0, a lemma

Tag 0BQ0 in the Stacks Project.

isUnibranch_iff_henselization_has_unique_minimalPrime, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Henselization/Unibranch.lean, line 132. References: EGA IV (18.6.12), “a) ⇔ c), irreducible”.

A local ring is unibranch if and only if a henselization of it has a unique minimal prime.

theorem isUnibranch_iff_henselization_has_unique_minimalPrime
    (A Ah : Type u) [CommRing A] [IsLocalRing A] [CommRing Ah] [IsLocalRing Ah]
    [Algebra A Ah]
    [IsLocalRing.IsHenselization A (IsLocalRing.ResidueField A) Ah] :
    IsUnibranch A ↔ ∃! p : Ideal Ah, p ∈ minimalPrimes Ah

Tag 0C25, a lemma

Tag 0C25 in the Stacks Project.

geometric_branches_of_strictHenselization, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Henselization/BranchCorrespondence.lean, line 538.

Let A' be the integral closure of a local ring A in the total ring of fractions of its reduction, and let Aˢʰ be a strict henselization with separably closed residue field ksep inside an algebraic closure kbar. The residue extension at every maximal ideal of A' is algebraic, and its embeddings into kbar canonically parametrize the fibre in Spec (A' ⊗[A] Aˢʰ). Contraction to Aˢʰ is bijective on minimal primes, and containment gives mutually unique correspondences between minimal and maximal primes of the tensor product.

theorem geometric_branches_of_strictHenselization
    (A : Type u) (Ash : Type u') (ksep : Type v) (kbar : Type w)
    [CommRing A] [IsLocalRing A] [CommRing Ash] [IsLocalRing Ash] [Algebra A Ash]
    [Field ksep] [Algebra (IsLocalRing.ResidueField A) ksep]
    [IsSepClosure (IsLocalRing.ResidueField A) ksep]
    [IsLocalRing.IsHenselization A ksep Ash]
    [Field kbar] [Algebra (IsLocalRing.ResidueField A) kbar]
    [IsAlgClosure (IsLocalRing.ResidueField A) kbar]
    [Algebra ksep kbar] [IsScalarTower (IsLocalRing.ResidueField A) ksep kbar]
    (e : ksep ≃+* IsLocalRing.ResidueField Ash)
    (he : ∀ x : A,
      e (algebraMap (IsLocalRing.ResidueField A) ksep (IsLocalRing.residue A x)) =
        IsLocalRing.residue Ash (algebraMap A Ash x))
    (hmin : (minimalPrimes A).Finite) :
    let A' := integralClosure A (FractionRing (A ⧸ nilradical A))
    let T := TensorProduct A A' Ash
    (∀ m' : MaximalSpectrum A',
      ∃ h : IsLocalRing.maximalIdeal A = m'.asIdeal.comap (algebraMap A A'),
        let f := Ideal.quotientMap m'.asIdeal (algebraMap A A') (le_of_eq h)
        f.IsIntegral ∧
          ∃ E : ({σ : (A' ⧸ m'.asIdeal) →+* kbar //
                    σ.comp f = algebraMap (IsLocalRing.ResidueField A) kbar} ≃
                  {q : PrimeSpectrum T //
                    PrimeSpectrum.comap
                      (Algebra.TensorProduct.includeLeftRingHom : A' →+* T) q =
                        m'.toPrimeSpectrum}),
            ∀ σ, ∃ φ : T →+* kbar,
              (∀ (a : A') (b : Ash),
                  φ (a ⊗ₜ[A] b) =
                    σ.1 (Ideal.Quotient.mk m'.asIdeal a) *
                      algebraMap ksep kbar (e.symm (IsLocalRing.residue Ash b))) ∧
                (E σ).1.asIdeal = RingHom.ker φ ∧
                ∀ ψ : T →+* kbar,
                  (∀ (a : A') (b : Ash),
                    ψ (a ⊗ₜ[A] b) =
                      σ.1 (Ideal.Quotient.mk m'.asIdeal a) *
                        algebraMap ksep kbar (e.symm (IsLocalRing.residue Ash b))) →
                  ψ = φ) ∧
      Set.BijOn
        (Ideal.comap
          (Algebra.TensorProduct.includeRight : Ash →ₐ[A] T).toRingHom)
        (minimalPrimes T) (minimalPrimes Ash) ∧
      (∀ q ∈ minimalPrimes T, ∃! m : Ideal T, m.IsMaximal ∧ q ≤ m) ∧
      (∀ m : Ideal T, m.IsMaximal → ∃! q : Ideal T, q ∈ minimalPrimes T ∧ q ≤ m)

Tag 06DM, a lemma

Tag 06DM in the Stacks Project.

isGeometricallyUnibranch_iff_strictHenselization_has_unique_minimalPrime, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Henselization/Unibranch.lean, line 425. References: EGA IV (18.8.15), “a) ⇔ c), irreducible”.

A local ring is geometrically unibranch if and only if a strict henselization of it has a unique minimal prime.

theorem isGeometricallyUnibranch_iff_strictHenselization_has_unique_minimalPrime
    (A Ash : Type u) [CommRing A] [IsLocalRing A] [CommRing Ash] [IsLocalRing Ash]
    [Algebra A Ash]
    (ksep : Type v) [Field ksep] [Algebra (IsLocalRing.ResidueField A) ksep]
    [IsSepClosure (IsLocalRing.ResidueField A) ksep]
    [IsLocalRing.IsHenselization A ksep Ash] :
    IsGeometricallyUnibranch A ↔
      ∃! p : Ideal Ash, p ∈ minimalPrimes Ash

Tag 0C26, a definition

Tag 0C26 in the Stacks Project.

numberOfBranches, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Defs.lean, line 96. The attribute's comment: “number of branches”.

The number of branches of a local ring is the extended-natural cardinality of the set of minimal primes of a henselization.

noncomputable def numberOfBranches (A Ah : Type u) [CommRing A] [IsLocalRing A]
    [CommRing Ah]
    [IsLocalRing Ah] [Algebra A Ah]
    [IsLocalRing.IsHenselization A (IsLocalRing.ResidueField A) Ah] : ℕ∞

numberOfGeometricBranches, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Defs.lean, line 105. The attribute's comment: “number of geometric branches”.

The number of geometric branches of a local ring is the extended-natural cardinality of the set of minimal primes of a strict henselization.

noncomputable def numberOfGeometricBranches (A : Type u) (ksep : Type v) (Ash : Type u)
    [CommRing A] [IsLocalRing A] [Field ksep] [Algebra (IsLocalRing.ResidueField A) ksep]
    [IsSepClosure (IsLocalRing.ResidueField A) ksep] [CommRing Ash] [IsLocalRing Ash]
    [Algebra A Ash] [IsLocalRing.IsHenselization A ksep Ash] : ℕ∞

Tag 0C37, a lemma

Tag 0C37 in the Stacks Project.

numberOfBranches_characterizations, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Henselization/BranchNumbers.lean, line 204.

The numbers of branches and geometric branches of a local ring are infinite if the ring has infinitely many minimal primes, and they equal one exactly for unibranch and geometrically unibranch rings, respectively. If the ring has finitely many minimal primes, its branches are the maximal ideals of the normalization of its reduction, with geometric branches weighted by the separable degrees of the corresponding residue-field extensions.

theorem numberOfBranches_characterizations
    (A Ah Ash : Type u) (ksep : Type v)
    [CommRing A] [IsLocalRing A]
    [CommRing Ah] [IsLocalRing Ah] [Algebra A Ah]
    [IsLocalRing.IsHenselization A (IsLocalRing.ResidueField A) Ah]
    [Field ksep] [Algebra (IsLocalRing.ResidueField A) ksep]
    [IsSepClosure (IsLocalRing.ResidueField A) ksep]
    [CommRing Ash] [IsLocalRing Ash] [Algebra A Ash]
    [IsLocalRing.IsHenselization A ksep Ash] :
    ((minimalPrimes A).Infinite →
        numberOfBranches A Ah = ⊤ ∧
          numberOfGeometricBranches A ksep Ash = ⊤) ∧
      (numberOfBranches A Ah = 1 ↔ IsUnibranch A) ∧
      (numberOfGeometricBranches A ksep Ash = 1 ↔
        IsGeometricallyUnibranch A) ∧
      ((minimalPrimes A).Finite →
        let A' := integralClosure A (FractionRing (A ⧸ nilradical A))
        numberOfBranches A Ah =
            Set.encard {m' : Ideal A' | m'.IsMaximal} ∧
          ∃ hcontract : ∀ m' : MaximalSpectrum A',
              IsLocalRing.maximalIdeal A = m'.asIdeal.comap (algebraMap A A'),
            numberOfGeometricBranches A ksep Ash =
              ENat.card (Σ m' : MaximalSpectrum A',
                letI : Field (A' ⧸ m'.asIdeal) := Ideal.Quotient.field m'.asIdeal
                let f := Ideal.quotientMap m'.asIdeal (algebraMap A A')
                  (le_of_eq (hcontract m'))
                letI : Algebra (IsLocalRing.ResidueField A) (A' ⧸ m'.asIdeal) := f.toAlgebra
                (Field.sepDegree (IsLocalRing.ResidueField A)
                  (A' ⧸ m'.asIdeal)).ord.ToType))

Tag 0DQ1, a lemma

Tag 0DQ1 in the Stacks Project.

numberOfBranches_eq_of_isLocalization_smooth, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Miscellaneous/SmoothInvariance.lean, line 402.

A local homomorphism of local rings which is a localization of a smooth ring map preserves the number of geometric branches. If its residue-field extension is purely inseparable, it also preserves the number of branches.

The source's two rings carry independent universes, together with their henselizations and strict henselizations: A, Aʰ, Aˢʰ in one and B, Bʰ, Bˢʰ in the other, and the two separable closures in two more. The universe work is done by StacksDraft.henselization_tensorProduct_at_unique_prime and StacksDraft.strictHenselization_tensorProduct_at_induced_prime, the source-strength forms of tags 08HU and 08HV, and by the ind-étale layer they are built on, which lets the branch-counting lemmas above be stated with the base and its henselization in independent universes.

theorem numberOfBranches_eq_of_isLocalization_smooth
    (A Ah Ash : Type uA) (B Bh Bsh : Type uB) (kAsep : Type vA) (kBsep : Type vB)
    [CommRing A] [IsLocalRing A] [CommRing B] [IsLocalRing B] [Algebra A B]
    [IsLocalHom (algebraMap A B)]
    (hsm : ∃ (C : Type uC) (_ : CommRing C) (_ : Algebra A C) (_ : Algebra C B)
      (_ : IsScalarTower A C B) (S : Submonoid C),
        Algebra.Smooth A C ∧ IsLocalization S B)
    [CommRing Ah] [IsLocalRing Ah] [Algebra A Ah]
    [IsLocalRing.IsHenselization A (IsLocalRing.ResidueField A) Ah]
    [CommRing Bh] [IsLocalRing Bh] [Algebra B Bh]
    [IsLocalRing.IsHenselization B (IsLocalRing.ResidueField B) Bh]
    [Field kAsep] [Algebra (IsLocalRing.ResidueField A) kAsep]
    [IsSepClosure (IsLocalRing.ResidueField A) kAsep]
    [CommRing Ash] [IsLocalRing Ash] [Algebra A Ash]
    [IsLocalRing.IsHenselization A kAsep Ash]
    [Field kBsep] [Algebra (IsLocalRing.ResidueField B) kBsep]
    [IsSepClosure (IsLocalRing.ResidueField B) kBsep]
    [CommRing Bsh] [IsLocalRing Bsh] [Algebra B Bsh]
    [IsLocalRing.IsHenselization B kBsep Bsh] :
    numberOfGeometricBranches A kAsep Ash = numberOfGeometricBranches B kBsep Bsh ∧
      ((IsLocalRing.ResidueField.map (algebraMap A B)).IsPurelyInseparable →
        numberOfBranches A Ah = numberOfBranches B Bh)

Sources

• Section 15.108, Tag 06DT
• AxiomMath/Stacks at 5ed82d14

LeanIsUnibranch · IsGeometricallyUnibranch · bijective_maximalIdeals_and_minimalPrimes_of_henselization · isUnibranch_iff_henselization_has_unique_minimalPrime · geometric_branches_of_strictHenselization · isGeometricallyUnibranch_iff_strictHenselization_has_unique_minimalPrime · numberOfBranches · numberOfGeometricBranches

Section15.109 Miscellaneous on branches#

Section 15.109 of the Stacks Project, Tag 0GS4. 4 tags here are carried by 4 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0GS5, a lemma

Tag 0GS5 in the Stacks Project.

comap_ne_bot_of_localization_or_discrete_fibers, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/UnibranchEtale.lean, line 153.

Let A → B be a map of domains. Suppose that it is a localization of an étale map; or that it is flat and a localization of an unramified, quasi-finite, or integral map; or that it is flat and its fibers have no nontrivial specializations; or that it maps the generic point to the generic point and its fibers have no nontrivial specializations; or that exactly one point of Spec(B) maps to the generic point of Spec(A). Then every nonzero ideal of B has nonzero contraction to A.

theorem comap_ne_bot_of_localization_or_discrete_fibers
    (A : Type u) (B : Type v) [CommRing A] [CommRing B] [IsDomain A] [IsDomain B]
    [Algebra A B]
    (h :
      (∃ (T : Type (max u v)) (_ : CommRing T) (_ : Algebra A T) (_ : Algebra T B)
          (_ : IsScalarTower A T B), Algebra.Etale A T ∧
            ∃ M : Submonoid T, IsLocalization M B) ∨
      (Module.Flat A B ∧
        ∃ (T : Type (max u v)) (_ : CommRing T) (_ : Algebra A T) (_ : Algebra T B)
          (_ : IsScalarTower A T B), Algebra.Unramified A T ∧
            ∃ M : Submonoid T, IsLocalization M B) ∨
      (Module.Flat A B ∧
        ∃ (T : Type (max u v)) (_ : CommRing T) (_ : Algebra A T) (_ : Algebra T B)
          (_ : IsScalarTower A T B), Algebra.QuasiFinite A T ∧
            ∃ M : Submonoid T, IsLocalization M B) ∨
      (Module.Flat A B ∧
        ∃ (T : Type (max u v)) (_ : CommRing T) (_ : Algebra A T) (_ : Algebra T B)
          (_ : IsScalarTower A T B), Algebra.IsIntegral A T ∧
            ∃ M : Submonoid T, IsLocalization M B) ∨
      (Module.Flat A B ∧ NoNontrivialSpecializationsInFibers A B) ∨
      ((⊥ : Ideal B).comap (algebraMap A B) = ⊥ ∧
        NoNontrivialSpecializationsInFibers A B) ∨
      (∃! q : Ideal B, q.IsPrime ∧ q.comap (algebraMap A B) = ⊥)) :
    ∀ J : Ideal B, J ≠ ⊥ → J.comap (algebraMap A B) ≠ ⊥

Tag 0GSC, a lemma

Tag 0GSC in the Stacks Project.

exists_etale_away_of_isGeometricallyUnibranch_of_isUnramifiedAt, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Miscellaneous/UnramifiedNeighborhood.lean, line 251. References: EGA IV (18.10.1), “(i)+(ii) ⇒ étale”.

Let A → B be a ring map from a domain, and let q be a prime of B lying over a prime p of A. If Aₚ is geometrically unibranch, the map is unramified at q, and the canonical map Aₚ → B_q is injective, then some principal localization B_g, with g ∉ q, is étale over A.

theorem exists_etale_away_of_isGeometricallyUnibranch_of_isUnramifiedAt
    {A : Type u} {B : Type v} [CommRing A] [CommRing B] [Algebra A B] [IsDomain A]
    (p : Ideal A) [p.IsPrime] (q : Ideal B) [q.IsPrime] [q.LiesOver p]
    (hgeom : IsGeometricallyUnibranch (Localization.AtPrime p))
    (hunram : ∃ f : B, f ∉ q ∧ Algebra.Unramified A (Localization.Away f))
    (hinj : letI := Localization.AtPrime.algebraOfLiesOver p q
      Function.Injective
        (algebraMap (Localization.AtPrime p) (Localization.AtPrime q))) :
    ∃ g : B, g ∉ q ∧ Algebra.Etale A (Localization.Away g)

Tag 0GS6, a lemma

Tag 0GS6 in the Stacks Project.

exists_etale_isLocalization_of_isGeometricallyUnibranch, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/UnibranchEtale.lean, line 336. References: EGA IV (18.10.1), “(i)+(ii) ⇒ étale”; SGA 1 I (9.5), “(ii)”.

Stacks 0GS6. An injective, essentially finite type local map from a geometrically unibranch local domain is a localization of an étale map when its target maximal ideal is extended from the source and its residue-field extension is separable.

theorem exists_etale_isLocalization_of_isGeometricallyUnibranch
    (A : Type u) (B : Type v)
    [CommRing A] [IsLocalRing A] [IsDomain A]
    [CommRing B] [IsLocalRing B] [Algebra A B]
    [IsLocalHom (algebraMap A B)] [Algebra.EssFiniteType A B]
    (hinjective : Function.Injective (algebraMap A B))
    (hunibranch : IsGeometricallyUnibranch A)
    (hmaximal : Ideal.map (algebraMap A B) (IsLocalRing.maximalIdeal A) =
      IsLocalRing.maximalIdeal B)
    (hseparable : Algebra.IsSeparable (IsLocalRing.ResidueField A)
      (IsLocalRing.ResidueField B)) :
    ∃ (C : Type (max u v)) (_ : CommRing C) (_ : Algebra A C)
      (_ : Algebra C B) (_ : IsScalarTower A C B) (M : Submonoid C),
        Algebra.Etale A C ∧ IsLocalization M B

Tag 06DU, a lemma

Tag 06DU in the Stacks Project.

minimalPrimes_strictHenselization_tensorProduct_bijOn, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Miscellaneous/StrictHenselizationTensorProduct.lean, line 905.

The minimal primes of the strict henselization of the localization of A ⊗[k] B at the maximal ideal induced by the residue maps correspond bijectively, by contraction, to pairs of minimal primes of A and B.

theorem minimalPrimes_strictHenselization_tensorProduct_bijOn
    (k A B : Type u) [Field k] [IsAlgClosed k]
    [CommRing A] [Algebra k A] [HenselianLocalRing A] [IsSepClosed (ResidueField A)]
    [CommRing B] [Algebra k B] [HenselianLocalRing B] [IsSepClosed (ResidueField B)]
    (resA : A →ₐ[k] k) (resB : B →ₐ[k] k)
    (hresA : RingHom.ker resA.toRingHom = maximalIdeal A)
    (hresB : RingHom.ker resB.toRingHom = maximalIdeal B)
    (R C : Type u) [CommRing R] [IsLocalRing R] [CommRing C] [IsLocalRing C]
    [Algebra (A ⊗[k] B) R]
    [IsLocalization
      (Submonoid.comap
        (Algebra.TensorProduct.lift resA resB (fun _ _ ↦ Commute.all _ _)).toRingHom
        (nonZeroDivisors k)) R]
    [Algebra R C]
    (ksep : Type v) [Field ksep] [Algebra (ResidueField R) ksep]
    [IsSepClosure (ResidueField R) ksep]
    [IsLocalRing.IsHenselization R ksep C] :
    Set.BijOn
      (fun q : Ideal C ↦
        (q.comap ((algebraMap R C).comp
          ((algebraMap (A ⊗[k] B) R).comp
            (Algebra.TensorProduct.includeLeft : A →ₐ[k] A ⊗[k] B).toRingHom)),
         q.comap ((algebraMap R C).comp
          ((algebraMap (A ⊗[k] B) R).comp
            (Algebra.TensorProduct.includeRight : B →ₐ[k] A ⊗[k] B).toRingHom))))
      (minimalPrimes C) ((minimalPrimes A) ×ˢ (minimalPrimes B))

Sources

• Section 15.109, Tag 0GS4
• AxiomMath/Stacks at 5ed82d14

Leancomap_ne_bot_of_localization_or_discrete_fibers · exists_etale_away_of_isGeometricallyUnibranch_of_isUnramifiedAt · exists_etale_isLocalization_of_isGeometricallyUnibranch · minimalPrimes_strictHenselization_tensorProduct_bijOn

Section15.110 Branches of the completion#

Section 15.110 of the Stacks Project, Tag 0C27. 8 tags here are carried by 8 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0C28, a lemma

Tag 0C28 in the Stacks Project.

numberOfBranches_le_and_numberOfGeometricBranches_le_completion, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Completion/BranchCounts.lean, line 425.

For a Noetherian local ring A with henselization Aʰ and strict henselization Aˢʰ, there is a map Aʰ → Â along which contraction is surjective on minimal primes,  ⊗[Aʰ] Aˢʰ is a strict henselization of Â, and completion decreases neither the number of branches nor the number of geometric branches.

theorem numberOfBranches_le_and_numberOfGeometricBranches_le_completion
    (A Ah Ash : Type u)
    [CommRing A] [IsLocalRing A] [IsNoetherianRing A]
    [CommRing Ah] [IsLocalRing Ah] [Algebra A Ah]
    [IsHenselization A (ResidueField A) Ah]
    (κsep : Type v) [Field κsep] [Algebra (ResidueField A) κsep]
    [IsSepClosure (ResidueField A) κsep]
    [CommRing Ash] [IsLocalRing Ash] [Algebra A Ash] [Algebra Ah Ash]
    [IsScalarTower A Ah Ash] [IsHenselization A κsep Ash]
    [IsLocalHom (algebraMap Ah Ash)] :
    let Ahat := AdicCompletion (maximalIdeal A) A
    ∃ hAhatLocal : IsLocalRing Ahat,
      letI : IsLocalRing Ahat := hAhatLocal
      ∃ hAhatMapLocal : IsLocalHom (algebraMap A Ahat),
        letI : IsLocalHom (algebraMap A Ahat) := hAhatMapLocal
        ∃ f : Ah →+* Ahat,
          (∀ x : A, f (algebraMap A Ah x) = algebraMap A Ahat x) ∧
          Set.SurjOn (fun q => Ideal.comap f q) (minimalPrimes Ahat) (minimalPrimes Ah) ∧
          letI : Algebra Ah Ahat := f.toAlgebra
          ∃ hAhatHenselization : IsHenselization Ahat (ResidueField Ahat) Ahat,
            letI : IsHenselization Ahat (ResidueField Ahat) Ahat := hAhatHenselization
            numberOfBranches A Ah ≤ numberOfBranches Ahat Ahat ∧
            ∃ algKhat : Algebra (ResidueField Ahat) κsep,
              letI : Algebra (ResidueField Ahat) κsep := algKhat
              (∀ x : ResidueField A,
                algebraMap (ResidueField Ahat) κsep
                    (ResidueField.map (algebraMap A Ahat) x) =
                  algebraMap (ResidueField A) κsep x) ∧
              ∃ hKsep : IsSepClosure (ResidueField Ahat) κsep,
                letI : IsSepClosure (ResidueField Ahat) κsep := hKsep
                ∃ hTensorLocal : IsLocalRing (Ahat ⊗[Ah] Ash),
                  letI : IsLocalRing (Ahat ⊗[Ah] Ash) := hTensorLocal
                  ∃ hTensorHenselization : IsHenselization Ahat κsep (Ahat ⊗[Ah] Ash),
                    letI : IsHenselization Ahat κsep (Ahat ⊗[Ah] Ash) := hTensorHenselization
                    numberOfGeometricBranches A κsep Ash ≤
                      numberOfGeometricBranches Ahat κsep (Ahat ⊗[Ah] Ash)

Tag 0C29, a lemma

Tag 0C29 in the Stacks Project.

numberOfBranches_eq_completion_iff_radical_map_minimalPrime_isPrime, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Completion/BranchCounts.lean, line 288.

A Noetherian local ring and its completion have the same number of branches if and only if the radical of the extension of every minimal prime of a henselization to the completion is prime.

theorem numberOfBranches_eq_completion_iff_radical_map_minimalPrime_isPrime
    (A Ah : Type u)
    [CommRing A] [IsLocalRing A] [IsNoetherianRing A]
    [CommRing Ah] [IsLocalRing Ah] [Algebra A Ah]
    [IsHenselization A (ResidueField A) Ah] :
    let Ahat := AdicCompletion (maximalIdeal A) A
    ∃ hAhatLocal : IsLocalRing Ahat,
      letI : IsLocalRing Ahat := hAhatLocal
      ∃ hAhatMapLocal : IsLocalHom (algebraMap A Ahat),
        letI : IsLocalHom (algebraMap A Ahat) := hAhatMapLocal
        ∃ f : Ah →+* Ahat,
          (∀ x : A, f (algebraMap A Ah x) = algebraMap A Ahat x) ∧
          ∃ hAhatHenselization : IsHenselization Ahat (ResidueField Ahat) Ahat,
            letI : IsHenselization Ahat (ResidueField Ahat) Ahat := hAhatHenselization
            (numberOfBranches A Ah =
                numberOfBranches Ahat Ahat ↔
              ∀ q ∈ minimalPrimes Ah, (Ideal.map f q).radical.IsPrime)

Tag 0C2A, a lemma

Tag 0C2A in the Stacks Project.

exists_surjective_bijective_away_prod, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Localization/Idempotent.lean, line 844.

Stacks Tag 0C2A, Lemma 15.110.3. Let A be a ring, I ⊆ A a finitely generated ideal, and A → C a ring map such that for every f ∈ I the induced map A_f → C_f is localization at an idempotent. Then there is a surjective ring map A → C' such that for every f ∈ I the canonical ring map A_f → (C × C')_f is an isomorphism (here expressed as a bijective ring homomorphism).

Note: C' is constructed as a quotient of A, so it lives in the universe u of A.

theorem exists_surjective_bijective_away_prod {A : Type u} {C : Type v} [CommRing A] [CommRing C]
  [Algebra A C]
    (I : Ideal A) (hI : I.FG)
    (h : ∀ f ∈ I, IsLocalizationAtIdempotent
      (IsLocalization.Away.map (Localization.Away f)
        (Localization.Away (algebraMap A C f)) (algebraMap A C) f)) :
    ∃ (C' : Type u) (_ : CommRing C') (_ : Algebra A C'),
      Function.Surjective (algebraMap A C') ∧
      ∀ f ∈ I, Function.Bijective
        (IsLocalization.Away.map (Localization.Away f)
          (Localization.Away (algebraMap A (C × C') f)) (algebraMap A (C × C')) f)

Tag 0ALR, a lemma

Tag 0ALR in the Stacks Project.

exists_finiteType_algEquiv_adicCompletion_of_cotangent_annihilated, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/AdicCompletion/Algebraization.lean, line 521.

Stacks Tag 0ALR. A quotient of the adic completion of a finite-type algebra is itself the adic completion of a finite-type algebra if its conormal module is annihilated by a power of the defining ideal.

theorem exists_finiteType_algEquiv_adicCompletion_of_cotangent_annihilated
    (A : Type u) (B : Type v) (C : Type w) [CommRing A] [IsNoetherianRing A]
    [CommRing B] [Algebra A B] [Algebra.FiniteType A B]
    [CommRing C] [Algebra A C] (I : Ideal A)
    (φ : AdicCompletion (I.map (algebraMap A B)) B →ₐ[A] C)
    (hφ : Function.Surjective φ) (c : ℕ)
    (hJ : I ^ c • (⊤ : Submodule A (RingHom.ker φ.toRingHom).Cotangent) = ⊥) :
    ∃ (D : Type (max u v w)) (_ : CommRing D) (_ : Algebra A D),
      Algebra.FiniteType A D ∧
        Nonempty (C ≃ₐ[A] AdicCompletion (I.map (algebraMap A D)) D)

Tag 0C2B, a lemma

Tag 0C2B in the Stacks Project.

exists_minimalPrime_henselization_radical_map_eq_of_quotient_dimension_one, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Completion/HenselizationToCompletion.lean, line 716.

A minimal prime defining a one-dimensional component of the completion of a Noetherian local ring is the radical of the extension of a minimal prime of its henselization.

theorem exists_minimalPrime_henselization_radical_map_eq_of_quotient_dimension_one
    (A : Type u) (Ah : Type v)
    [CommRing A] [IsLocalRing A] [IsNoetherianRing A]
    [CommRing Ah] [IsLocalRing Ah] [Algebra A Ah]
    [IsHenselization A (ResidueField A) Ah] :
    let Ahat := AdicCompletion (maximalIdeal A) A
    ∃ f : Ah →+* Ahat,
      (∀ x : A, f (algebraMap A Ah x) = algebraMap A Ahat x) ∧
        ∀ q : Ideal Ahat, q ∈ minimalPrimes Ahat →
          ringKrullDim (Ahat ⧸ q) = 1 →
            ∃ qh : Ideal Ah, qh ∈ minimalPrimes Ah ∧ q = (Ideal.map f qh).radical

Tag 0C2C, a lemma

Tag 0C2C in the Stacks Project.

not_isConnected_puncturedSpectrum_adicCompletion_iff_henselization, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Completion/PuncturedSpectrum.lean, line 122.

The punctured spectrum of the completion of a Noetherian local ring is disconnected if and only if the punctured spectrum of its henselization is disconnected.

theorem not_isConnected_puncturedSpectrum_adicCompletion_iff_henselization
    (A : Type u) (Ah : Type v)
    [CommRing A] [IsLocalRing A] [IsNoetherianRing A]
    [CommRing Ah] [IsLocalRing Ah] [Algebra A Ah]
    [IsHenselization A (ResidueField A) Ah] :
    let Ahat := AdicCompletion (maximalIdeal A) A
    (¬ _root_.IsConnected {p : PrimeSpectrum Ahat | p.asIdeal ≠ maximalIdeal Ahat}) ↔
      ¬ _root_.IsConnected {p : PrimeSpectrum Ah | p.asIdeal ≠ maximalIdeal Ah}

Tag 0C2D, a lemma

Tag 0C2D in the Stacks Project.

numberOfBranches_and_numberOfGeometricBranches_eq_adicCompletion_of_dimension_one, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Completion/BranchCounts.lean, line 680.

Completion preserves the ordinary and the geometric number of branches of a one-dimensional Noetherian local ring.

theorem numberOfBranches_and_numberOfGeometricBranches_eq_adicCompletion_of_dimension_one
    (A Ah AhatH Ash AhatSh : Type u) [CommRing A] [IsNoetherianRing A] [IsLocalRing A]
    [CommRing Ah] [IsLocalRing Ah] [Algebra A Ah]
    [IsHenselization A (ResidueField A) Ah]
    [CommRing AhatH] [IsLocalRing AhatH]
    [Algebra (AdicCompletion (maximalIdeal A) A) AhatH]
    [IsHenselization (AdicCompletion (maximalIdeal A) A)
      (ResidueField (AdicCompletion (maximalIdeal A) A)) AhatH]
    (ksep : Type v) [Field ksep] [Algebra (ResidueField A) ksep]
    [IsSepClosure (ResidueField A) ksep]
    [CommRing Ash] [IsLocalRing Ash] [Algebra A Ash] [IsHenselization A ksep Ash]
    (khatsep : Type w) [Field khatsep]
    [Algebra (ResidueField (AdicCompletion (maximalIdeal A) A)) khatsep]
    [IsSepClosure (ResidueField (AdicCompletion (maximalIdeal A) A)) khatsep]
    [CommRing AhatSh] [IsLocalRing AhatSh]
    [Algebra (AdicCompletion (maximalIdeal A) A) AhatSh]
    [IsHenselization (AdicCompletion (maximalIdeal A) A) khatsep AhatSh]
    (hdim : ringKrullDim A = 1) :
    numberOfBranches A Ah =
        numberOfBranches (AdicCompletion (maximalIdeal A) A) AhatH ∧
      numberOfGeometricBranches A ksep Ash =
        numberOfGeometricBranches
          (AdicCompletion (maximalIdeal A) A) khatsep AhatSh

Tag 0C2E, a lemma

Tag 0C2E in the Stacks Project.

isNagataRing_and_numberOfBranches_and_numberOfGeometricBranches_eq_adicCompletion, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Branches/Completion/GeometricallyNormalFormalFibres.lean, line 444. References: EGA IV (7.6.1), “A' finite over A”; EGA IV (7.8.3), “(vi) local case”.

A Noetherian local ring whose formal fibres are geometrically normal is Nagata, and its ordinary and geometric branch counts agree with those of its completion.

theorem isNagataRing_and_numberOfBranches_and_numberOfGeometricBranches_eq_adicCompletion
    (A Ah AhatH Ash AhatSh : Type u) [CommRing A] [IsNoetherianRing A] [IsLocalRing A]
    [CommRing Ah] [IsLocalRing Ah] [Algebra A Ah]
    [IsHenselization A (ResidueField A) Ah]
    [CommRing AhatH] [IsLocalRing AhatH]
    [Algebra (AdicCompletion (maximalIdeal A) A) AhatH]
    [IsHenselization (AdicCompletion (maximalIdeal A) A)
      (ResidueField (AdicCompletion (maximalIdeal A) A)) AhatH]
    (ksep : Type v) [Field ksep] [Algebra (ResidueField A) ksep]
    [IsSepClosure (ResidueField A) ksep]
    [CommRing Ash] [IsLocalRing Ash] [Algebra A Ash] [IsHenselization A ksep Ash]
    (khatsep : Type w) [Field khatsep]
    [Algebra (ResidueField (AdicCompletion (maximalIdeal A) A)) khatsep]
    [IsSepClosure (ResidueField (AdicCompletion (maximalIdeal A) A)) khatsep]
    [CommRing AhatSh] [IsLocalRing AhatSh]
    [Algebra (AdicCompletion (maximalIdeal A) A) AhatSh]
    [IsHenselization (AdicCompletion (maximalIdeal A) A) khatsep AhatSh]
    (hformal : ∀ q : PrimeSpectrum A,
      Algebra.IsGeometricallyNormal q.asIdeal.ResidueField
        (q.asIdeal.Fiber (AdicCompletion (maximalIdeal A) A))) :
    IsNagataRing A ∧
      numberOfBranches A Ah =
        numberOfBranches (AdicCompletion (maximalIdeal A) A) AhatH ∧
      numberOfGeometricBranches A ksep Ash =
        numberOfGeometricBranches
          (AdicCompletion (maximalIdeal A) A) khatsep AhatSh

Sources

• Section 15.110, Tag 0C27
• AxiomMath/Stacks at 5ed82d14

LeannumberOfBranches_le_and_numberOfGeometricBranches_le_completion · numberOfBranches_eq_completion_iff_radical_map_minimalPrime_isPrime · exists_surjective_bijective_away_prod · exists_finiteType_algEquiv_adicCompletion_of_cotangent_annihilated · exists_minimalPrime_henselization_radical_map_eq_of_quotient_dimension_one · not_isConnected_puncturedSpectrum_adicCompletion_iff_henselization · numberOfBranches_and_numberOfGeometricBranches_eq_adicCompletion_of_dimension_one · isNagataRing_and_numberOfBranches_and_numberOfGeometricBranches_eq_adicCompletion

Section15.111 Formally catenary rings#

Section 15.111 of the Stacks Project, Tag 0AW1. 6 tags here are carried by 7 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0AW2, a definition

Tag 0AW2 in the Stacks Project.

IsFormallyCatenary, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Catenary/Formal.lean, line 116. References: EGA IV (7.1.9).

A local ring A is formally catenary if, for every minimal prime p of A, the spectrum of the quotient of its maximal-ideal-adic completion by the extension of p is equidimensional.

def IsFormallyCatenary (A : Type*) [CommRing A] [IsLocalRing A] : Prop

Tag 0AW3, a lemma

Tag 0AW3 in the Stacks Project.

not_isUniversallyCatenary_of_not_isFormallyCatenary, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Catenary/Formal.lean, line 1715.

Stacks 0AW3. A Noetherian local ring which is not formally catenary is not universally catenary.

theorem not_isUniversallyCatenary_of_not_isFormallyCatenary
    (A : Type u) [CommRing A] [IsNoetherianRing A] [IsLocalRing A]
    (hA : ¬ IsFormallyCatenary A) : ¬ Ring.IsUniversallyCatenary A

Tag 0AW4, a lemma

Tag 0AW4 in the Stacks Project.

flat_local_catenary_equidim_fibre, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Equidimensional/FlatLocal.lean, line 557. The attribute's comment: “(1)”. References: EGA IV (7.1.3), “(ii) 'in particular'”.

Stacks 0AW4 (1). For a flat local map A → B of Noetherian local rings with B catenary and Spec(B) equidimensional, the fibre Spec(B / pB) is equidimensional for every prime p ⊆ A.

theorem flat_local_catenary_equidim_fibre (A B : Type*) [CommRing A] [CommRing B]
    [IsNoetherianRing A] [IsNoetherianRing B] [IsLocalRing A] [IsLocalRing B]
    [Algebra A B] [IsLocalHom (algebraMap A B)] [Module.Flat A B]
    (hcat : IrreducibleClosedsCodim.IsCatenarySpace (PrimeSpectrum B))
    (hequi : SpecEquidimensional B)
    (p : Ideal A) [p.IsPrime] :
    SpecEquidimensional (B ⧸ p.map (algebraMap A B))

flat_local_catenary_equidim_base, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Equidimensional/FlatLocal.lean, line 570. The attribute's comment: “(2)”. References: EGA IV (7.1.3), “(i)”; EGA IV (7.8.3), “(x)”; EGA IV (7.8.4), “partial”.

Stacks 0AW4 (2). For a flat local map A → B of Noetherian local rings with B catenary and Spec(B) equidimensional, the base A is catenary and Spec(A) is equidimensional.

theorem flat_local_catenary_equidim_base (A B : Type*) [CommRing A] [CommRing B]
    [IsNoetherianRing A] [IsNoetherianRing B] [IsLocalRing A] [IsLocalRing B]
    [Algebra A B] [IsLocalHom (algebraMap A B)] [Module.Flat A B]
    (hcat : IrreducibleClosedsCodim.IsCatenarySpace (PrimeSpectrum B))
    (hequi : SpecEquidimensional B) :
    IrreducibleClosedsCodim.IsCatenarySpace (PrimeSpectrum A) ∧ SpecEquidimensional A

Tag 0AW5, a lemma

Tag 0AW5 in the Stacks Project.

isUniversallyCatenary_of_isFormallyCatenary, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Catenary/Formal.lean, line 652. References: EGA IV (7.1.11), “(i)”.

Stacks 0AW5. A formally catenary Noetherian local ring is universally catenary.

theorem isUniversallyCatenary_of_isFormallyCatenary
    (A : Type u) [CommRing A] [IsNoetherianRing A] [IsLocalRing A]
    (hA : IsFormallyCatenary A) : Ring.IsUniversallyCatenary A

Tag 0AW6, a proposition

Tag 0AW6 in the Stacks Project.

isUniversallyCatenary_iff_isFormallyCatenary, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Catenary/Formal.lean, line 1742. References: EGA IV (7.1.11), “(i)”.

Ratliff's theorem (Stacks 0AW6). A Noetherian local ring is universally catenary if and only if it is formally catenary.

theorem isUniversallyCatenary_iff_isFormallyCatenary
    (A : Type u) [CommRing A] [IsNoetherianRing A] [IsLocalRing A] :
    Ring.IsUniversallyCatenary A ↔ IsFormallyCatenary A

Tag 0C2F, a lemma

Tag 0C2F in the Stacks Project.

StacksDraft.Catenary.isUniversallyCatenary_henselization_and_of_isUnibranch, a lemma in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Catenary/Henselization.lean, line 400.

Stacks 0C2F. A Noetherian local ring with geometrically normal formal fibres has a universally catenary henselization, and is itself universally catenary if it is unibranch.

public lemma isUniversallyCatenary_henselization_and_of_isUnibranch
    (A : Type u) (Ah : Type v)
    [CommRing A] [IsNoetherianRing A] [IsLocalRing A]
    [CommRing Ah] [IsLocalRing Ah] [Algebra A Ah]
    [IsHenselization A (ResidueField A) Ah]
    (hformal : ∀ q : PrimeSpectrum A,
      Algebra.IsGeometricallyNormal q.asIdeal.ResidueField
        (q.asIdeal.Fiber (AdicCompletion (maximalIdeal A) A))) :
    Ring.IsUniversallyCatenary Ah ∧ (IsUnibranch A → Ring.IsUniversallyCatenary A)

Sources

• Section 15.111, Tag 0AW1
• AxiomMath/Stacks at 5ed82d14

LeanIsFormallyCatenary · not_isUniversallyCatenary_of_not_isFormallyCatenary · flat_local_catenary_equidim_fibre · flat_local_catenary_equidim_base · isUniversallyCatenary_of_isFormallyCatenary · isUniversallyCatenary_iff_isFormallyCatenary · StacksDraft.Catenary.isUniversallyCatenary_henselization_and_of_isUnibranch

Section15.112 Group actions and integral closure#

Section 15.112 of the Stacks Project, Tag 0BRE. 11 tags here are carried by 16 declarations of the repository (all in StacksFrontier) and 3 registered declarations, quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0BRF, a lemma

Tag 0BRF in the Stacks Project.

MulSemiringAction.exists_monic_map_eq_X_sub_C_pow_card, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/Quotient.lean, line 1050.

Stacks 0BRF. Let φ : A → B be a surjective G-equivariant ring homomorphism for a finite group G. Every b ∈ B^G is matched by a monic polynomial P over A^G whose image under the restriction A^G → B^G of φ is (X - C b) ^ |G|.

The witness is the orbit polynomial ∏_{g ∈ G} (X - C (g • a)) of any lift a of b: its coefficients are G-invariant, and each factor maps to X - C b, since φ (g • a) = g • b = b.

theorem exists_monic_map_eq_X_sub_C_pow_card
    {G : Type*} [Group G] [Finite G]
    {A B : Type*} [CommRing A] [CommRing B]
    [MulSemiringAction G A] [MulSemiringAction G B]
    (φ : A →+*[G] B) (hφ : Function.Surjective φ)
    (b : FixedPoints.subring B G) :
    ∃ P : Polynomial (FixedPoints.subring A G), P.Monic ∧
      Polynomial.map
          ((φ : A →+* B).restrict (FixedPoints.subring A G) (FixedPoints.subring B G)
            (by
              intro a ha g
              change g • φ a = φ a
              rw [← map_smul φ, ha g])) P =
        (Polynomial.X - Polynomial.C b) ^ Nat.card G

Tag 09EG, a lemma

Tag 09EG in the Stacks Project.

FixedPoints.quotientComparison_injective, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/Quotient.lean, line 376.

(Well-definedness of R^G/I^G ⊆ (R/I)^G.) The canonical comparison map φ is injective, so R^G / I^G is (isomorphic to) a subring of (R/I)^G.

Proof: φ is Ideal.Quotient.lift of toFixedQuot : R^G → (R/I)^G, so it suffices to show ker(toFixedQuot) = I^G = invariantIdeal. If a ∈ R^G maps to 0, then mk_I (a : R) = 0 in R/I, i.e. a ∈ I; and a ∈ R^G, so a ∈ invariantIdeal R G I by definition (I.comap (R^G).subtype). Conversely the quotient by invariantIdeal already kills these, so the lifted map is injective.

theorem quotientComparison_injective :
    Function.Injective (compMap R G I hcompat)

FixedPoints.compMap_isIntegral, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/Quotient.lean, line 461.

(Integrality.) Via the comparison map φ, the ring (R/I)^G is an integral extension of R^G / I^G: every element of (R/I)^G is a root of a monic polynomial with coefficients in the image of R^G / I^G.

Proof: by Algebra.IsIntegral.mk, fix b ∈ (R/I)^G. Take the monic witness P over R^G/I^G from compMap_polyWitness b, with map φ P = (X - C b)^{|G|}. Then under the algebra structure φ.toAlgebra, aeval b P = (map φ P).eval b = ((X - C b)^\{|G|}).eval b = (b - b)^\{|G|} = 0 (using Polynomial.aeval_def, Polynomial.eval_map, Polynomial.eval_pow, Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C), and P is monic, so b is integral.

theorem compMap_isIntegral :
    letI := (compMap R G I hcompat).toAlgebra
    Algebra.IsIntegral (FixedPoints.subring R G ⧸ invIdeal R G I)
      (FixedPoints.subring (R ⧸ I) G)

FixedPoints.quotientComparison_comap_isHomeomorph, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/Quotient.lean, line 475.

(Homeomorphism on spectra.) The induced map on prime spectra Spec(φ) : Spec((R/I)^G) → Spec(R^G/I^G) is a homeomorphism for the Zariski topologies. This is PrimeSpectrum.isHomeomorph_comap (the abstract form of Stacks 09EF) applied to the two facts read off from quotientComparison_polyWitness and quotientComparison_injective: every b ∈ (R/I)^G has b ^ |G| in the range of φ, and ker φ = 0 is nil.

theorem quotientComparison_comap_isHomeomorph :
    IsHomeomorph (PrimeSpectrum.comap (compMap R G I hcompat))

FixedPoints.quotientComparison_residueField_isPurelyInseparable, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/Quotient.lean, line 504.

(Purely inseparable residue extensions.) For every prime 𝔮 ⊆ (R/I)^G with contraction 𝔭 = φ⁻¹(𝔮) ⊆ R^G/I^G, the induced extension of residue fields κ(𝔭) → κ(𝔮) is purely inseparable. This is the abstract form of Stacks 09EF, isPurelyInseparable_residueField_of_sub_C_pow_witness, applied to the monic witness quotientComparison_polyWitness.

theorem quotientComparison_residueField_isPurelyInseparable
    (q : Ideal (FixedPoints.subring (R ⧸ I) G)) [q.IsPrime] :
    letI := (Ideal.ResidueField.map (q.comap (compMap R G I hcompat)) q
      (compMap R G I hcompat) rfl).toAlgebra
    IsPurelyInseparable (Ideal.ResidueField (q.comap (compMap R G I hcompat)))
      (Ideal.ResidueField q)

Tag 0H34, a lemma

Tag 0H34 in the Stacks Project.

Stacks.Invariant.QuotientComparison.charpolyFixed_map_mk_of_mem, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/Quotient.lean, line 861.

Stacks 0H34 (tag 0H34 = Lemma 15.112.3). For x ∈ JR, the characteristic polynomial ∏_{σ ∈ G} (T - σ x) = T^n + a₁T^{n-1} + … + aₙ has all its non-leading coefficients aᵢ in J; equivalently, reducing the coefficients of charpolyFixed G x modulo J gives X ^ n.

Implementation note: the coefficients aᵢ lie in R^G, and what has to be shown is that they lie in J — the ideal of R^G — and not merely in the a priori larger JR ∩ R^G. Writing x = Σⱼ fⱼ bⱼ with fⱼ ∈ J ⊆ R^G and bⱼ ∈ R, invariance of fⱼ gives σ x = Σⱼ fⱼ (σ bⱼ), so each elementary symmetric coefficient is a sum of products each carrying a factor ∏ fⱼ ∈ J, and symmetrizing over the orbit keeps the remaining cofactor in R^G; hence aᵢ ∈ J · R^G = J.

theorem charpolyFixed_map_mk_of_mem [Fintype G] (J : Ideal (FixedPoints.subring R G))
    (x : R) (hx : x ∈ JRideal J) :
    Polynomial.map (Ideal.Quotient.mk J) (charpolyFixed G x)
      = (Polynomial.X : Polynomial ((FixedPoints.subring R G) ⧸ J)) ^ (Fintype.card G)

Tag 0H35, a lemma

Tag 0H35 in the Stacks Project.

Stacks.Invariant.QuotientComparison.exists_monic_map_eq_X_sub_C_pow, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/Quotient.lean, line 898. The attribute's comment: “(1)”.

(Lifting characteristic-polynomial form.) For every b ∈ (R / JR)^G there is a monic polynomial P ∈ (R^G / J)[T] whose image under Polynomial.map (invariantComparison J) equals (X - C b) ^ n, where n = Fintype.card G.

theorem exists_monic_map_eq_X_sub_C_pow [Fintype G] (J : Ideal (FixedPoints.subring R G))
    (b : FixedPoints.subring (R ⧸ JRideal J) G) :
    ∃ P : Polynomial ((FixedPoints.subring R G) ⧸ J),
      P.Monic ∧
        P.map (invariantComparison J)
          = (Polynomial.X - Polynomial.C b) ^ (Fintype.card G)

Stacks.Invariant.QuotientComparison.X_sub_C_pow_eq_X_pow_of_mem_ker, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/Quotient.lean, line 928. The attribute's comment: “(2)”.

(Kernel elements are nilpotent-like.) For every a ∈ ker(invariantComparison J) we have (X - C a) ^ n = X ^ n in (R^G / J)[T], where n = Fintype.card G.

theorem X_sub_C_pow_eq_X_pow_of_mem_ker [Fintype G] (J : Ideal (FixedPoints.subring R G))
    (a : (FixedPoints.subring R G) ⧸ J) (ha : a ∈ RingHom.ker (invariantComparison J)) :
    (Polynomial.X - Polynomial.C a) ^ (Fintype.card G)
      = (Polynomial.X : Polynomial ((FixedPoints.subring R G) ⧸ J)) ^ (Fintype.card G)

Stacks.Invariant.QuotientComparison.invariantComparison_isIntegral, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/Quotient.lean, line 974. The attribute's comment: “final statement, integrality”.

(In particular: integrality.) The comparison map Φ : R^G / J → (R / JR)^G is an integral ring map: every b ∈ (R / JR)^G is a root of the monic polynomial over R^G / J supplied by exists_monic_map_eq_X_sub_C_pow, because that polynomial maps to (X - C b) ^ n, which vanishes at b.

theorem invariantComparison_isIntegral [Fintype G] (J : Ideal (FixedPoints.subring R G)) :
    (invariantComparison J).IsIntegral

Stacks.Invariant.QuotientComparison.invariantComparison_comap_isHomeomorph, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/Quotient.lean, line 986. The attribute's comment: “final statement, homeomorphism on spectra”.

(In particular: homeomorphism on spectra.) Spec(Φ) : Spec((R / JR)^G) → Spec(R^G / J) is a homeomorphism. This is PrimeSpectrum.isHomeomorph\_comap (Stacks 0BR8) applied to the two facts extracted from parts (1) and (2): every b ∈ (R / JR)^G has b ^ n in the range of Φ (exists_apply_eq_pow), and ker Φ is nil (ker_invariantComparison_le_nilradical).

theorem invariantComparison_comap_isHomeomorph [Fintype G] (J : Ideal (FixedPoints.subring R G)) :
    IsHomeomorph (PrimeSpectrum.comap (invariantComparison J))

Stacks.Invariant.QuotientComparison.invariantComparison_residueField_isPurelyInseparable, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/Quotient.lean, line 1000. The attribute's comment: “final statement, purely inseparable residue fields”.

(In particular: purely inseparable residue extensions.) For every prime 𝔮 ⊆ (R / JR)^G with contraction 𝔭 = Φ⁻¹(𝔮) ⊆ R^G / J, the induced extension of residue fields κ(𝔭) → κ(𝔮) is purely inseparable. This is the abstract form of Stacks 09EF (isPurelyInseparable_residueField_of_sub_C_pow_witness, proved with Stacks 09EG) applied to the monic witness of part (1).

theorem invariantComparison_residueField_isPurelyInseparable [Fintype G]
    (J : Ideal (FixedPoints.subring R G))
    (q : Ideal (FixedPoints.subring (R ⧸ JRideal J) G)) [q.IsPrime] :
    letI := (Ideal.ResidueField.map (q.comap (invariantComparison J)) q
      (invariantComparison J) rfl).toAlgebra
    IsPurelyInseparable (Ideal.ResidueField (q.comap (invariantComparison J)))
      (Ideal.ResidueField q)

Tag 0H36, a remark

Tag 0H36 in the Stacks Project.

Stacks.Invariant.QuotientComparison.invariantComparison_bijective_of_isUnit_card, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/Quotient.lean, line 1107.

Stacks 0H36. If the order of the finite group G acting on R is invertible in R, the canonical comparison map Φ : R^G / J → (R / JR)^G is an isomorphism.

Injectivity: for a ∈ ker Φ the identity (X - C a) ^ n = X ^ n of X_sub_C_pow_eq_X_pow_of_mem_ker has vanishing coefficient in degree n - 1, i.e. n • a = 0, and n is invertible. Surjectivity: n⁻¹ ∑_{g ∈ G} g • r is an invariant lift of any invariant class r̄.

theorem invariantComparison_bijective_of_isUnit_card
    {R : Type*} [CommRing R] {G : Type*} [Group G] [Fintype G] [MulSemiringAction G R]
    (J : Ideal (FixedPoints.subring R G)) (hG : IsUnit (Fintype.card G : R)) :
    Function.Bijective (invariantComparison J)

Tag 0BRG, a lemma

Tag 0BRG in the Stacks Project.

FixedPoints.exists_monic_polynomial_and_pow_eq_of_invariant_tensorProduct, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/BaseChange.lean, line 399.

Stacks 0BRG. Invariants in A ⊗[R^G] R lift to monic degree-|G| polynomial relations over A; moreover, an element in the kernel of the canonical map from A to those invariants satisfies (X - C a) ^ |G| = X ^ |G|.

Present A as a quotient of MvPolynomial A R^G. Base change along R^G → R presents A ⊗[R^G] R as the corresponding quotient of MvPolynomial A R, whose invariant subring is MvPolynomial A R^G (fixedPolynomialEquiv), and the kernel of the base-changed presentation is the extension JR of the kernel J of the original one (baseChangePresentation_ker_eq_JRideal). Both assertions are therefore the corresponding assertions of Stacks 0H35 for J, transported along the presentation.

theorem exists_monic_polynomial_and_pow_eq_of_invariant_tensorProduct
    (R : Type u) [CommRing R]
    (G : Type v) [Group G] [Finite G] [MulSemiringAction G R]
    (A : Type w) [CommRing A] [Algebra (FixedPoints.subring R G) A]
    [MulSemiringAction G (A ⊗[(FixedPoints.subring R G)] R)]
    (smul_tmul : ∀ (g : G) (a : A) (r : R),
      g • (a ⊗ₜ[(FixedPoints.subring R G)] r) = a ⊗ₜ[(FixedPoints.subring R G)] (g • r)) :
    let ι : A →+* FixedPoints.subring (A ⊗[(FixedPoints.subring R G)] R) G :=
      RingHom.codRestrict
        (Algebra.TensorProduct.includeLeftRingHom (R := FixedPoints.subring R G))
        (FixedPoints.subring (A ⊗[(FixedPoints.subring R G)] R) G)
        (fun a g => by
          simpa [Algebra.TensorProduct.includeLeftRingHom_apply] using smul_tmul g a 1)
    (∀ b : FixedPoints.subring (A ⊗[(FixedPoints.subring R G)] R) G,
      ∃ P : Polynomial A, P.Monic ∧
        Polynomial.map ι P = (Polynomial.X - Polynomial.C b) ^ Nat.card G) ∧
    ∀ a : A, a ∈ RingHom.ker ι →
      (Polynomial.X - Polynomial.C a) ^ Nat.card G =
        (Polynomial.X : Polynomial A) ^ Nat.card G

Tag 0BRH, a lemma

Tag 0BRH in the Stacks Project.

FixedPoints.canonicalMap_to_invariants_bijective_of_flat_and_isIntegral_isHomeomorph, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/BaseChange.lean, line 500.

Stacks 0BRH. The canonical map from a flat algebra over an invariant ring to the invariants after extension of scalars is bijective. In general, the map is integral, is a homeomorphism on prime spectra, and induces purely inseparable residue field extensions.

The unconditional half follows from Stacks 0BRG: the monic witnesses generate the target over A, and the kernel is nil, so PrimeSpectrum.isHomeomorph_comap_and_isPurelyInseparable_and_baseChange_of_adjoin_eq_top applies. In the flat case A ⊗[R^G] R → ∏_{g ∈ G} (A ⊗[R^G] R) has A as the equalizer of the two maps r ↦ (g • r) and r ↦ r, because taking equalizers commutes with a flat base change.

theorem canonicalMap_to_invariants_bijective_of_flat_and_isIntegral_isHomeomorph
    (R : Type u) [CommRing R]
    (G : Type v) [Group G] [Finite G] [MulSemiringAction G R]
    (A : Type w) [CommRing A] [Algebra (FixedPoints.subring R G) A]
    [MulSemiringAction G (A ⊗[(FixedPoints.subring R G)] R)]
    (smul_tmul : ∀ (g : G) (a : A) (r : R),
      g • (a ⊗ₜ[(FixedPoints.subring R G)] r) =
        a ⊗ₜ[(FixedPoints.subring R G)] (g • r)) :
    let ι : A →+* FixedPoints.subring (A ⊗[(FixedPoints.subring R G)] R) G :=
      RingHom.codRestrict
        (Algebra.TensorProduct.includeLeftRingHom (R := FixedPoints.subring R G))
        (FixedPoints.subring (A ⊗[(FixedPoints.subring R G)] R) G)
        (fun a g => by
          simpa [Algebra.TensorProduct.includeLeftRingHom_apply] using smul_tmul g a 1)
    (Module.Flat (FixedPoints.subring R G) A → Function.Bijective ι) ∧
      ι.IsIntegral ∧
      IsHomeomorph (PrimeSpectrum.comap ι) ∧
      ∀ (p : Ideal A) [p.IsPrime]
        (q : Ideal (FixedPoints.subring (A ⊗[(FixedPoints.subring R G)] R) G)) [q.IsPrime]
        (hpq : p = q.comap ι),
        (Ideal.ResidueField.map p q ι hpq).IsPurelyInseparable

Tag 0BRI, a lemma

Tag 0BRI in the Stacks Project.

Declarations registered for the tag:

• Algebra.IsInvariant.exists_smul_of_under_eq registered with add_to_stacks in StacksFrontier/RingTheory/Invariant/Primes.lean, line 60 (StacksFrontier).

Tag 0BRJ, a lemma

Tag 0BRJ in the Stacks Project.

Declarations registered for the tag:

• Ideal.IsFractionRing.normal registered with add_to_stacks in StacksFrontier/RingTheory/Invariant/Primes.lean, line 62 (StacksFrontier): “normality, hence algebraicity, of the residue field extension”.
• IsFractionRing.stabilizerHom_surjective registered with add_to_stacks in StacksFrontier/RingTheory/Invariant/Primes.lean, line 65 (StacksFrontier): “surjectivity of the decomposition group onto the relative automorphisms”. References: SGA 1 V (1.1), “more general”; SGA 1 V (1.3), “special case”.

Tag 0BRK, a lemma

Tag 0BRK in the Stacks Project.

IsIntegralClosure.exists_smul_of_under_eq_and_residue_normal_stabilizerHom_surjective, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/Primes.lean, line 337. References: SGA 1 V (1.1), “special case”; SGA 1 V (1.3), “special case”.

In a possibly infinite Galois extension, the primes of an integral closure above a fixed prime form one Galois orbit. The induced residue-field extension is normal, and its decomposition group surjects onto the relative automorphism group.

theorem exists_smul_of_under_eq_and_residue_normal_stabilizerHom_surjective
    (A : Type uA) (K : Type uK) (L : Type uL) (B : Type uB)
    [CommRing A] [CommRing B] [Field K] [Field L]
    [Algebra A K] [Algebra B L] [IsFractionRing A K] [IsFractionRing B L]
    [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L]
    [IsScalarTower A B L] [IsIntegrallyClosed A] [IsIntegralClosure B A L]
    [IsGalois K L] :
    letI := IsIntegralClosure.MulSemiringAction A K L B
    letI : SMulCommClass Gal(L/K) A B :=
      ⟨fun σ a b ↦ by
        simp only [Algebra.smul_def, smul_mul']
        congr 1
        exact (galRestrict A K L B σ).commutes a⟩
    (∀ (Q Q' : Ideal B) [Q.IsPrime] [Q'.IsPrime],
      Q.under A = Q'.under A → ∃ σ : Gal(L/K), Q' = σ • Q) ∧
    ∀ (P : Ideal A) (Q : Ideal B) [Q.IsPrime] [Q.LiesOver P],
      ∀ (k l : Type*) [Field k] [Field l] [Algebra k l]
        [Algebra (A ⧸ P) k] [IsFractionRing (A ⧸ P) k]
        [Algebra (B ⧸ Q) l] [IsFractionRing (B ⧸ Q) l]
        [Algebra (A ⧸ P) l] [IsScalarTower (A ⧸ P) (B ⧸ Q) l]
        [IsScalarTower (A ⧸ P) k l],
        Normal k l ∧
          Function.Surjective (IsFractionRing.stabilizerHom Gal(L/K) P Q k l)

Tag 0BSX, a lemma

Tag 0BSX in the Stacks Project.

IsIntegralClosure.image_stabilizer_and_inertia_restrictNormalHom, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Invariant/Primes.lean, line 403.

In a tower of possibly infinite Galois extensions, restriction maps the decomposition group and the inertia group at a prime of the larger integral closure onto the corresponding groups at its contraction to the smaller integral closure.

theorem image_stabilizer_and_inertia_restrictNormalHom
    (A : Type uA) (K : Type uK) (L : Type uL) (M : Type uM)
    (B : Type uB) (C : Type uC)
    [CommRing A] [Field K] [Field L] [Field M] [CommRing B] [CommRing C]
    [Algebra A K] [Algebra K L] [Algebra L M] [Algebra K M]
    [Algebra A L] [Algebra A M] [IsScalarTower A K L] [IsScalarTower K L M]
    [IsScalarTower A K M] [IsScalarTower A L M]
    [Algebra A B] [Algebra B L] [IsScalarTower A B L]
    [Algebra A C] [Algebra C M] [IsScalarTower A C M]
    [Algebra B C] [Algebra B M] [IsScalarTower A B C]
    [IsScalarTower B L M] [IsScalarTower B C M]
    [IsFractionRing A K] [IsFractionRing B L] [IsFractionRing C M]
    [IsIntegrallyClosed A] [IsIntegralClosure B A L] [IsIntegralClosure C A M]
    [IsGalois K L] [IsGalois K M]
    (r : Ideal C) [r.IsPrime] :
    letI := IsIntegralClosure.MulSemiringAction A K M C
    letI : SMulCommClass Gal(M/K) A C :=
      ⟨fun σ a c ↦ by
        simp only [Algebra.smul_def, smul_mul']
        congr 1
        exact (galRestrict A K M C σ).commutes a⟩
    letI := IsIntegralClosure.MulSemiringAction A K L B
    letI : SMulCommClass Gal(L/K) A B :=
      ⟨fun σ a b ↦ by
        simp only [Algebra.smul_def, smul_mul']
        congr 1
        exact (galRestrict A K L B σ).commutes a⟩
    let q := r.under B
    let res : Gal(M/K) →* Gal(L/K) := AlgEquiv.restrictNormalHom L
    res '' (MulAction.stabilizer Gal(M/K) r : Set Gal(M/K)) =
        (MulAction.stabilizer Gal(L/K) q : Set Gal(L/K)) ∧
      res '' (r.inertia Gal(M/K) : Set Gal(M/K)) =
        (q.inertia Gal(L/K) : Set Gal(L/K))

Sources

• Section 15.112, Tag 0BRE
• AxiomMath/Stacks at 5ed82d14

LeanMulSemiringAction.exists_monic_map_eq_X_sub_C_pow_card · FixedPoints.quotientComparison_injective · FixedPoints.compMap_isIntegral · FixedPoints.quotientComparison_comap_isHomeomorph · FixedPoints.quotientComparison_residueField_isPurelyInseparable · Stacks.Invariant.QuotientComparison.charpolyFixed_map_mk_of_mem · Stacks.Invariant.QuotientComparison.exists_monic_map_eq_X_sub_C_pow · Stacks.Invariant.QuotientComparison.X_sub_C_pow_eq_X_pow_of_mem_ker

Section15.113 Extensions of discrete valuation rings#

Section 15.113 of the Stacks Project, Tag 0EXQ. 9 tags here are carried by 15 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 09E4, a definition

Tag 09E4 in the Stacks Project.

IsDiscreteValuationRing.IsExtension, a class in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Solution.lean, line 60.

An extension of discrete valuation rings is an injective local algebra map.

class IsExtension (A B : Type*) [CommRing A] [IsDomain A]
    [IsDiscreteValuationRing A] [CommRing B] [IsDomain B]
    [IsDiscreteValuationRing B] [Algebra A B] : Prop extends
    FaithfulSMul A B, IsLocalHom (algebraMap A B)

IsDiscreteValuationRing.ramificationIndex, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Solution.lean, line 67. References: SGA 7 IX (10.3); SGA 7 IX (9.7).

The ramification index of an extension of discrete valuation rings.

noncomputable abbrev ramificationIndex (A B : Type*) [CommRing A] [IsDomain A]
    [IsDiscreteValuationRing A] [CommRing B] [IsDomain B]
    [IsDiscreteValuationRing B] [Algebra A B] [IsExtension A B] : ℕ

IsDiscreteValuationRing.IsWeaklyUnramified, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Solution.lean, line 74.

An extension of DVRs is weakly unramified when its ramification index is one.

abbrev IsWeaklyUnramified (A B : Type*) [CommRing A] [IsDomain A]
    [IsDiscreteValuationRing A] [CommRing B] [IsDomain B]
    [IsDiscreteValuationRing B] [Algebra A B] [IsExtension A B] : Prop

IsDiscreteValuationRing.residueDegree, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Solution.lean, line 81.

The residue degree of an extension of discrete valuation rings.

noncomputable abbrev residueDegree (A B : Type*) [CommRing A] [IsDomain A]
    [IsDiscreteValuationRing A] [CommRing B] [IsDomain B]
    [IsDiscreteValuationRing B] [Algebra A B] [IsExtension A B] : ℕ

IsDiscreteValuationRing.ramificationIndex_characterization, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Solution.lean, line 213. References: SGA 7 IX (10.3), “special case”.

The ramification index is positive, is the unique positive exponent expressing the extended maximal ideal as a power of the maximal ideal upstairs, and is the exponent relating any choices of uniformizers up to a unit.

theorem ramificationIndex_characterization (A B : Type*) [CommRing A] [IsDomain A]
    [IsDiscreteValuationRing A] [CommRing B] [IsDomain B] [IsDiscreteValuationRing B]
    [Algebra A B] [IsExtension A B] :
    1 ≤ ramificationIndex A B ∧
      (IsLocalRing.maximalIdeal A).map (algebraMap A B) =
        IsLocalRing.maximalIdeal B ^ ramificationIndex A B ∧
      (∀ e : ℕ, 1 ≤ e →
        (IsLocalRing.maximalIdeal A).map (algebraMap A B) =
          IsLocalRing.maximalIdeal B ^ e →
        e = ramificationIndex A B) ∧
      ∀ πA : A, Irreducible πA → ∀ πB : B, Irreducible πB →
        ∃ u : Bˣ, algebraMap A B πA = (u : B) * πB ^ ramificationIndex A B

Tag 09E5, a lemma

Tag 09E5 in the Stacks Project.

IsDiscreteValuationRing.residueField_finite_and_ramificationIdx_mul_residueDegree_le_finrank, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/RamificationIndex.lean, line 213.

Let A → B be an injective local map of discrete valuation rings, with compatible fraction fields K → L. If L / K is finite, then the residue-field extension is finite and its degree, multiplied by the ramification index of B / A, is at most [L : K].

theorem residueField_finite_and_ramificationIdx_mul_residueDegree_le_finrank
    (A B K L : Type*)
    [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [CommRing B] [IsDomain B] [IsDiscreteValuationRing B]
    [Algebra A B] [FaithfulSMul A B] [IsLocalHom (algebraMap A B)]
    [Field K] [Algebra A K] [IsFractionRing A K]
    [Field L] [Algebra B L] [IsFractionRing B L]
    [Algebra A L] [Algebra K L]
    [IsScalarTower A B L] [IsScalarTower A K L]
    [Module.Finite K L] :
    Module.Finite (IsLocalRing.ResidueField A) (IsLocalRing.ResidueField B) ∧
      (IsLocalRing.maximalIdeal B).ramificationIdx A *
          Module.finrank (IsLocalRing.ResidueField A) (IsLocalRing.ResidueField B) ≤
        Module.finrank K L

Tag 0BRL, a lemma

Tag 0BRL in the Stacks Project.

IsDiscreteValuationRing.ramificationIdx_and_inertiaDeg_tower, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/RamificationIndex.lean, line 280.

Ramification indices and residue degrees multiply in a tower of extensions of discrete valuation rings.

theorem ramificationIdx_and_inertiaDeg_tower
    (A B C : Type*)
    [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [CommRing B] [IsDomain B] [IsDiscreteValuationRing B]
    [CommRing C] [IsDomain C] [IsDiscreteValuationRing C]
    [Algebra A B] [Algebra B C] [Algebra A C] [IsScalarTower A B C]
    [FaithfulSMul A B] [FaithfulSMul B C]
    [IsLocalHom (algebraMap A B)] [IsLocalHom (algebraMap B C)] :
    (IsLocalRing.maximalIdeal C).ramificationIdx A =
        (IsLocalRing.maximalIdeal B).ramificationIdx A *
          (IsLocalRing.maximalIdeal C).ramificationIdx B ∧
      (IsLocalRing.maximalIdeal C).inertiaDeg A =
        (IsLocalRing.maximalIdeal B).inertiaDeg A *
          (IsLocalRing.maximalIdeal C).inertiaDeg B

Tag 09E6, a lemma

Tag 09E6 in the Stacks Project.

Stacks.DiscreteValuationRing.Ramification.ramificationIdx_eq_prime_pow_of_purelyInseparable, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/RamificationIndex.lean, line 113.

Stacks 09E6. Let A ⊂ B be an extension of discrete valuation rings inducing the field extension K ⊂ L. If K has characteristic p with p prime and L / K is purely inseparable, then the ramification index e of B over A is a power of p: there is m ≥ 0 with e = p ^ m.

theorem ramificationIdx_eq_prime_pow_of_purelyInseparable (A B : Type*) [CommRing A] [IsDomain A]
    [IsDiscreteValuationRing A] [CommRing B] [IsDomain B] [IsDiscreteValuationRing B] [Algebra A B]
    [FaithfulSMul A B] [IsLocalHom (algebraMap A B)] (K L : Type*) [Field K] [Algebra A K]
    [IsFractionRing A K] [Field L] [Algebra B L] [IsFractionRing B L] [Algebra A L] [Algebra K L]
    [IsScalarTower A B L] [IsScalarTower A K L] (p : ℕ) (hp : p.Prime) [CharP K p]
    [IsPurelyInseparable K L] : ∃ m : ℕ, ramificationIdx A B = p ^ m

Tag 09E7, a lemma

Tag 09E7 in the Stacks Project.

IsDiscreteValuationRing.formallySmooth_iff_weaklyUnramified_and_isSeparableExtension, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/FormallySmoothExtension.lean, line 200.

An extension of discrete valuation rings is formally smooth for the maximal-ideal-adic topology if and only if it is weakly unramified and its possibly nonalgebraic residue-field extension is separable.

theorem formallySmooth_iff_weaklyUnramified_and_isSeparableExtension
    (A : Type u) (B : Type v)
    [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [CommRing B] [IsDomain B] [IsDiscreteValuationRing B]
    [Algebra A B] [IsExtension A B] :
    RingHom.FormallySmoothNAdic.{max u v} (algebraMap A B)
        (IsLocalRing.maximalIdeal B) ↔
      IsWeaklyUnramified A B ∧
        Algebra.IsSeparableExtension (IsLocalRing.ResidueField A)
          (IsLocalRing.ResidueField B)

Tag 09E8, a remark

Tag 09E8 in the Stacks Project.

IsDiscreteValuationRing.integralClosure_finite_dedekind_and_degree_eq_sum_ramification_inertia, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Unramified.lean, line 328.

Let A be a discrete valuation ring with fraction field K, and let L / K be a finite separable extension. The integral closure B of A in L is finite and Noetherian over A, has dimension one, and is Dedekind. Its maximal ideals m i are precisely the primes over the maximal ideal of A; every B_(m i) is a discrete valuation ring, and [L : K] = ∑ i, e(m i / m_A) * f(m i / m_A). If A is henselian—or, in particular, maximal-ideal-adically complete—there is exactly one such maximal ideal.

theorem integralClosure_finite_dedekind_and_degree_eq_sum_ramification_inertia
    (A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    (K : Type*) [Field K] [Algebra A K] [IsFractionRing A K]
    (L : Type*) [Field L] [Algebra K L] [FiniteDimensional K L]
    [Algebra.IsSeparable K L] [Algebra A L] [IsScalarTower A K L] :
    let B := integralClosure A L
    Module.Finite A B ∧
      IsNoetherianRing B ∧
      ringKrullDim B = 1 ∧
      IsDedekindDomain B ∧
      ∃ (n : ℕ) (m : Fin n → Ideal B),
        ∃ hmax : ∀ i, (m i).IsMaximal,
          (∀ q : Ideal B, q.IsMaximal ↔ ∃! i, m i = q) ∧
          (∀ i, (m i).LiesOver (IsLocalRing.maximalIdeal A)) ∧
          (∀ i, letI : (m i).IsPrime := (hmax i).isPrime
            IsDiscreteValuationRing (Localization.AtPrime (m i))) ∧
          Module.finrank K L = ∑ i, (m i).ramificationIdx A * (m i).inertiaDeg A ∧
          (HenselianLocalRing A → n = 1) ∧
          (IsAdicComplete (IsLocalRing.maximalIdeal A) A → n = 1)

Tag 09E9, a definition

Tag 09E9 in the Stacks Project.

IsDiscreteValuationRing.IsUnramifiedWithRespectTo, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Unramified.lean, line 416. The attribute's comment: “(1)”.

An extension is unramified with respect to a discrete valuation ring A if, at every maximal ideal of the integral closure lying over the maximal ideal of A, the ramification index is one and the induced residue field extension is separable.

def IsUnramifiedWithRespectTo
    (A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    (K : Type*) [Field K] [Algebra A K] [IsFractionRing A K]
    (L : Type*) [Field L] [Algebra K L] [FiniteDimensional K L]
    [Algebra.IsSeparable K L] [Algebra A L] [IsScalarTower A K L] : Prop

IsDiscreteValuationRing.IsTamelyRamifiedWithRespectTo, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Unramified.lean, line 431. The attribute's comment: “(2)”. References: SGA 1 XIII (2.0), “L a field extension”.

An extension is tamely ramified with respect to A if every induced residue field extension is separable and, in positive residue characteristic, every ramification index is relatively prime to that characteristic. This is Stacks.KummerExtension.IsTamelyRamified.

abbrev IsTamelyRamifiedWithRespectTo
    (A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    (K : Type*) [Field K] [Algebra A K] [IsFractionRing A K]
    (L : Type*) [Field L] [Algebra K L] [FiniteDimensional K L]
    [Algebra.IsSeparable K L] [Algebra A L] [IsScalarTower A K L] : Prop

IsDiscreteValuationRing.IsTotallyRamifiedWithRespectTo, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Unramified.lean, line 442. The attribute's comment: “(3)”.

An extension is totally ramified with respect to A if its integral closure has a unique maximal ideal above that of A and its induced residue field extension is trivial. This is Stacks.KummerExtension.IsTotallyRamified.

abbrev IsTotallyRamifiedWithRespectTo
    (A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    (K : Type*) [Field K] [Algebra A K] [IsFractionRing A K]
    (L : Type*) [Field L] [Algebra K L] [FiniteDimensional K L]
    [Algebra.IsSeparable K L] [Algebra A L] [IsScalarTower A K L] : Prop

Tag 0EXR, a lemma

Tag 0EXR in the Stacks Project.

IsDiscreteValuationRing.isUnramifiedWithRespectTo_of_tower_and_exists_isGalois_and_common_extension, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Unramified.lean, line 772.

Stacks 0EXR. Unramified finite separable extensions of the fraction field of a discrete valuation ring are closed under passage to intermediate fields, admit finite Galois overextensions which remain unramified, and any two admit a common finite separable overextension which remains unramified.

theorem isUnramifiedWithRespectTo_of_tower_and_exists_isGalois_and_common_extension
    (A : Type u) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    (K : Type v) [Field K] [Algebra A K] [IsFractionRing A K] :
    (∀ (L : Type w) [Field L] [Algebra K L] [FiniteDimensional K L]
        [Algebra.IsSeparable K L] [Algebra A L] [IsScalarTower A K L]
        (M : Type x) [Field M] [Algebra L M] [FiniteDimensional L M]
        [Algebra.IsSeparable L M] [Algebra K M] [IsScalarTower K L M]
        [Algebra A M] [IsScalarTower A K M],
      letI : FiniteDimensional K M := FiniteDimensional.trans K L M
      letI : Algebra.IsSeparable K M := Algebra.IsSeparable.trans K L M
      IsUnramifiedWithRespectTo A K M →
        IsUnramifiedWithRespectTo A K L) ∧
    (∀ (L : Type w) [Field L] [Algebra K L] [FiniteDimensional K L]
        [Algebra.IsSeparable K L] [Algebra A L] [IsScalarTower A K L],
      IsUnramifiedWithRespectTo A K L →
        ∃ (M : Type (max u v w)) (_ : Field M) (_ : Algebra K M)
          (_ : FiniteDimensional K M) (_ : Algebra.IsSeparable K M)
          (_ : Algebra A M) (_ : IsScalarTower A K M)
          (_ : Algebra L M) (_ : IsScalarTower K L M),
          IsGalois K M ∧ IsUnramifiedWithRespectTo A K M) ∧
    (∀ (L₁ : Type w) [Field L₁] [Algebra K L₁] [FiniteDimensional K L₁]
        [Algebra.IsSeparable K L₁] [Algebra A L₁] [IsScalarTower A K L₁]
        (L₂ : Type x) [Field L₂] [Algebra K L₂] [FiniteDimensional K L₂]
        [Algebra.IsSeparable K L₂] [Algebra A L₂] [IsScalarTower A K L₂],
      IsUnramifiedWithRespectTo A K L₁ →
        IsUnramifiedWithRespectTo A K L₂ →
        ∃ (L : Type (max u v w x)) (_ : Field L) (_ : Algebra K L)
          (_ : FiniteDimensional K L) (_ : Algebra.IsSeparable K L)
          (_ : Algebra A L) (_ : IsScalarTower A K L)
          (_ : Algebra L₁ L) (_ : IsScalarTower K L₁ L)
          (_ : Algebra L₂ L) (_ : IsScalarTower K L₂ L),
          IsUnramifiedWithRespectTo A K L)

Tag 0EXS, a lemma

Tag 0EXS in the Stacks Project.

IsDiscreteValuationRing.isUnramifiedIn_comp, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Unramified.lean, line 871. References: SGA 1 I (10.4), “(ii) transitivity”.

If p is unramified in B and C is unramified over every prime of B lying over p, then p is unramified in C.

theorem isUnramifiedIn_comp
    (A B C : Type*) [CommRing A] [CommRing B] [CommRing C]
    [Algebra A B] [Algebra B C] [Algebra A C] [IsScalarTower A B C]
    (p : Ideal A) (hAB : Algebra.IsUnramifiedIn B p)
    (hBC : ∀ (q : Ideal B) (_ : q.IsPrime), q.LiesOver p → Algebra.IsUnramifiedIn C q) :
    Algebra.IsUnramifiedIn C p

Sources

• Section 15.113, Tag 0EXQ
• AxiomMath/Stacks at 5ed82d14

LeanIsDiscreteValuationRing.IsExtension · IsDiscreteValuationRing.ramificationIndex · IsDiscreteValuationRing.IsWeaklyUnramified · IsDiscreteValuationRing.residueDegree · IsDiscreteValuationRing.ramificationIndex_characterization · IsDiscreteValuationRing.residueField_finite_and_ramificationIdx_mul_residueDegree_le_finrank · IsDiscreteValuationRing.ramificationIdx_and_inertiaDeg_tower · Stacks.DiscreteValuationRing.Ramification.ramificationIdx_eq_prime_pow_of_purelyInseparable

Section15.114 Galois extensions and ramification#

Section 15.114 of the Stacks Project, Tag 09E3. 12 tags here are carried by 17 declarations of the repository (all in StacksFrontier) and 4 registered declarations, quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 09EA, a lemma

Tag 09EA in the Stacks Project.

IsDiscreteValuationRing.galAction_transitive_on_maximalIdeals_integralClosure, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Galois.lean, line 157.

For a finite Galois extension L / K, the canonical action of Gal(L/K) on the integral closure of a discrete valuation ring A in L is transitive on its maximal ideals.

theorem galAction_transitive_on_maximalIdeals_integralClosure
    (A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    (K : Type*) [Field K] [Algebra A K] [IsFractionRing A K]
    (L : Type*) [Field L] [Algebra K L] [FiniteDimensional K L]
    [IsGalois K L] [Algebra A L] [IsScalarTower A K L] :
    let B := integralClosure A L
    letI := IsIntegralClosure.MulSemiringAction A K L B
    ∀ P Q : Ideal B, P.IsMaximal → Q.IsMaximal →
      ∃ σ : Gal(L/K), σ • P = Q

Tag 09EB, a lemma

Tag 09EB in the Stacks Project.

IsDiscreteValuationRing.exists_uniform_ramificationIdx_inertiaDeg_and_finrank_eq, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Galois.lean, line 182.

In a finite Galois extension of the fraction field of a discrete valuation ring, all primes of the integral closure over the maximal ideal have the same positive ramification index and inertia degree. The field degree is the number of these primes times the two common values.

theorem exists_uniform_ramificationIdx_inertiaDeg_and_finrank_eq
    (A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    (K : Type*) [Field K] [Algebra A K] [IsFractionRing A K]
    (L : Type*) [Field L] [Algebra K L] [FiniteDimensional K L]
    [IsGalois K L] [Algebra A L] [IsScalarTower A K L] :
    let B := integralClosure A L
    let p := IsLocalRing.maximalIdeal A
    ∃ e f : ℕ,
      1 ≤ e ∧
      1 ≤ f ∧
      (∀ P : p.primesOver B, P.1.ramificationIdx A = e ∧ P.1.inertiaDeg A = f) ∧
      (p.primesOver B).ncard * e * f = Module.finrank K L

Tag 09EC, a definition

Tag 09EC in the Stacks Project.

IsDiscreteValuationRing.under_eq_maximalIdeal_and_ker_stabilizerHom_eq_inertia, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Galois.lean, line 235. The attribute's comment: “The decomposition group is MulAction.stabilizer and the inertia group is Ideal.inertia, both of which Mathlib already has for an arbitrary action; what is recorded here is that they are the source's two groups, the ideal below P being the maximal ideal of A so that the extension acted on is κ(P)/κ_A. Stated without the fraction-field, finiteness and Galois hypotheses, which the comparison does not use, so the source's setting is a special case.”. References: SGA 7 I (0.3), “partial”.

The decomposition and the inertia group at a maximal ideal P of the integral closure B of a discrete valuation ring A in a field extension L / K.

The decomposition group is the stabilizer {σ ∈ Gal(L/K) | σ P = P}, and the ideal of A below P is the maximal ideal, so the residue extension the decomposition group acts on really is κ(P) / κ_A. The inertia group is the kernel of that action: it is Ideal.inertia, sitting inside the decomposition group, and an element of the decomposition group lies in it exactly when it acts trivially on κ(P).

theorem under_eq_maximalIdeal_and_ker_stabilizerHom_eq_inertia
    (A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    (K : Type*) [Field K] [Algebra A K]
    (L : Type*) [Field L] [Algebra K L] [Algebra A L] [IsScalarTower A K L]
    (P : Ideal (integralClosure A L)) [P.IsMaximal] :
    let B := integralClosure A L
    let D := MulAction.stabilizer Gal(L/K) P
    let I := P.inertia Gal(L/K)
    (∀ σ : Gal(L/K), σ ∈ D ↔ σ • P = P) ∧
      P.under A = IsLocalRing.maximalIdeal A ∧
      I ≤ D ∧
      (Ideal.Quotient.stabilizerHom P (P.under A) Gal(L/K)).ker = I.subgroupOf D ∧
      (∀ σ : D, σ ∈ (Ideal.Quotient.stabilizerHom P (P.under A) Gal(L/K)).ker ↔
        ∀ b : B, Ideal.Quotient.mk P ((σ : Gal(L/K)) • b) = Ideal.Quotient.mk P b)

Declarations registered for the tag:

• MulAction.stabilizer registered with add_to_stacks in StacksFrontier/RingTheory/DiscreteValuationRing/Galois.lean, line 102 (StacksFrontier): “(1), the decomposition group”. References: SGA 1 V (6.13); SGA 3 I (2.3.3), “special case”; SGA 3 VIIB (2.5.A); SGA 3 XXI (6.7.2), “more general”; SGA 5 IIIB (5.13), “more general”; SGA 5 X (4.2).
• Ideal.inertia registered with add_to_stacks in StacksFrontier/RingTheory/DiscreteValuationRing/Galois.lean, line 110 (StacksFrontier): “(2), the inertia group”. References: SGA 7 I (0.3); SGA 7 XIII (0.2).
• Ideal.Quotient.stabilizerHom registered with add_to_stacks in StacksFrontier/RingTheory/DiscreteValuationRing/Galois.lean, line 114 (StacksFrontier): “The action of the decomposition group on the residue algebra”.
• Ideal.Quotient.ker_stabilizerHom registered with add_to_stacks in StacksFrontier/RingTheory/DiscreteValuationRing/Galois.lean, line 117 (StacksFrontier): “The inertia group as the kernel of the residue-algebra action”.

Tag 09ED, a lemma

Tag 09ED in the Stacks Project.

IsDiscreteValuationRing.residueField_normal_and_stabilizerHom_surjective_of_isGalois, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Galois.lean, line 263. References: SGA 1 V (1.1), “special case”; SGA 1 V (1.3), “special case”.

In a finite Galois extension of the fraction field of a discrete valuation ring, every residue extension at a maximal ideal of the integral closure is normal, and its decomposition group surjects onto the relative automorphism group. Separability of one such residue extension is equivalent to separability of all of them; when this holds, the fixed residue extension is Galois and the same canonical map is surjective onto its Galois group.

theorem residueField_normal_and_stabilizerHom_surjective_of_isGalois
    (A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    (K : Type*) [Field K] [Algebra A K] [IsFractionRing A K]
    (L : Type*) [Field L] [Algebra K L] [FiniteDimensional K L]
    [IsGalois K L] [Algebra A L] [IsScalarTower A K L]
    (P : Ideal (integralClosure A L)) [P.IsMaximal] :
    let B := integralClosure A L
    let p := P.under A
    letI : Field (A ⧸ p) := Ideal.Quotient.field p
    letI : Field (B ⧸ P) := Ideal.Quotient.field P
    Normal (A ⧸ p) (B ⧸ P) ∧
      Function.Surjective (Ideal.Quotient.stabilizerHom P p Gal(L/K)) ∧
      ((∃ (Q : Ideal B) (_hQ : Q.IsMaximal),
          let q := Q.under A
          letI : Field (A ⧸ q) := Ideal.Quotient.field q
          letI : Field (B ⧸ Q) := Ideal.Quotient.field Q
          Algebra.IsSeparable (A ⧸ q) (B ⧸ Q)) ↔
        (∀ (Q : Ideal B) (_hQ : Q.IsMaximal),
          let q := Q.under A
          letI : Field (A ⧸ q) := Ideal.Quotient.field q
          letI : Field (B ⧸ Q) := Ideal.Quotient.field Q
          Algebra.IsSeparable (A ⧸ q) (B ⧸ Q))) ∧
      ((∃ (Q : Ideal B) (_hQ : Q.IsMaximal),
          let q := Q.under A
          letI : Field (A ⧸ q) := Ideal.Quotient.field q
          letI : Field (B ⧸ Q) := Ideal.Quotient.field Q
          Algebra.IsSeparable (A ⧸ q) (B ⧸ Q)) →
        IsGalois (A ⧸ p) (B ⧸ P) ∧
          Function.Surjective (Ideal.Quotient.stabilizerHom P p Gal(L/K)))

Tag 09EE, a lemma

Tag 09EE in the Stacks Project.

IsDiscreteValuationRing.exists_wildInertia_exact_and_tameInertia_equiv_rootsOfUnity, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Galois.lean, line 847. References: SGA 1 XIII (2.0), “inertia structure”; SGA 7 I (0.3), “finite Galois case”.

The inertia group at a maximal ideal of the integral closure in a finite Galois extension has a canonical wild–tame exact sequence. Wild inertia consists precisely of the elements of inertia acting trivially on the maximal ideal modulo its square; it is normal in the decomposition group and is a p-group in positive residue characteristic (and trivial in characteristic zero). The tame quotient is cyclic of prime-to-p order and is canonically isomorphic to the corresponding roots of unity in the residue field. If the residue extension is separable, wild inertia is also the kernel of the action on B / P².

theorem exists_wildInertia_exact_and_tameInertia_equiv_rootsOfUnity
    (A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    (K : Type*) [Field K] [Algebra A K] [IsFractionRing A K]
    (L : Type*) [Field L] [Algebra K L] [FiniteDimensional K L]
    [IsGalois K L] [Algebra A L] [IsScalarTower A K L]
    (P : Ideal (integralClosure A L)) [P.IsMaximal] :
    let B := integralClosure A L
    letI := IsIntegralClosure.MulSemiringAction A K L B
    let D := MulAction.stabilizer Gal(L/K) P
    let I := P.inertia Gal(L/K)
    let inclusion : I →* D := I.subtype.codRestrict D
      (fun σ ↦ Ideal.inertia_le_stabilizer (M := Gal(L/K)) P σ.property)
    let e := P.ramificationIdx A
    let p := ringChar (IsLocalRing.ResidueField A)
    ∃ W : Subgroup I, ∃ _hW : W.Normal,
      (W.map inclusion).Normal ∧
      (∀ σ : I, σ ∈ W ↔ ∀ b : B, b ∈ P → (σ : Gal(L/K)) • b - b ∈ P ^ 2) ∧
      (∀ σ : D, σ ∈ W.map inclusion ↔
        (σ : Gal(L/K)) ∈ I ∧ ∀ b : B, b ∈ P → σ • b - b ∈ P ^ 2) ∧
      (0 < p → IsPGroup p W) ∧
      (p = 0 → W = ⊥) ∧
      (letI : Field (A ⧸ P.under A) := Ideal.Quotient.field (P.under A)
       Algebra.IsSeparable (A ⧸ P.under A) (B ⧸ P) →
         ∀ σ : D, σ ∈ W.map inclusion ↔ ∀ b : B, σ • b - b ∈ P ^ 2) ∧
      ∃ θ : I →* rootsOfUnity e (B ⧸ P),
        Function.Injective W.subtype ∧
        Function.MulExact W.subtype θ ∧
        Function.Surjective θ ∧
        IsCyclic (rootsOfUnity e (B ⧸ P)) ∧
        Nat.card (rootsOfUnity e (B ⧸ P)) = e / p ^ e.factorization p ∧
        (∀ (σ : I) (x a : B), x ∈ P → x ∉ P ^ 2 →
            Ideal.Quotient.mk P a = (θ σ).1.1 →
            (σ : Gal(L/K)) • x - a * x ∈ P ^ 2) ∧
        ∀ θ' : I →* rootsOfUnity e (B ⧸ P),
          (∀ (σ : I) (x a : B), x ∈ P → x ∉ P ^ 2 →
            Ideal.Quotient.mk P a = (θ' σ).1.1 →
            (σ : Gal(L/K)) • x - a * x ∈ P ^ 2) →
          θ' = θ

Tag 0BU3, an equation

Tag 0BU3 in the Stacks Project.

IsDiscreteValuationRing.exists_uniformizerRatioMap_to_rootsOfUnity, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Galois.lean, line 992.

Let a group act on a discrete valuation ring, trivially on its residue field. Suppose an invariant element is a unit times the e-th power of a uniformizer. For every group element, its action on the uniformizer differs from the uniformizer by a unit; the residues of these units are e-th roots of unity and hence define a map to the group of such roots.

theorem exists_uniformizerRatioMap_to_rootsOfUnity
    (C : Type u) [CommRing C] [IsDomain C] [IsDiscreteValuationRing C]
    (I : Type v) [Group I] [MulSemiringAction I C]
    (e : ℕ) (π a : C) (u : Cˣ)
    (hπ : Irreducible π)
    (ha : a = (u : C) * π ^ e)
    (ha_fixed : ∀ σ : I, σ • a = a)
    (hres : ∀ (σ : I) (x : C),
      Ideal.Quotient.mk (IsLocalRing.maximalIdeal C) (σ • x) =
        Ideal.Quotient.mk (IsLocalRing.maximalIdeal C) x) :
    ∃ (θ : I → rootsOfUnity e (C ⧸ IsLocalRing.maximalIdeal C))
      (θunit : I → Cˣ),
      (∀ σ : I, (θunit σ : C) * π = σ • π) ∧
        ∀ σ : I,
          (θ σ : (C ⧸ IsLocalRing.maximalIdeal C)ˣ) =
            Units.map (Ideal.Quotient.mk (IsLocalRing.maximalIdeal C)) (θunit σ)

Tag 0BU4, a definition

Tag 0BU4 in the Stacks Project.

IsDiscreteValuationRing.inertiaCharacter, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Inertia/Character.lean, line 176.

Stacks Project, Tag 0BU4. The canonical character from inertia to the roots of unity of order the ramification index in the residue field.

noncomputable def inertiaCharacter

IsDiscreteValuationRing.wildInertiaGroup, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Inertia/Character.lean, line 180. References: SGA 1 XIII (2.0), “partial”.

Stacks Project, Tag 0BU4. The wild inertia group at P.

noncomputable def wildInertiaGroup

IsDiscreteValuationRing.tameInertiaGroup, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Inertia/Character.lean, line 188. References: SGA 7 I (0.3).

Stacks Project, Tag 0BU4. The tame inertia group, namely inertia modulo wild inertia.

abbrev tameInertiaGroup

IsDiscreteValuationRing.tameInertiaMap, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Inertia/Character.lean, line 192.

Stacks Project, Tag 0BU4. The quotient map from inertia to tame inertia.

noncomputable def tameInertiaMap : P.inertia Gal(L/K) →* tameInertiaGroup A K L P

IsDiscreteValuationRing.tameInertiaEquivRootsOfUnity, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Inertia/Character.lean, line 205. References: SGA 7 I (0.3), “finite Galois case”.

Stacks Project, Tag 0BU4. The canonical isomorphism from tame inertia to the roots of unity induced by the inertia character.

noncomputable def tameInertiaEquivRootsOfUnity :
    tameInertiaGroup A K L P ≃*
      rootsOfUnity (P.ramificationIdx A) (integralClosure A L ⧸ P)

Tag 0BU5, a lemma

Tag 0BU5 in the Stacks Project.

IsDiscreteValuationRing.inertiaCharacter_conj, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Inertia/Character.lean, line 238. References: SGA 7 I (0.3), “finite Galois case”.

Stacks Tag 0BU5. The inertia character is equivariant for conjugation by the decomposition group: conjugating an inertia element by a decomposition element has character equal to the image of the original character under the induced residue-field automorphism.

theorem inertiaCharacter_conj
    (A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    (K : Type*) [Field K] [Algebra A K] [IsFractionRing A K]
    (L : Type*) [Field L] [Algebra K L] [FiniteDimensional K L]
    [IsGalois K L] [Algebra A L] [IsScalarTower A K L]
    (P : Ideal (integralClosure A L)) [P.IsMaximal] :
    let B := integralClosure A L
    letI := IsIntegralClosure.MulSemiringAction A K L B
    let D := MulAction.stabilizer Gal(L/K) P
    let I := P.inertia Gal(L/K)
    let e := P.ramificationIdx A
    let conjugate : D → I → I := fun τ σ ↦
      ⟨(τ : Gal(L/K)) * (σ : Gal(L/K)) * (τ : Gal(L/K))⁻¹, by
        rw [Ideal.mem_inertia]
        intro x
        have hσ := (Ideal.mem_inertia.mp σ.property) ((τ : Gal(L/K))⁻¹ • x)
        have hτ := Ideal.smul_mem_pointwise_smul (τ : Gal(L/K)) _ P hσ
        rw [τ.property] at hτ
        simpa only [mul_smul, smul_sub, smul_inv_smul] using hτ⟩
    ∀ (θ : I →* rootsOfUnity e (B ⧸ P)),
      (∀ (σ : I) (x a : B), x ∈ P → x ∉ P ^ 2 →
        Ideal.Quotient.mk P a = (θ σ).1.1 →
        (σ : Gal(L/K)) • x - a * x ∈ P ^ 2) →
      ∀ (τ : D) (σ : I),
        θ (conjugate τ σ) =
          (Ideal.quotientEquiv P P
            ((MulSemiringAction.toRingEquiv Gal(L/K) B) (τ : Gal(L/K)))
            τ.property.symm).toMulEquiv.restrictRootsOfUnity e (θ σ)

Tag 09EH, a lemma

Tag 09EH in the Stacks Project.

IsDiscreteValuationRing.inertia_fixedSubalgebra_isIntegralClosure_and_etale_localization, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Inertia/InvariantRing.lean, line 229. References: SGA 7 I (0.3), “partial”.

Stacks Tag 09EH. The subring of the integral closure fixed by the inertia group is the integral closure in the fixed field, and its localization at the prime below the chosen maximal ideal is étale over the original discrete valuation ring.

theorem inertia_fixedSubalgebra_isIntegralClosure_and_etale_localization
    (A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    (K : Type*) [Field K] [Algebra A K] [IsFractionRing A K]
    (L : Type*) [Field L] [Algebra K L] [FiniteDimensional K L]
    [IsGalois K L] [Algebra A L] [IsScalarTower A K L]
    (P : Ideal (integralClosure A L)) [P.IsMaximal] :
    let B := integralClosure A L
    letI := IsIntegralClosure.MulSemiringAction A K L B
    letI : SMulCommClass Gal(L/K) A B :=
      ⟨fun g a b ↦ by
        simp only [Algebra.smul_def]
        change galRestrict A K L B g (algebraMap A B a * b) =
          algebraMap A B a * galRestrict A K L B g b
        rw [map_mul, AlgEquiv.commutes]⟩
    let I := P.inertia Gal(L/K)
    let B' := FixedPoints.subalgebra A B I
    let L' := IntermediateField.fixedField I
    let inclusion : B' →ₐ[A] L := (integralClosure A L).val.comp B'.val
    let iota : B' →ₐ[A] L' :=
      inclusion.codRestrict (L'.toSubalgebra.restrictScalars A) (fun b ↦ by
        change (inclusion b : L) ∈ L'
        rw [IntermediateField.mem_fixedField_iff]
        intro g hg
        exact (algebraMap_galRestrict_apply A g (b : B)).symm.trans
          (congrArg (algebraMap B L) (b.property ⟨g, hg⟩)))
    letI := iota.toAlgebra
    IsIntegralClosure B' A L' ∧
      Algebra.Etale A (Localization.AtPrime (P.under B'))

Tag 0BU6, a remark

Tag 0BU6 in the Stacks Project.

IsDiscreteValuationRing.decomposition_inertia_wildInertia_tower, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Inertia/DecompositionTower.lean, line 622.

Stacks Tag 0BU6. The fixed fields, invariant rings, contracted primes, localized rings, and residue fields attached to the wild inertia, inertia, and decomposition groups of a maximal ideal in a finite Galois extension form the standard ramification tower.

theorem decomposition_inertia_wildInertia_tower
    (A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    (K : Type*) [Field K] [Algebra A K] [IsFractionRing A K]
    (L : Type*) [Field L] [Algebra K L] [FiniteDimensional K L]
    [IsGalois K L] [Algebra A L] [IsScalarTower A K L]
    (m : Ideal (integralClosure A L)) [m.IsMaximal] :
    let B := integralClosure A L
    letI := IsIntegralClosure.MulSemiringAction A K L B
    letI : SMulCommClass Gal(L/K) A B :=
      ⟨fun g a b ↦ by
        simp only [Algebra.smul_def]
        change galRestrict A K L B g (algebraMap A B a * b) =
          algebraMap A B a * galRestrict A K L B g b
        rw [map_mul, AlgEquiv.commutes]⟩
    let D := MulAction.stabilizer Gal(L/K) m
    let I := m.inertia Gal(L/K)
    let hID : I ≤ D := Ideal.inertia_le_stabilizer (M := Gal(L/K)) m
    let iID : I →* D := I.subtype.codRestrict D (fun σ ↦ hID σ.property)
    ∃ (P : Subgroup I) (hP : P.Normal),
      let PG : Subgroup Gal(L/K) := P.map I.subtype
      let PD : Subgroup D := P.map iID
      (∀ σ : D, σ ∈ PD ↔ (σ : Gal(L/K)) ∈ I ∧
        ∀ b : B, b ∈ m → (σ : Gal(L/K)) • b - b ∈ m ^ 2) ∧
      let ID : Subgroup D := I.subgroupOf D
      let LD := IntermediateField.fixedField D
      let LI := IntermediateField.fixedField I
      let LP := IntermediateField.fixedField PG
      let BD := FixedPoints.subalgebra A B D
      let BI := FixedPoints.subalgebra A B I
      let BP := FixedPoints.subalgebra A B PG
      let hPGI : PG ≤ I := fun σ hσ ↦ by
        obtain ⟨τ, -, rfl⟩

Tag 0BU7, a lemma

Tag 0BU7 in the Stacks Project.

IsDiscreteValuationRing.surjective_restrictions_commute_with_residue_and_inertiaCharacter, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Inertia/GaloisTower.lean, line 197.

Stacks Project, Tag 0BU7. In a tower of finite Galois extensions of the fraction field of a discrete valuation ring, restriction surjects on decomposition, inertia, and wild inertia groups. The induced maps commute with residue-field automorphisms and with the canonical inertia characters; on roots of unity the latter vertical map is the e' / e-power map.

theorem surjective_restrictions_commute_with_residue_and_inertiaCharacter
    (A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    (K : Type*) [Field K] [Algebra A K] [IsFractionRing A K]
    (L : Type*) [Field L] [Algebra K L] [FiniteDimensional K L]
    [IsGalois K L] [Algebra A L] [IsScalarTower A K L]
    (M : Type*) [Field M] [Algebra K M] [FiniteDimensional K M]
    [IsGalois K M] [Algebra L M] [IsScalarTower K L M]
    [Algebra A M] [IsScalarTower A K M] [IsScalarTower A L M]
    (r : Ideal (integralClosure A M)) [r.IsMaximal] :
    let B := integralClosure A L
    let C := integralClosure A M
    let fBC : B →ₐ[A] C := (IsScalarTower.toAlgHom A L M).mapIntegralClosure
    letI : Algebra B C := fBC.toRingHom.toAlgebra
    letI : IsScalarTower A B C :=
      IsScalarTower.of_algebraMap_eq fun x ↦ (fBC.commutes x).symm
    letI : Algebra.IsIntegral B C :=
      ⟨fun x ↦ IsIntegral.tower_top (IsIntegralClosure.isIntegral A M x)⟩
    letI : MulSemiringAction Gal(M/K) C :=
      IsIntegralClosure.MulSemiringAction A K M C
    letI : MulSemiringAction Gal(L/K) B :=
      IsIntegralClosure.MulSemiringAction A K L B
    let q := r.under B
    let p := r.under A
    letI : r.LiesOver q := ⟨rfl⟩
    letI : q.LiesOver p := ⟨r.under_under.symm⟩
    letI : Field (A ⧸ p) := Ideal.Quotient.field p
    letI : Field (B ⧸ q) := Ideal.Quotient.field q
    letI : Field (C ⧸ r) := Ideal.Quotient.field r
    letI : Algebra (B ⧸ q) (C ⧸ r) := Ideal.Quotient.algebraOfLiesOver r q
    letI : Algebra (A ⧸ p) (B ⧸ q) := Ideal.Quotient.algebraOfLiesOver q p
    letI : Algebra (A ⧸ p) (C ⧸ r) := Ideal.Quotient.algebraOfLiesOver r p
    let res : Gal(M/K) →* Gal(L/K) := AlgEquiv.restrictNormalHom L
    let D' := MulAction.stabilizer Gal(M/K) r
    let D := MulAction.stabilizer Gal(L/K) q
    let e' := r.ramificationIdx A
    let e := q.ramificationIdx A
    ∃ (resD : D' →* D)
      (resI : r.inertia Gal(M/K) →* q.inertia Gal(L/K))
      (resP : wildInertiaGroup A K M r →*
        wildInertiaGroup A K L q),
      Function.Surjective resD ∧ Function.Surjective resI ∧ Function.Surjective resP ∧
      (∀ σ, ((resD σ : D) : Gal(L/K)) = res ((σ : D') : Gal(M/K))) ∧
      (∀ σ, (resI σ).1 = res σ.1) ∧
      (∀ σ, (resP σ).1.1 = res σ.1.1) ∧
      e ∣ e' ∧
      ∃ (residueRestriction : Gal((C ⧸ r)/(A ⧸ p)) →* Gal((B ⧸ q)/(A ⧸ p)))
        (rootRestriction : rootsOfUnity e' (C ⧸ r) →* rootsOfUnity e (B ⧸ q)),
        (∀ τ b, algebraMap (B ⧸ q) (C ⧸ r) (residueRestriction τ b) =
          τ (algebraMap (B ⧸ q) (C ⧸ r) b)) ∧
        (∀ ζ, algebraMap (B ⧸ q) (C ⧸ r) (rootRestriction ζ).1.1 =
          (ζ ^ (e' / e)).1.1) ∧
        (∀ σ b, algebraMap (B ⧸ q) (C ⧸ r)
            (Ideal.Quotient.mk q ((resD σ).1 • b)) =
          Ideal.Quotient.mk r (σ.1 • fBC b)) ∧
        ∀ σ, rootRestriction (inertiaCharacter A K M r σ) =
          inertiaCharacter A K L q (resI σ)

Tag 0BU8, a remark

Tag 0BU8 in the Stacks Project.

IsDiscreteValuationRing.canonicalInertiaCharacter, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Inertia/GaloisTower.lean, line 486.

Stacks Project, Tag 0BU8. The canonical inertia character, obtained by raising the ordinary inertia character to the residue-characteristic-primary part of the ramification index.

noncomputable def canonicalInertiaCharacter :
    P.inertia Gal(L/K) →*
      rootsOfUnity (tameRamificationIndex A L P) (integralClosure A L ⧸ P)

IsDiscreteValuationRing.canonicalInertiaCharacter_spec, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Inertia/GaloisTower.lean, line 518. References: SGA 7 I (0.3), “finite Galois case”.

Stacks Project, Tag 0BU8. The scaled inertia character identifies tame inertia with the roots of unity of order |I_t|. Its order is the prime-to-residue-characteristic part of the ramification index, it is equivariant under the decomposition group, and these canonical characters form the stated power-map square in every finite Galois tower.

theorem canonicalInertiaCharacter_spec :
    let B := integralClosure A L
    letI := IsIntegralClosure.MulSemiringAction A K L B
    let D := MulAction.stabilizer Gal(L/K) P
    let I := P.inertia Gal(L/K)
    let e := P.ramificationIdx A
    let p := ringChar (IsLocalRing.ResidueField A)
    let q := wildRamificationFactor A L P
    let n := tameRamificationIndex A L P
    let conjugate : D → I → I := fun τ σ ↦
      ⟨(τ : Gal(L/K)) * (σ : Gal(L/K)) * (τ : Gal(L/K))⁻¹, by
        rw [Ideal.mem_inertia]
        intro x
        have hσ := (Ideal.mem_inertia.mp σ.property) ((τ : Gal(L/K))⁻¹ • x)
        have hτ := Ideal.smul_mem_pointwise_smul (τ : Gal(L/K)) _ P hσ
        rw [τ.property] at hτ
        simpa only [mul_smul, smul_sub, smul_inv_smul] using hτ⟩
    e = q * n ∧
    (p = 0 → q = 1) ∧
    (0 < p → ∃ a : ℕ, q = p ^ a) ∧
    rootsOfUnity e (B ⧸ P) = rootsOfUnity n (B ⧸ P) ∧
    Nat.card (tameInertiaGroup A K L P) = n ∧
    ∃ θcan : tameInertiaGroup A K L P ≃* rootsOfUnity n (B ⧸ P),
      (∀ σ : I, θcan (tameInertiaMap A K L P σ) =
        canonicalInertiaCharacter A K L P σ) ∧
      (∀ (τ : D) (σ : I),
        canonicalInertiaCharacter A K L P (conjugate τ σ) =
          (Ideal.quotientEquiv P P
            ((MulSemiringAction.toRingEquiv Gal(L/K) B) (τ : Gal(L/K)))
            τ.property.symm).toMulEquiv.restrictRootsOfUnity n
              (canonicalInertiaCharacter A K L P σ)) ∧
      ∀ (M : Type*) [Field M] [Algebra K M] [FiniteDimensional K M]
        [IsGalois K M] [Algebra L M] [IsScalarTower K L M]
        [Algebra A M] [IsScalarTower A K M] [IsScalarTower A L M]
        (r : Ideal (integralClosure A M)) [r.IsMaximal],
        let C := integralClosure A M
        let fBC : B →ₐ[A] C := (IsScalarTower.toAlgHom A L M).mapIntegralClosure
        letI : Algebra B C := fBC.toRingHom.toAlgebra
        letI : IsScalarTower A B C :=
          IsScalarTower.of_algebraMap_eq fun x ↦ (fBC.commutes x).symm
        letI : Algebra.IsIntegral B C :=
          ⟨fun x ↦ IsIntegral.tower_top (IsIntegralClosure.isIntegral A M x)⟩
        letI : MulSemiringAction Gal(M/K) C :=
          IsIntegralClosure.MulSemiringAction A K M C
        let Q := r.under B
        let pA := r.under A
        letI : r.LiesOver Q := ⟨rfl⟩
        letI : Q.LiesOver pA := ⟨r.under_under.symm⟩
        letI : Field (A ⧸ pA) := Ideal.Quotient.field pA
        letI : Field (B ⧸ Q) := Ideal.Quotient.field Q
        letI : Field (C ⧸ r) := Ideal.Quotient.field r
        letI : Algebra (B ⧸ Q) (C ⧸ r) := Ideal.Quotient.algebraOfLiesOver r Q
        let _e' := r.ramificationIdx A
        let n' := tameRamificationIndex A M r
        let n := tameRamificationIndex A L Q
        let res : Gal(M/K) →* Gal(L/K) := AlgEquiv.restrictNormalHom L
        ∃ (resI : r.inertia Gal(M/K) →* Q.inertia Gal(L/K))
          (rootRestriction : rootsOfUnity n' (C ⧸ r) →* rootsOfUnity n (B ⧸ Q)),
          (∀ σ, (resI σ).1 = res σ.1) ∧
          n ∣ n' ∧
          (∀ ζ, algebraMap (B ⧸ Q) (C ⧸ r) (rootRestriction ζ).1.1 =
            (ζ ^ (n' / n)).1.1) ∧
          ∀ σ, rootRestriction (canonicalInertiaCharacter A K M r σ) =
            canonicalInertiaCharacter A K L Q (resI σ)

Sources

• Section 15.114, Tag 09E3
• AxiomMath/Stacks at 5ed82d14

LeanIsDiscreteValuationRing.galAction_transitive_on_maximalIdeals_integralClosure · IsDiscreteValuationRing.exists_uniform_ramificationIdx_inertiaDeg_and_finrank_eq · IsDiscreteValuationRing.under_eq_maximalIdeal_and_ker_stabilizerHom_eq_inertia · MulAction.stabilizer · Ideal.inertia · Ideal.Quotient.stabilizerHom · Ideal.Quotient.ker_stabilizerHom · IsDiscreteValuationRing.residueField_normal_and_stabilizerHom_surjective_of_isGalois

Section15.115 Krasner's lemma#

Section 15.115 of the Stacks Project, Tag 0BU9. 3 tags here are carried by 7 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 09EI, a lemma

Tag 09EI in the Stacks Project.

krasner_lemma, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Krasner.lean, line 295.

Krasner's lemma (Stacks Tag 09EI). Let A be a complete local Noetherian domain of Krull dimension 1 with maximal ideal m. If α is a root of P that is not a root of P', then for every c there is an n such that any polynomial Q whose coefficients all lie in m ^ n makes P + Q have a root β with β - α ∈ m ^ c.

theorem krasner_lemma
    (A : Type*) [CommRing A] [IsDomain A] [IsLocalRing A] [IsNoetherianRing A]
    (hdim : ringKrullDim A = 1)
    [IsAdicComplete (maximalIdeal A) A]
    (P : A[X]) (α : A)
    (hroot : P.eval α = 0)
    (hderiv : (derivative P).eval α ≠ 0) :
    ∀ c : ℕ, ∃ n : ℕ, ∀ Q : A[X], (∀ i, Q.coeff i ∈ (maximalIdeal A) ^ n) →
      ∃ β : A, (P + Q).eval β = 0 ∧ β - α ∈ (maximalIdeal A) ^ c

Tag 09EJ, a lemma

Tag 09EJ in the Stacks Project.

Krasner.exists_finite_separable_extension_tensorProduct_equiv, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Krasner.lean, line 851.

Stacks 09EJ. Let A be a Noetherian local ring whose completion at its maximal ideal is a one-dimensional domain. A finite separable extension of the fraction field of the completion is obtained by scalar extension from a finite separable extension of the fraction field of A.

theorem exists_finite_separable_extension_tensorProduct_equiv
    {A K Khat M : Type*} [CommRing A] [IsNoetherianRing A] [IsLocalRing A]
    [IsDomain (AdicCompletion (IsLocalRing.maximalIdeal A) A)]
    (hAhat_dim : ringKrullDim (AdicCompletion (IsLocalRing.maximalIdeal A) A) = 1)
    [Field K] [Algebra A K] [IsFractionRing A K]
    [Field Khat]
    [Algebra (AdicCompletion (IsLocalRing.maximalIdeal A) A) Khat]
    [IsFractionRing (AdicCompletion (IsLocalRing.maximalIdeal A) A) Khat]
    [Field M] [Algebra Khat M] [FiniteDimensional Khat M]
    [Algebra.IsSeparable Khat M] :
    let Ahat := AdicCompletion (IsLocalRing.maximalIdeal A) A
    letI : Algebra A Khat :=
      ((algebraMap Ahat Khat).comp (algebraMap A Ahat)).toAlgebra
    letI : Algebra K Khat :=
      (IsFractionRing.lift (A := A) (K := K) (L := Khat) (g := algebraMap A Khat)
        ((IsFractionRing.injective Ahat Khat).comp
          (by
            dsimp only [Ahat]
            exact AdicCompletion.of_injective (IsLocalRing.maximalIdeal A) A))).toAlgebra
    ∃ L : IntermediateField K (AlgebraicClosure K),
      FiniteDimensional K L ∧ Algebra.IsSeparable K L ∧
        Nonempty (TensorProduct K Khat L ≃ₐ[Khat] M)

Tag 09EK, a definition

Tag 09EK in the Stacks Project.

IsDiscreteValuationRing.HasMixedCharacteristic, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/MixedCharacteristic.lean, line 52.

A discrete valuation ring has mixed characteristic (0, p) if p is positive, its residue field has characteristic p, and its fraction field has characteristic zero.

abbrev HasMixedCharacteristic (A : Type*) [CommRing A] [IsDomain A]
    [IsDiscreteValuationRing A] (p : ℕ) : Prop

IsDiscreteValuationRing.integerLocalization, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/MixedCharacteristic.lean, line 58.

The localization ℤ_(p) associated to a mixed-characteristic discrete valuation ring.

noncomputable abbrev integerLocalization (A : Type*) [CommRing A] [IsDomain A]
    [IsDiscreteValuationRing A] {p : ℕ} (h : HasMixedCharacteristic A p) : Type

IsDiscreteValuationRing.mixedCharacteristicRingHom, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/MixedCharacteristic.lean, line 67.

The canonical map ℤ_(p) → A attached to a mixed-characteristic discrete valuation ring.

noncomputable def mixedCharacteristicRingHom (A : Type*) [CommRing A] [IsDomain A]
    [IsDiscreteValuationRing A] {p : ℕ} (h : HasMixedCharacteristic A p) :
    integerLocalization A h →+* A

IsDiscreteValuationRing.absoluteRamificationIndex, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/MixedCharacteristic.lean, line 93.

The absolute ramification index of a mixed-characteristic discrete valuation ring, namely the ramification index of its canonical extension ℤ_(p) → A.

noncomputable def absoluteRamificationIndex (A : Type*) [CommRing A] [IsDomain A]
    [IsDiscreteValuationRing A] {p : ℕ} (h : HasMixedCharacteristic A p) : ℕ

IsDiscreteValuationRing.hasMixedCharacteristic_iff_mixedCharZero_and_integerLocalization_extension, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/MixedCharacteristic.lean, line 102.

For a discrete valuation ring, the residue-field formulation of mixed characteristic agrees with Mathlib's MixedCharZero; moreover, the maximal ideal lies over (p) and the canonical map from ℤ_(p) is an injective local homomorphism whose source is a discrete valuation ring.

theorem hasMixedCharacteristic_iff_mixedCharZero_and_integerLocalization_extension
    (A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A] (p : ℕ) :
    (HasMixedCharacteristic A p ↔
      0 < p ∧ CharP (IsLocalRing.ResidueField A) p ∧ MixedCharZero A p) ∧
      ∀ h : HasMixedCharacteristic A p,
        (IsLocalRing.maximalIdeal A).under ℤ = Ideal.span {(p : ℤ)} ∧
          Function.Injective (mixedCharacteristicRingHom A h) ∧
          IsLocalHom (mixedCharacteristicRingHom A h) ∧
          IsDiscreteValuationRing (integerLocalization A h)

Sources

• Section 15.115, Tag 0BU9
• AxiomMath/Stacks at 5ed82d14

Leankrasner_lemma · Krasner.exists_finite_separable_extension_tensorProduct_equiv · IsDiscreteValuationRing.HasMixedCharacteristic · IsDiscreteValuationRing.integerLocalization · IsDiscreteValuationRing.mixedCharacteristicRingHom · IsDiscreteValuationRing.absoluteRamificationIndex · IsDiscreteValuationRing.hasMixedCharacteristic_iff_mixedCharZero_and_integerLocalization_extension

Section15.116 Abhyankar's lemma and tame ramification#

Section 15.116 of the Stacks Project, Tag 0EXT. 9 tags here are carried by 15 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 09EM, a remark

Tag 09EM in the Stacks Project.

IsDiscreteValuationRing.integralClosures_after_finite_fraction_field_extension, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/BaseChangeClosure.lean, line 744.

The integral closures attached to a finite extension K₁/K and an extension of discrete valuation rings A → B have the product, spectral, local, henselian, inseparable, separable, and Nagata properties listed in Stacks Project, Tag 09EM.

theorem integralClosures_after_finite_fraction_field_extension
    (A : Type uA) (B : Type uB) (K : Type uK) (L : Type uL)
    (K₁ : Type uK₁) (L₁ : Type uL₁) (A₁ : Type uA₁) (B₁ : Type uB₁)
    [CommRing A] [CommRing B] [Field K] [Field L] [Field K₁]
    [CommRing L₁] [CommRing A₁] [CommRing B₁]
    [IsDomain A] [IsDomain B] [IsDiscreteValuationRing A] [IsDiscreteValuationRing B]
    [Algebra A B] [IsExtension A B]
    [Algebra A K] [Algebra A L] [Algebra B L] [Algebra K L] [Algebra K K₁]
    [Algebra A K₁] [Algebra K L₁] [Algebra B L₁] [Algebra L L₁] [Algebra K₁ L₁]
    [Algebra A A₁] [Algebra B B₁] [Algebra A B₁] [Algebra A₁ B₁] [Algebra B₁ L₁]
    [IsScalarTower A B L] [IsScalarTower A K L] [IsScalarTower A K K₁]
    [IsScalarTower K L L₁] [IsScalarTower K K₁ L₁] [IsScalarTower B L L₁]
    [IsScalarTower A B B₁] [IsScalarTower A A₁ B₁] [IsScalarTower B B₁ L₁]
    [IsFractionRing A K] [IsFractionRing B L] [Module.Finite K K₁]
    (eL₁ : L₁ ≃ₐ[K] ((L ⊗[K] K₁) ⧸ nilradical (L ⊗[K] K₁)))
    (eA₁ : A₁ ≃ₐ[A] integralClosure A K₁)
    (eB₁ : B₁ ≃ₐ[B] integralClosure B L₁)
    (eL₁_left : ∀ x : L,
      eL₁ (algebraMap L L₁ x) = Ideal.Quotient.mk _
        ((Algebra.TensorProduct.includeLeft : L →ₐ[K] L ⊗[K] K₁) x))
    (eL₁_right : ∀ x : K₁,
      eL₁ (algebraMap K₁ L₁ x) = Ideal.Quotient.mk _
        ((Algebra.TensorProduct.includeRight : K₁ →ₐ[K] L ⊗[K] K₁) x))
    (eB₁_coe : ∀ x : B₁, ((eB₁ x : integralClosure B L₁) : L₁) = algebraMap B₁ L₁ x)
    (hfieldSquare : ∀ x : K, algebraMap L L₁ (algebraMap K L x) =
      algebraMap K₁ L₁ (algebraMap K K₁ x))
    (hringSquare : ∀ x : A₁, algebraMap B₁ L₁ (algebraMap A₁ B₁ x) =
      algebraMap K₁ L₁ ((eA₁ x : integralClosure A K₁) : K₁)) :
    IsNonemptyFiniteProductOfFiniteFieldExtensions L L₁ ∧
      IsDedekindDomain A₁ ∧
      IsFiniteProductOfDedekindDomains B₁ ∧
      (∃ π : A,
        IsLocalRing.maximalIdeal A = Ideal.span {π} ∧
        IsRegular (algebraMap A (B ⊗[A] A₁) π) ∧
        (∃ e : Localization.Away (algebraMap A (B ⊗[A] A₁) π) ≃+* (L ⊗[K] K₁),
          (∀ b : B, e (algebraMap (B ⊗[A] A₁) _
              ((Algebra.TensorProduct.includeLeft : B →ₐ[A] B ⊗[A] A₁) b)) =
            (Algebra.TensorProduct.includeLeft : L →ₐ[K] L ⊗[K] K₁) (algebraMap B L b)) ∧
          (∀ a : A₁, e (algebraMap (B ⊗[A] A₁) _
              ((Algebra.TensorProduct.includeRight : A₁ →ₐ[A] B ⊗[A] A₁) a)) =
            (Algebra.TensorProduct.includeRight : K₁ →ₐ[K] L ⊗[K] K₁)
              ((eA₁ a : integralClosure A K₁) : K₁))) ∧
        ∃ g : (B ⊗[A] A₁) →+* B₁,
          g.IsIntegral ∧ RingHom.ker g ≤ nilradical (B ⊗[A] A₁) ∧
          Function.Surjective (PrimeSpectrum.comap g) ∧
          (∀ b : B, g ((Algebra.TensorProduct.includeLeft : B →ₐ[A] B ⊗[A] A₁) b) =
            algebraMap B B₁ b) ∧
          (∀ a : A₁, g ((Algebra.TensorProduct.includeRight : A₁ →ₐ[A] B ⊗[A] A₁) a) =
            algebraMap A₁ B₁ a)) ∧
      Function.Surjective (PrimeSpectrum.comap (algebraMap A₁ B₁)) ∧
      Finite (MaximalSpectrum A₁) ∧
      Nonempty (MaximalSpectrum A₁) ∧
      (∀ p : MaximalSpectrum A₁,
        Finite {q : MaximalSpectrum B₁ //
          q.asIdeal.comap (algebraMap A₁ B₁) = p.asIdeal} ∧
        Nonempty {q : MaximalSpectrum B₁ //
          q.asIdeal.comap (algebraMap A₁ B₁) = p.asIdeal}) ∧
      (∀ p : Ideal A₁, p.IsMaximal →
        ∃ q : Ideal B₁, q.IsMaximal ∧ q.comap (algebraMap A₁ B₁) = p) ∧
      (∀ q : Ideal B₁, q.IsMaximal →
        (q.comap (algebraMap A₁ B₁)).IsMaximal ∧
        ∃ (_ : q.IsPrime) (_ : (q.comap (algebraMap A₁ B₁)).IsPrime),
          (∃ _ : IsDomain (Localization (q.comap (algebraMap A₁ B₁)).primeCompl),
            IsDiscreteValuationRing (Localization (q.comap (algebraMap A₁ B₁)).primeCompl)) ∧
          (∃ _ : IsDomain (Localization q.primeCompl),
            IsDiscreteValuationRing (Localization q.primeCompl)) ∧
          ∃ φ : Localization (q.comap (algebraMap A₁ B₁)).primeCompl →+*
              Localization q.primeCompl,
            Function.Injective φ ∧ IsLocalHom φ ∧
              ∀ x : A₁, φ (algebraMap A₁ _ x) = algebraMap B₁ _ (algebraMap A₁ B₁ x)) ∧
      (HenselianLocalRing A → ∃ _ : IsDomain A₁, IsDiscreteValuationRing A₁) ∧
      (IsAdicComplete (IsLocalRing.maximalIdeal A) A →
        ∃ _ : IsDomain A₁, IsDiscreteValuationRing A₁) ∧
      (HenselianLocalRing B →
        IsFiniteProductOfDiscreteValuationRings B₁) ∧
      (IsAdicComplete (IsLocalRing.maximalIdeal B) B →
        IsFiniteProductOfDiscreteValuationRings B₁) ∧
      (IsPurelyInseparable K K₁ →
        (∃ _ : IsDomain A₁, IsDiscreteValuationRing A₁) ∧
        (∃ _ : IsDomain B₁, IsDiscreteValuationRing B₁)) ∧
      (Algebra.IsSeparable K K₁ →
        IsReduced (L ⊗[K] K₁) ∧
        IsNonemptyFiniteProductOfFiniteSeparableFieldExtensions L L₁ ∧
        Module.Finite A A₁ ∧ Module.Finite B B₁) ∧
      (IsNagataRing A → Module.Finite A A₁) ∧
      (IsNagataRing B → Module.Finite B B₁)

Tag 09EV, a lemma

Tag 09EV in the Stacks Project.

Stacks.KummerExtension.intermediateField_eq_adjoin, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Kummer/IntermediateFields.lean, line 406. The attribute's comment: “(6)”.

If n is prime to the residue characteristic of A, then every intermediate field K ⊆ L ⊆ K_1 is of the form L = K[pi^{1/d}] for some positive divisor d | n: L is generated over K by the d-th root (pi')^{n/d} of pi, and [L : K] = d and [K_1 : L] = n / d.

theorem intermediateField_eq_adjoin [Fact (Irreducible (kummerPoly K π n))]
    (hn : 2 ≤ n) (hπ : Irreducible π)
    (hcop : ringChar (A ⧸ IsLocalRing.maximalIdeal A) = 0 ∨
      Nat.Coprime (ringChar (A ⧸ IsLocalRing.maximalIdeal A)) n)
    (L : IntermediateField K (K1 K π n)) :
    ∃ d : ℕ, 0 < d ∧ d ∣ n ∧
      L = IntermediateField.adjoin K {(piRoot K π n) ^ (n / d)} ∧
      Module.finrank K L = d ∧
      Module.finrank L (K1 K π n) = n / d

Stacks.KummerExtension.isDiscreteValuationRing_A1, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Kummer/Ramification.lean, line 184. The attribute's comment: “(3)”.

A_1 is a discrete valuation ring.

theorem isDiscreteValuationRing_A1 [Fact (Irreducible (kummerPoly K π n))]
    (hn : 2 ≤ n) (hπ : Irreducible π) :
    IsDiscreteValuationRing (A1 K π n)

Stacks.KummerExtension.exists_ramificationIdx_eq, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Kummer/Ramification.lean, line 221. The attribute's comment: “(4)”.

A_1 has a unique maximal ideal m lying over m_A, and the ramification index of m over m_A is n.

theorem exists_ramificationIdx_eq (hn : 2 ≤ n) (hπ : Irreducible π) :
    ∃ m : Ideal (A1 K π n), m.IsMaximal ∧
      m.LiesOver (IsLocalRing.maximalIdeal A) ∧
      (∀ m' : Ideal (A1 K π n), m'.IsMaximal →
        m'.LiesOver (IsLocalRing.maximalIdeal A) → m' = m) ∧
      Ideal.ramificationIdx' (IsLocalRing.maximalIdeal A) m = n

Stacks.KummerExtension.isTotallyRamified_K1, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Kummer/Ramification.lean, line 284. The attribute's comment: “(5)”.

K_1 / K is totally ramified with respect to A, whatever the residue characteristic of A.

theorem isTotallyRamified_K1 [Fact (Irreducible (kummerPoly K π n))]
    (hn : 2 ≤ n) (hπ : Irreducible π) :
    IsTotallyRamified A K (K1 K π n)

Stacks.KummerExtension.isTamelyRamified_K1, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Kummer/Ramification.lean, line 371. The attribute's comment: “(6)”.

If n is prime to the residue characteristic of A, then K_1 / K is tamely ramified with respect to A.

theorem isTamelyRamified_K1 [Fact (Irreducible (kummerPoly K π n))]
    [Algebra.IsSeparable K (K1 K π n)]
    (hn : 2 ≤ n) (hπ : Irreducible π)
    (hcop : ringChar (A ⧸ IsLocalRing.maximalIdeal A) = 0 ∨
      Nat.Coprime (ringChar (A ⧸ IsLocalRing.maximalIdeal A)) n) :
    IsTamelyRamified A K (K1 K π n)

Stacks.KummerExtension.finrank_K1, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Kummer/RingOfIntegers.lean, line 396. The attribute's comment: “(1)”.

[K_1 : K] = n.

theorem finrank_K1 (hn : 2 ≤ n) (hπ : Irreducible π) :
    Module.finrank K (K1 K π n) = n

Stacks.KummerExtension.integralClosure_eq_adjoin, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Kummer/RingOfIntegers.lean, line 555. The attribute's comment: “(2)”.

The integral closure A_1 of A in K_1 is the A-subalgebra A[pi'] = A[pi^{1/n}] generated by pi'.

theorem integralClosure_eq_adjoin (hn : 2 ≤ n) (hπ : Irreducible π) :
    (A1 K π n) = Algebra.adjoin A {piRoot K π n}

Tag 09EQ, a lemma

Tag 09EQ in the Stacks Project.

IsDiscreteValuationRing.formallySmooth_localizations_after_finite_fraction_field_extension, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Abhyankar/FiniteFractionFieldExtension.lean, line 348.

Formal smoothness of an extension of discrete valuation rings persists at every local factor after a finite extension of the fraction field. The integral closure upstairs is the reduced tensor product, and it is the unreduced tensor product whenever L ⊗_K K₁ is reduced.

theorem formallySmooth_localizations_after_finite_fraction_field_extension
    (A : Type uA) (B : Type uB) (K : Type uK) (L : Type uL)
    (K₁ : Type uK₁) (L₁ : Type uL₁) (A₁ : Type uA₁) (B₁ : Type uB₁)
    [CommRing A] [CommRing B] [Field K] [Field L] [Field K₁]
    [CommRing L₁] [CommRing A₁] [CommRing B₁]
    [IsDomain A] [IsDomain B] [IsDiscreteValuationRing A] [IsDiscreteValuationRing B]
    [Algebra A B] [IsExtension A B]
    [Algebra A K] [Algebra A L] [Algebra B L] [Algebra K L] [Algebra K K₁]
    [Algebra A K₁] [Algebra K L₁] [Algebra B L₁] [Algebra L L₁] [Algebra K₁ L₁]
    [Algebra A A₁] [Algebra B B₁] [Algebra A B₁] [Algebra A₁ B₁] [Algebra B₁ L₁]
    [IsScalarTower A B L] [IsScalarTower A K L] [IsScalarTower A K K₁]
    [IsScalarTower K L L₁] [IsScalarTower K K₁ L₁] [IsScalarTower B L L₁]
    [IsScalarTower A B B₁] [IsScalarTower A A₁ B₁] [IsScalarTower B B₁ L₁]
    [IsFractionRing A K] [IsFractionRing B L] [Module.Finite K K₁]
    (eL₁ : L₁ ≃ₐ[K] ((L ⊗[K] K₁) ⧸ nilradical (L ⊗[K] K₁)))
    (eA₁ : A₁ ≃ₐ[A] integralClosure A K₁)
    (eB₁ : B₁ ≃ₐ[B] integralClosure B L₁)
    (eL₁_left : ∀ x : L,
      eL₁ (algebraMap L L₁ x) = Ideal.Quotient.mk _
        ((Algebra.TensorProduct.includeLeft : L →ₐ[K] L ⊗[K] K₁) x))
    (eL₁_right : ∀ x : K₁,
      eL₁ (algebraMap K₁ L₁ x) = Ideal.Quotient.mk _
        ((Algebra.TensorProduct.includeRight : K₁ →ₐ[K] L ⊗[K] K₁) x))
    (eB₁_coe : ∀ x : B₁, ((eB₁ x : integralClosure B L₁) : L₁) = algebraMap B₁ L₁ x)
    (hfieldSquare : ∀ x : K, algebraMap L L₁ (algebraMap K L x) =
      algebraMap K₁ L₁ (algebraMap K K₁ x))
    (hringSquare : ∀ x : A₁, algebraMap B₁ L₁ (algebraMap A₁ B₁ x) =
      algebraMap K₁ L₁ ((eA₁ x : integralClosure A K₁) : K₁))
    (hformal : RingHom.FormallySmoothNAdic.{max uA uB} (algebraMap A B)
      (IsLocalRing.maximalIdeal B)) :
    (∃ eB : B₁ ≃ₐ[B] ((B ⊗[A] A₁) ⧸ nilradical (B ⊗[A] A₁)),
        (∀ x : A₁, eB (algebraMap A₁ B₁ x) = Ideal.Quotient.mk _
          ((Algebra.TensorProduct.includeRight : A₁ →ₐ[A] B ⊗[A] A₁) x))) ∧
      (IsReduced (L ⊗[K] K₁) →
        (∃ eL : L₁ ≃ₐ[L] (L ⊗[K] K₁),
        (∀ x : L₁, Ideal.Quotient.mk _ (eL x) = eL₁ x) ∧
        (∀ x : K₁, eL (algebraMap K₁ L₁ x) =
          (Algebra.TensorProduct.includeRight : K₁ →ₐ[K] L ⊗[K] K₁) x)) ∧
      (∃ eB : B₁ ≃ₐ[B] (B ⊗[A] A₁),
        (∀ x : A₁, eB (algebraMap A₁ B₁ x) =
          (Algebra.TensorProduct.includeRight : A₁ →ₐ[A] B ⊗[A] A₁) x))) ∧
      (∀ q : Ideal B₁, q.IsMaximal →
        (q.comap (algebraMap A₁ B₁)).IsMaximal ∧
        ∃ (_ : q.IsPrime) (_ : (q.comap (algebraMap A₁ B₁)).IsPrime),
          (∃ _ : IsDomain (Localization (q.comap (algebraMap A₁ B₁)).primeCompl),
            IsDiscreteValuationRing
              (Localization (q.comap (algebraMap A₁ B₁)).primeCompl)) ∧
          (∃ _ : IsDomain (Localization q.primeCompl),
            IsDiscreteValuationRing (Localization q.primeCompl)) ∧
          ∃ φ : Localization (q.comap (algebraMap A₁ B₁)).primeCompl →+*
              Localization q.primeCompl,
            Function.Injective φ ∧ IsLocalHom φ ∧
              (∀ x : A₁, φ (algebraMap A₁ _ x) =
                algebraMap B₁ _ (algebraMap A₁ B₁ x)) ∧
              RingHom.FormallySmoothNAdic.{max uA₁ uB₁} φ
                (IsLocalRing.maximalIdeal (Localization q.primeCompl)))

Tag 0BRM, a lemma

Tag 0BRM in the Stacks Project.

IsDiscreteValuationRing.formallySmooth_localizations_of_ramificationIndex_dvd, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Abhyankar/AbhyankarLemma.lean, line 97.

Abhyankar's lemma. A tamely ramified extension of discrete valuation rings becomes formally smooth at every local factor after a finite fraction-field extension whose local ramification index is divisible by the original one.

theorem formallySmooth_localizations_of_ramificationIndex_dvd
    (A : Type u) (B : Type v) (K : Type w) (L : Type x) (K₁ : Type y)
    [CommRing A] [CommRing B] [Field K] [Field L] [Field K₁]
    [IsDomain A] [IsDomain B] [IsDiscreteValuationRing A] [IsDiscreteValuationRing B]
    [Algebra A B] [IsExtension A B]
    [Algebra A K] [Algebra A L] [Algebra B L] [Algebra K L] [Algebra K K₁]
    [Algebra A K₁]
    [IsScalarTower A B L] [IsScalarTower A K L] [IsScalarTower A K K₁]
    [IsFractionRing A K] [IsFractionRing B L] [Module.Finite K K₁]
    (htame : ringChar (IsLocalRing.ResidueField A) = 0 ∨
      Nat.Coprime (ringChar (IsLocalRing.ResidueField A))
          (ramificationIndex A B) ∧
        Algebra.IsSeparableExtension (IsLocalRing.ResidueField A)
          (IsLocalRing.ResidueField B)) :
    ∃ (n : ℕ) (E : Fin n → Type (max u v w x y))
      (_ : ∀ i, Field (E i))
      (_ : ∀ i, Algebra K (E i))
      (_ : ∀ i, Algebra L (E i))
      (_ : ∀ i, Algebra K₁ (E i))
      (_ : ∀ i, Algebra B (E i))
      (_ : ∀ i, IsScalarTower K L (E i))
      (_ : ∀ i, IsScalarTower K K₁ (E i))
      (_ : ∀ i, Module.Finite L (E i))
      (decomp : ((L ⊗[K] K₁) ⧸ nilradical (L ⊗[K] K₁)) ≃ₐ[K] ∀ i, E i),
      0 < n ∧
      (∀ i (z : L),
        decomp (Ideal.Quotient.mk _
          ((Algebra.TensorProduct.includeLeft : L →ₐ[K] L ⊗[K] K₁) z)) i =
          algebraMap L (E i) z) ∧
      (∀ i (z : K₁),
        decomp (Ideal.Quotient.mk _
          ((Algebra.TensorProduct.includeRight : K₁ →ₐ[K] L ⊗[K] K₁) z)) i =
          algebraMap K₁ (E i) z) ∧
      (∀ i (z : B), algebraMap B (E i) z =
        algebraMap L (E i) (algebraMap B L z)) ∧
      ∀ i, let A₁ := integralClosure A K₁
        let B₁ := integralClosure B (E i)
        ∃ f : A₁ →+* B₁,
          (∀ z : A₁, ((f z : B₁) : E i) =
            algebraMap K₁ (E i) ((z : A₁) : K₁)) ∧
          (∀ z : A, f (algebraMap A A₁ z) =
            algebraMap B B₁ (algebraMap A B z)) ∧
          ∀ p : Ideal A₁, p.IsMaximal →
            ramificationIndex A B ∣ p.ramificationIdx A →
            ∀ q : Ideal B₁, q.IsMaximal → q.comap f = p →
              ∃ (_ : q.IsPrime) (_ : p.IsPrime),
                (∃ _ : IsDomain (Localization p.primeCompl),
                  IsDiscreteValuationRing (Localization p.primeCompl)) ∧
                (∃ _ : IsDomain (Localization q.primeCompl),
                  IsDiscreteValuationRing (Localization q.primeCompl)) ∧
                ∃ φ : Localization p.primeCompl →+* Localization q.primeCompl,
                  Function.Injective φ ∧ IsLocalHom φ ∧
                    (∀ z : A₁, φ (algebraMap A₁ _ z) =
                      algebraMap B₁ _ (f z)) ∧
                    RingHom.FormallySmoothNAdic φ
                      (IsLocalRing.maximalIdeal (Localization q.primeCompl))

Tag 0EXU, a lemma

Tag 0EXU in the Stacks Project.

IsDiscreteValuationRing.isTamelyRamified_tower, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/TameRamification.lean, line 307.

Tame ramification composes in a tower of finite separable extensions when the upper extension is tamely ramified at every maximal localization of the intermediate integral closure.

theorem isTamelyRamified_tower
    (A : Type u) (K : Type v) (L : Type w) (M : Type x)
    [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [Field K] [Field L] [Field M]
    [Algebra A K] [IsFractionRing A K]
    [Algebra K L] [Algebra L M] [Algebra K M] [IsScalarTower K L M]
    [Algebra A L] [IsScalarTower A K L]
    [Algebra A M] [IsScalarTower A K M] [IsScalarTower A L M]
    [Module.Finite K L] [Algebra.IsSeparable K L]
    [Module.Finite L M] [Algebra.IsSeparable L M]
    (hLK : Stacks.KummerExtension.IsTamelyRamified A K L)
    (hML : ∀ (m : Ideal (integralClosure A L)) (hm : m.IsMaximal),
      letI : m.IsPrime := hm.isPrime
      letI : IsFractionRing (integralClosure A L) L :=
        integralClosure.isFractionRing_of_finite_extension K L
      letI : Algebra (Localization.AtPrime m) L :=
        IsLocalization.localizationAlgebraOfSubmonoidLe _ _ m.primeCompl
          (nonZeroDivisors (integralClosure A L)) m.primeCompl_le_nonZeroDivisors
      letI : Algebra (Localization.AtPrime m) M :=
        ((algebraMap L M).comp (algebraMap (Localization.AtPrime m) L)).toAlgebra
      letI : IsScalarTower (Localization.AtPrime m) L M :=
        IsScalarTower.of_algebraMap_eq fun _ => rfl
      Stacks.KummerExtension.IsTamelyRamified (Localization.AtPrime m) L M) :
    letI : Module.Finite K M := Module.Finite.trans L M
    letI : Algebra.IsSeparable K M := Algebra.IsSeparable.trans K L M
    Stacks.KummerExtension.IsTamelyRamified A K M

Tag 0EXV, a lemma

Tag 0EXV in the Stacks Project.

IsDiscreteValuationRing.isTamelyRamified_of_isTamelyRamified_tower, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/TameRamification.lean, line 398. References: SGA 1 XIII (2.0.3), “subalgebra of tame”.

Every intermediate extension of a finite separable extension tamely ramified with respect to a discrete valuation ring is itself tamely ramified.

theorem isTamelyRamified_of_isTamelyRamified_tower
    (A : Type u) (K : Type v) (L : Type w) (M : Type x)
    [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [Field K] [Field L] [Field M]
    [Algebra A K] [IsFractionRing A K]
    [Algebra K L] [Algebra L M] [Algebra K M] [IsScalarTower K L M]
    [Algebra A L] [IsScalarTower A K L]
    [Algebra A M] [IsScalarTower A K M] [IsScalarTower A L M]
    [Module.Finite K L] [Algebra.IsSeparable K L]
    [Module.Finite L M] [Algebra.IsSeparable L M]
    (hM : letI : Module.Finite K M := Module.Finite.trans L M
      letI : Algebra.IsSeparable K M := Algebra.IsSeparable.trans K L M
      Stacks.KummerExtension.IsTamelyRamified A K M) :
    Stacks.KummerExtension.IsTamelyRamified A K L

Tag 0EXW, a lemma

Tag 0EXW in the Stacks Project.

IsDiscreteValuationRing.isTamelyRamifiedWithRespectTo_tfae_hasUnramifiedKummerOverfield, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Abhyankar/TameRamificationKummer.lean, line 764.

A finite separable extension of the fraction field of a discrete valuation ring is tamely ramified if and only if it embeds into an unramified extension after some Kummer base change; equivalently, such an unramified overfield exists after every prime-to-residue-characteristic multiple of one fixed exponent.

theorem isTamelyRamifiedWithRespectTo_tfae_hasUnramifiedKummerOverfield
    (A : Type u) (K : Type v) (L : Type w)
    [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [Field K] [Field L]
    [Algebra A K] [IsFractionRing A K]
    [Algebra K L] [Algebra A L] [IsScalarTower A K L]
    [FiniteDimensional K L] [Algebra.IsSeparable K L]
    (π : A) (hπ : Irreducible π) :
    List.TFAE
      [ IsTamelyRamifiedWithRespectTo A K L,
        ∃ e : ℕ, HasUnramifiedKummerOverfield A K L π hπ e,
        ∃ e₀ : ℕ, 1 ≤ e₀ ∧ IsUnit (e₀ : IsLocalRing.ResidueField A) ∧
          ∀ d : ℕ, 1 ≤ d → IsUnit (d : IsLocalRing.ResidueField A) →
            HasUnramifiedKummerOverfield A K L π hπ (d * e₀) ]

Tag 0EXX, a lemma

Tag 0EXX in the Stacks Project.

IsDiscreteValuationRing.exists_isGalois_and_common_extension_isTamelyRamifiedWithRespectTo, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Abhyankar/TameGaloisClosure.lean, line 249.

Stacks 0EXX. Every finite separable extension of the fraction field of a discrete valuation ring which is tamely ramified with respect to the ring is contained in a finite Galois extension with the same property. Any two such extensions are contained in a common finite separable extension which is again tamely ramified.

theorem exists_isGalois_and_common_extension_isTamelyRamifiedWithRespectTo
    (A : Type u) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    (K : Type v) [Field K] [Algebra A K] [IsFractionRing A K] :
    (∀ (L : Type w) [Field L] [Algebra K L] [FiniteDimensional K L]
        [Algebra.IsSeparable K L] [Algebra A L] [IsScalarTower A K L],
      IsTamelyRamifiedWithRespectTo A K L →
        ∃ (M : Type (max u v w)) (_ : Field M) (_ : Algebra K M)
          (_ : FiniteDimensional K M) (_ : Algebra.IsSeparable K M)
          (_ : Algebra A M) (_ : IsScalarTower A K M)
          (_ : Algebra L M) (_ : IsScalarTower K L M),
          IsGalois K M ∧ IsTamelyRamifiedWithRespectTo A K M) ∧
    (∀ (L₁ : Type w) [Field L₁] [Algebra K L₁] [FiniteDimensional K L₁]
        [Algebra.IsSeparable K L₁] [Algebra A L₁] [IsScalarTower A K L₁]
        (L₂ : Type x) [Field L₂] [Algebra K L₂] [FiniteDimensional K L₂]
        [Algebra.IsSeparable K L₂] [Algebra A L₂] [IsScalarTower A K L₂],
      IsTamelyRamifiedWithRespectTo A K L₁ →
        IsTamelyRamifiedWithRespectTo A K L₂ →
        ∃ (L : Type (max u v w x)) (_ : Field L) (_ : Algebra K L)
          (_ : FiniteDimensional K L) (_ : Algebra.IsSeparable K L)
          (_ : Algebra A L) (_ : IsScalarTower A K L)
          (_ : Algebra L₁ L) (_ : IsScalarTower K L₁ L)
          (_ : Algebra L₂ L) (_ : IsScalarTower K L₂ L),
          IsTamelyRamifiedWithRespectTo A K L)

Tag 0EXY, a lemma

Tag 0EXY in the Stacks Project.

IsDiscreteValuationRing.exists_algEquiv_pi_field_and_ramification_goes_up, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Abhyankar/RamificationGoesUp.lean, line 783. References: SGA 1 XIII (2.0.3), “tame after DVR extension”; SGA 1 XIII (2.0.2), “⇒: tameness ascends”.

Stacks 0EXY. A finite separable field extension remains a finite product of fields after scalar extension along an extension of discrete valuation rings. If the original extension is unramified, respectively tamely ramified, then every resulting field factor has the same property over the larger discrete valuation ring.

theorem exists_algEquiv_pi_field_and_ramification_goes_up
    (A : Type u) (B : Type v) (K : Type w) (L : Type x) (K' : Type y)
    [CommRing A] [CommRing B] [IsDomain A] [IsDomain B]
    [IsDiscreteValuationRing A] [IsDiscreteValuationRing B]
    [Field K] [Field L] [Field K']
    [Algebra A B] [IsExtension A B]
    [Algebra A K] [IsFractionRing A K]
    [Algebra B L] [IsFractionRing B L]
    [Algebra K L] [Algebra A L]
    [IsScalarTower A B L] [IsScalarTower A K L]
    [Algebra K K'] [Algebra A K'] [IsScalarTower A K K']
    [Module.Finite K K'] [Algebra.IsSeparable K K'] :
    ∃ (n : ℕ) (E : Fin n → Type (max u v w x y))
        (_ : ∀ i, Field (E i))
        (_ : ∀ i, Algebra L (E i))
        (_ : ∀ i, Algebra B (E i))
        (_ : ∀ i, IsScalarTower B L (E i))
        (_ : ∀ i, Module.Finite L (E i))
        (_ : ∀ i, Algebra.IsSeparable L (E i)),
      0 < n ∧
        Nonempty ((L ⊗[K] K') ≃ₐ[L] ∀ i, E i) ∧
        (IsUnramifiedWithRespectTo A K K' →
          ∀ i, IsUnramifiedWithRespectTo B L (E i)) ∧
        (IsTamelyRamifiedWithRespectTo A K K' →
          ∀ i, IsTamelyRamifiedWithRespectTo B L (E i))

Sources

• Section 15.116, Tag 0EXT
• AxiomMath/Stacks at 5ed82d14

LeanIsDiscreteValuationRing.integralClosures_after_finite_fraction_field_extension · Stacks.KummerExtension.intermediateField_eq_adjoin · Stacks.KummerExtension.isDiscreteValuationRing_A1 · Stacks.KummerExtension.exists_ramificationIdx_eq · Stacks.KummerExtension.isTotallyRamified_K1 · Stacks.KummerExtension.isTamelyRamified_K1 · Stacks.KummerExtension.finrank_K1 · Stacks.KummerExtension.integralClosure_eq_adjoin

Section15.117 Eliminating ramification#

Section 15.117 of the Stacks Project, Tag 09EL. 19 tags here are carried by 23 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 09EN, a definition

Tag 09EN in the Stacks Project.

IsDiscreteValuationRing.IntegralClosureBaseChange.IsWeakSolution, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Solution.lean, line 144. The attribute's comment: “(1), weak solution”.

Every localization arising from the base change has ramification index one.

def IsWeakSolution
    {A : Type uA} {B : Type uB} {K : Type uK} {L : Type uL}
    {K₁ : Type uK₁}
    [CommRing A] [CommRing B] [Field K] [Field L] [Field K₁]
    [IsDomain A] [IsDomain B] [IsDiscreteValuationRing A] [IsDiscreteValuationRing B]
    [Algebra A B] [Algebra A K] [Algebra A L] [Algebra B L]
    [Algebra K L] [Algebra K K₁] [Algebra A K₁]
    [IsExtension A B] [IsFractionRing A K] [IsFractionRing B L]
    [IsScalarTower A B L] [IsScalarTower A K L] [IsScalarTower A K K₁]
    [Module.Finite K K₁]
    (D : IntegralClosureBaseChange A B K L K₁) : Prop

IsDiscreteValuationRing.IntegralClosureBaseChange.IsSolution, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Solution.lean, line 163. The attribute's comment: “(2), solution”.

Every localization arising from the base change is formally smooth for the adic topology defined by the maximal ideal of the target localization.

def IsSolution.{v}
    {A : Type uA} {B : Type uB} {K : Type uK} {L : Type uL}
    {K₁ : Type uK₁}
    [CommRing A] [CommRing B] [Field K] [Field L] [Field K₁]
    [IsDomain A] [IsDomain B] [IsDiscreteValuationRing A] [IsDiscreteValuationRing B]
    [Algebra A B] [Algebra A K] [Algebra A L] [Algebra B L]
    [Algebra K L] [Algebra K K₁] [Algebra A K₁]
    [IsExtension A B] [IsFractionRing A K] [IsFractionRing B L]
    [IsScalarTower A B L] [IsScalarTower A K L] [IsScalarTower A K K₁]
    [Module.Finite K K₁]
    (D : IntegralClosureBaseChange.{uA, uB, uK, uL, uK₁, uL₁, uA₁, uB₁}
      A B K L K₁) : Prop

IsDiscreteValuationRing.IntegralClosureBaseChange.IsSeparableSolution, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Solution.lean, line 188. The attribute's comment: “separable solution”.

A solution whose finite extension K₁/K is separable.

def IsSeparableSolution.{v}
    {A : Type uA} {B : Type uB} {K : Type uK} {L : Type uL}
    {K₁ : Type uK₁}
    [CommRing A] [CommRing B] [Field K] [Field L] [Field K₁]
    [IsDomain A] [IsDomain B] [IsDiscreteValuationRing A] [IsDiscreteValuationRing B]
    [Algebra A B] [Algebra A K] [Algebra A L] [Algebra B L]
    [Algebra K L] [Algebra K K₁] [Algebra A K₁]
    [IsExtension A B] [IsFractionRing A K] [IsFractionRing B L]
    [IsScalarTower A B L] [IsScalarTower A K L] [IsScalarTower A K K₁]
    [Module.Finite K K₁]
    (D : IntegralClosureBaseChange.{uA, uB, uK, uL, uK₁, uL₁, uA₁, uB₁}
      A B K L K₁) : Prop

Tag 09EP, an example

Tag 09EP in the Stacks Project.

IsDiscreteValuationRing.weakSolution_isInseparable_and_inseparable_isSolution, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/WeakSolution/Inseparability.lean, line 925.

For k[[x]] → k[[x^(1/p)]] in positive characteristic p, every weak solution is inseparable, while every finite inseparable extension of k((x)) is a solution.

theorem weakSolution_isInseparable_and_inseparable_isSolution
    (p : ℕ) [Fact p.Prime]
    (k : Type uk) (A : Type uA) (B : Type uB) (K : Type uK) (L : Type uL)
    [Field k] [PerfectField k] [CharP k p]
    [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [CommRing B] [IsDomain B] [IsDiscreteValuationRing B]
    [Field K] [Field L]
    [Algebra k A] [Algebra k B] [Algebra k K] [Algebra k L]
    [Algebra A B] [Algebra A K] [Algebra A L] [Algebra B L] [Algebra K L]
    [IsScalarTower k A B] [IsScalarTower k A K] [IsScalarTower k B L]
    [IsScalarTower k K L] [IsScalarTower A B L] [IsScalarTower A K L]
    [IsExtension A B] [IsFractionRing A K] [IsFractionRing B L]
    (eA : A ≃ₐ[k] PowerSeries k) (eB : B ≃ₐ[k] PowerSeries k)
    (eK : K ≃ₐ[k] LaurentSeries k) (eL : L ≃ₐ[k] LaurentSeries k)
    (hAB : ∀ a : A, eB (algebraMap A B a) =
      PowerSeries.subst (PowerSeries.X ^ p) (eA a))
    (_hAK : ∀ a : A, eK (algebraMap A K a) = (eA a : LaurentSeries k))
    (_hBL : ∀ b : B, eL (algebraMap B L b) = (eB b : LaurentSeries k)) :
    (∀ (K₁ : Type uK₁) [Field K₁] [Algebra K K₁] [Algebra A K₁]
        [IsScalarTower A K K₁] [Module.Finite K K₁]
        (D : IntegralClosureBaseChange.{uA, uB, uK, uL, uK₁, uL₁, uA₁, uB₁}
          A B K L K₁),
        D.IsWeakSolution → ¬ Algebra.IsSeparable K K₁) ∧
      ∀ (K₁ : Type uK₁) [Field K₁] [Algebra K K₁] [Algebra A K₁]
        [IsScalarTower A K K₁] [Module.Finite K K₁],
        ¬ Algebra.IsSeparable K K₁ →
          ∃ D : IntegralClosureBaseChange.{uA, uB, uK, uL, uK₁,
              max uL uK₁, uK₁, max uL uK₁} A B K L K₁,
            D.IsSolution.{uA, uB, uK, uL, uK₁, max uL uK₁, uK₁, max uL uK₁, v}

Tag 09ER, a lemma

Tag 09ER in the Stacks Project.

IsDiscreteValuationRing.isField_tensorProduct_and_isDVR_baseChange_of_totallyRamified, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/WeakSolution/TotallyRamified.lean, line 273.

A finite separable extension totally ramified with respect to the lower discrete valuation ring preserves a weakly unramified extension after tensor-product base change. The tensor product is canonically the integral closure in the base-changed fraction field.

theorem isField_tensorProduct_and_isDVR_baseChange_of_totallyRamified
    (A : Type uA) (B : Type uB) (K : Type uK) (L : Type uL) (K₁ : Type uK₁)
    [CommRing A] [CommRing B] [Field K] [Field L] [Field K₁]
    [IsDomain A] [IsDomain B] [IsDiscreteValuationRing A] [IsDiscreteValuationRing B]
    [Algebra A B] [FaithfulSMul A B] [IsLocalHom (algebraMap A B)]
    [Algebra A K] [Algebra B L] [Algebra A L] [Algebra K L]
    [IsScalarTower A B L] [IsScalarTower A K L]
    [IsFractionRing A K] [IsFractionRing B L]
    [Algebra K K₁] [Algebra A K₁] [IsScalarTower A K K₁]
    [FiniteDimensional K K₁] [Algebra.IsSeparable K K₁]
    (hweak : Ideal.map (algebraMap A B) (IsLocalRing.maximalIdeal A) =
      IsLocalRing.maximalIdeal B)
    (htot : IsTotallyRamifiedWithRespectTo A K K₁) :
    let A₁ := integralClosure A K₁
    let B₁ := B ⊗[A] A₁
    let L₁ := L ⊗[K] K₁
    let C₁ := integralClosure B L₁
    ∃ hL₁ : IsField L₁,
      letI := hL₁.toField
      ∃ hA₁ : IsDiscreteValuationRing A₁,
        letI := hA₁
        ∃ hBdom : IsDomain B₁,
          letI := hBdom
          ∃ hB₁ : IsDiscreteValuationRing B₁,
            letI := hB₁
            let f : A₁ →+* B₁ := Algebra.TensorProduct.includeRight.toRingHom
            Function.Injective f ∧
              Ideal.map f (IsLocalRing.maximalIdeal A₁) = IsLocalRing.maximalIdeal B₁ ∧
              ∃ hCdom : IsDomain C₁,
                letI := hCdom
                ∃ hC₁ : IsDiscreteValuationRing C₁,
                  letI := hC₁
                  ∃ _ : IsFractionRing C₁ L₁,
                    ∃ e : B₁ ≃ₐ[B] C₁,
                      (∀ a : A₁,
                        ((e ((Algebra.TensorProduct.includeRight : A₁ →ₐ[A] B₁) a) : C₁)
                            : L₁) =
                          (Algebra.TensorProduct.includeRight : K₁ →ₐ[K] L₁) (a : K₁)) ∧
                        let f' : A₁ →+* C₁ :=
                          e.toRingHom.comp Algebra.TensorProduct.includeRight.toRingHom
                        Function.Injective f' ∧
                          Ideal.map f' (IsLocalRing.maximalIdeal A₁) =
                            IsLocalRing.maximalIdeal C₁

Tag 09ES, a lemma

Tag 09ES in the Stacks Project.

IsDiscreteValuationRing.weakSolutions_and_solutions_go_down_and_compose, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/EppSolution.lean, line 1213.

Weak solutions and solutions descend along a tower of extensions of discrete valuation rings. Conversely, they compose provided every field factor of the intermediate reduced tensor product is respectively a weak solution or a solution for the upper extension.

theorem weakSolutions_and_solutions_go_down_and_compose
    (A : Type uA) (B : Type uB) (C : Type uC)
    (K : Type uK) (L : Type uL) (M : Type uM) (K₁ : Type uK₁)
    [CommRing A] [CommRing B] [CommRing C]
    [IsDomain A] [IsDomain B] [IsDomain C]
    [IsDiscreteValuationRing A] [IsDiscreteValuationRing B]
    [IsDiscreteValuationRing C]
    [Field K] [Field L] [Field M] [Field K₁]
    [Algebra A B] [Algebra B C] [Algebra A C]
    [Algebra A K] [Algebra A L] [Algebra A M]
    [Algebra B L] [Algebra B M] [Algebra C M]
    [Algebra K L] [Algebra K M] [Algebra L M]
    [Algebra K K₁] [Algebra A K₁]
    [IsExtension A B] [IsExtension B C]
    [IsExtension A C]
    [IsFractionRing A K] [IsFractionRing B L] [IsFractionRing C M]
    [IsScalarTower A B C] [IsScalarTower A B L] [IsScalarTower A B M]
    [IsScalarTower B C M] [IsScalarTower A C M]
    [IsScalarTower A K L] [IsScalarTower A K M] [IsScalarTower K L M]
    [IsScalarTower B L M] [IsScalarTower A K K₁] [Module.Finite K K₁]
    (DAB : IntegralClosureBaseChange.{uA, uB, uK, uL, uK₁, uL₁, uA₁, uB₁}
      A B K L K₁)
    (DAC : IntegralClosureBaseChange.{uA, uC, uK, uM, uK₁, uM₁, uA₂, uC₁}
      A C K M K₁) :
    letI := DAB.commRingL₁
    letI := DAB.algebraLL₁
    ((DAC.IsWeakSolution → DAB.IsWeakSolution) ∧
      (DAC.IsSolution.{uA, uC, uK, uM, uK₁, uM₁, uA₂, uC₁, max uA₂ uC₁} →
        DAB.IsSolution.{uA, uB, uK, uL, uK₁, uL₁, uA₁, uB₁, max uA₁ uB₁})) ∧
      ∀ (n : ℕ) (E : Fin n → Type uE)
        [∀ i, Field (E i)] [∀ i, Algebra L (E i)] [∀ i, Algebra B (E i)]
        [∀ i, IsScalarTower B L (E i)] [∀ i, Module.Finite L (E i)],
        0 < n → Nonempty (DAB.L₁ ≃ₐ[L] ∀ i, E i) →
          ∀ DBC : ∀ i, IntegralClosureBaseChange.{uB, uC, uL, uM, uE, uN₁, uB₂, uC₂}
              B C L M (E i),
            ((DAB.IsWeakSolution ∧ ∀ i, (DBC i).IsWeakSolution) →
              DAC.IsWeakSolution) ∧
            ((DAB.IsSolution.{uA, uB, uK, uL, uK₁, uL₁, uA₁, uB₁, max uA₁ uB₁} ∧
                ∀ i, (DBC i).IsSolution.{uB, uC, uL, uM, uE, uN₁, uB₂, uC₂, max uB₂ uC₂}) →
              DAC.IsSolution.{uA, uC, uK, uM, uK₁, uM₁, uA₂, uC₁, max uA₂ uC₁})

Tag 09EU, a lemma

Tag 09EU in the Stacks Project.

IsDiscreteValuationRing.normal_extension_acts_transitively_on_maximalSpectrum_integralClosure, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/WeakSolution/GaloisAction.lean, line 397.

After a finite normal extension K₁/K, the automorphisms of K₁ over K act compatibly on the reduced base change L₁, on the integral closure A₁ of A in K₁, and on the integral closure B₁ of B in L₁. The resulting action on the maximal ideals of B₁ is transitive.

theorem normal_extension_acts_transitively_on_maximalSpectrum_integralClosure
    (A B K L K₁ L₁ A₁ B₁ : Type u)
    [CommRing A] [CommRing B] [Field K] [Field L] [Field K₁]
    [CommRing L₁] [CommRing A₁] [CommRing B₁]
    [IsDomain A] [IsDomain B] [IsDiscreteValuationRing A] [IsDiscreteValuationRing B]
    [Algebra A B] [FaithfulSMul A B] [IsLocalHom (algebraMap A B)]
    [Algebra A K] [Algebra A L] [Algebra B L] [Algebra K L] [Algebra K K₁]
    [Algebra A K₁] [Algebra K L₁] [Algebra B L₁] [Algebra L L₁] [Algebra K₁ L₁]
    [Algebra A A₁] [Algebra B B₁] [Algebra A B₁] [Algebra A₁ B₁] [Algebra B₁ L₁]
    [IsScalarTower A B L] [IsScalarTower A K L] [IsScalarTower A K K₁]
    [IsScalarTower K L L₁] [IsScalarTower K K₁ L₁] [IsScalarTower B L L₁]
    [IsScalarTower A B B₁] [IsScalarTower A A₁ B₁] [IsScalarTower B B₁ L₁]
    [IsFractionRing A K] [IsFractionRing B L] [Module.Finite K K₁] [Normal K K₁]
    (eL₁ : L₁ ≃ₐ[K] ((L ⊗[K] K₁) ⧸ nilradical (L ⊗[K] K₁)))
    (eA₁ : A₁ ≃ₐ[A] integralClosure A K₁)
    (eB₁ : B₁ ≃ₐ[B] integralClosure B L₁)
    (eL₁_left : ∀ x : L,
      eL₁ (algebraMap L L₁ x) = Ideal.Quotient.mk _
        ((Algebra.TensorProduct.includeLeft : L →ₐ[K] L ⊗[K] K₁) x))
    (eL₁_right : ∀ x : K₁,
      eL₁ (algebraMap K₁ L₁ x) = Ideal.Quotient.mk _
        ((Algebra.TensorProduct.includeRight : K₁ →ₐ[K] L ⊗[K] K₁) x))
    (eB₁_coe : ∀ x : B₁, ((eB₁ x : integralClosure B L₁) : L₁) = algebraMap B₁ L₁ x)
    (_hfieldSquare : ∀ x : K, algebraMap L L₁ (algebraMap K L x) =
      algebraMap K₁ L₁ (algebraMap K K₁ x))
    (hringSquare : ∀ x : A₁, algebraMap B₁ L₁ (algebraMap A₁ B₁ x) =
      algebraMap K₁ L₁ ((eA₁ x : integralClosure A K₁) : K₁)) :
    ∃ (ρL₁ : (K₁ ≃ₐ[K] K₁) →* (L₁ ≃ₐ[L] L₁))
        (ρA₁ : (K₁ ≃ₐ[K] K₁) →* (A₁ ≃ₐ[A] A₁))
        (ρB₁ : (K₁ ≃ₐ[K] K₁) →* (B₁ ≃ₐ[B] B₁)),
      (∀ (g : K₁ ≃ₐ[K] K₁) (x : K₁),
        ρL₁ g (algebraMap K₁ L₁ x) = algebraMap K₁ L₁ (g x)) ∧
      (∀ (g : K₁ ≃ₐ[K] K₁) (x : A₁),
        (((eA₁ (ρA₁ g x) : integralClosure A K₁) : K₁)) =
          g ((eA₁ x : integralClosure A K₁) : K₁)) ∧
      (∀ (g : K₁ ≃ₐ[K] K₁) (x : B₁),
        algebraMap B₁ L₁ (ρB₁ g x) = ρL₁ g (algebraMap B₁ L₁ x)) ∧
      (∀ (g : K₁ ≃ₐ[K] K₁) (x : A₁),
        ρB₁ g (algebraMap A₁ B₁ x) = algebraMap A₁ B₁ (ρA₁ g x)) ∧
      ∀ q q' : MaximalSpectrum B₁, ∃ g : K₁ ≃ₐ[K] K₁,
        q'.asIdeal = q.asIdeal.comap (ρB₁ g).symm.toRingHom

Tag 09EW, a lemma

Tag 09EW in the Stacks Project.

Stacks.EisensteinExtension.exists_totallyRamifiedExtension_with_uniformizer_eq, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/Kummer/Eisenstein.lean, line 636.

Let A be a discrete valuation ring with fraction field K and residue characteristic p > 0, with uniformizer π, and let n > 1. For every power q of p there is a separable extension L / K of degree q, totally ramified with respect to A with ramification index q, whose integral closure B of A is a discrete valuation ring with a uniformizer π_B satisfying π_B^q = π + π^n b and π_B^q = π + π_B^{nq} b' for some b, b' in B.

theorem exists_totallyRamifiedExtension_with_uniformizer_eq
    (A : Type u) (K : Type v) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [Field K] [Algebra A K] [IsFractionRing A K]
    (π : A) (hπ : Irreducible π) (p n q : ℕ)
    (hp : 0 < p)
    (_hres : ringChar (A ⧸ IsLocalRing.maximalIdeal A) = p)
    (hn : 1 < n) (hq : ∃ r : ℕ, q = p ^ r) :
    ∃ (L : Type v) (_ : Field L) (_ : Algebra K L) (_ : Algebra A L)
      (_ : IsScalarTower A K L) (_ : Module.Finite K L)
      (_ : Algebra.IsSeparable K L)
      (_ : IsDiscreteValuationRing (integralClosure A L)),
      Module.finrank K L = q ∧
        Stacks.KummerExtension.IsTotallyRamified A K L ∧
        Ideal.ramificationIdx' (IsLocalRing.maximalIdeal A)
            (IsLocalRing.maximalIdeal (integralClosure A L)) = q ∧
        ∃ πB b b' : integralClosure A L,
          Irreducible πB ∧
            πB ^ q = algebraMap A (integralClosure A L) π +
              (algebraMap A (integralClosure A L) π) ^ n * b ∧
            πB ^ q = algebraMap A (integralClosure A L) π + πB ^ (n * q) * b'

Tag 09EX, a lemma

Tag 09EX in the Stacks Project.

Stacks.DiscreteValuationRing.AdjoinPthRoot.statement1_not_pth_power, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/AdjoinPthRoot.lean, line 184. The attribute's comment: “(1)”.

If the residue class abar of a is not a p-th power in the residue field kappa_A, then a is not a p-th power in K.

theorem statement1_not_pth_power
    (ha : ¬ ∃ b : IsLocalRing.ResidueField A, IsLocalRing.residue A a = b ^ p) :
    ¬ ∃ b : K, algebraMap A K a = b ^ p

Stacks.DiscreteValuationRing.AdjoinPthRoot.statement2_integralClosure, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/AdjoinPthRoot.lean, line 451. The attribute's comment: “(2)”.

The integral closure of A in K[a^{1/p}] equals A[a^{1/p}] = A[alpha].

theorem statement2_integralClosure
    (ha : ¬ ∃ b : IsLocalRing.ResidueField A, IsLocalRing.residue A a = b ^ p) :
    integralClosure A (FieldExt (A := A) (K := K) p a)
      = ringExt (A := A) (K := K) p a

Stacks.DiscreteValuationRing.AdjoinPthRoot.statement3_dvr_weaklyUnramified, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/AdjoinPthRoot.lean, line 480. The attribute's comment: “(3)”.

The ring A[a^{1/p}] = A[alpha] is a discrete valuation ring, and the extension A ⊆ A[a^{1/p}] is weakly unramified: every uniformizer (irreducible element) of A maps to a uniformizer (irreducible element) of A[a^{1/p}].

theorem statement3_dvr_weaklyUnramified
    (ha : ¬ ∃ b : IsLocalRing.ResidueField A, IsLocalRing.residue A a = b ^ p) :
    haveI : Fact (Irreducible (polyK (A := A) (K := K) p a)) :=
      ⟨statement1_irreducible p a ha⟩
    letI B := ringExt (A := A) (K := K) p a
    haveI : IsDomain B := inferInstance
    IsDiscreteValuationRing B
      ∧ (∀ ϖ : A, Irreducible ϖ → Irreducible (algebraMap A B ϖ))

Tag 09EY, a lemma

Tag 09EY in the Stacks Project.

IsDiscreteValuationRing.weaklyUnramified_or_adjoin_root_or_after_totallyRamified_baseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/PurelyInseparable.lean, line 900.

For a weakly unramified extension A → B of discrete valuation rings with B Nagata, followed by a purely inseparable extension of fraction fields of prime degree p, either the composite extension is weakly unramified, the upper DVR is generated by a p-th root of a uniformizer of A, or a separable totally ramified degree-p base change makes both maps between the resulting integral closures weakly unramified.

theorem weaklyUnramified_or_adjoin_root_or_after_totallyRamified_baseChange
    (A B C K L M : Type u)
    [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [CommRing B] [IsDomain B] [IsDiscreteValuationRing B]
    [CommRing C] [IsDomain C] [IsDiscreteValuationRing C]
    [Field K] [Field L] [Field M]
    [Algebra A B] [Algebra B C] [Algebra A C]
    [FaithfulSMul A B] [FaithfulSMul B C]
    [IsScalarTower A B C]
    [IsLocalHom (algebraMap A B)] [IsLocalHom (algebraMap B C)]
    [Algebra A K] [Algebra B L] [Algebra C M]
    [Algebra A L] [Algebra A M] [Algebra B M]
    [Algebra K L] [Algebra K M] [Algebra L M]
    [IsScalarTower A B L] [IsScalarTower B C M]
    [IsScalarTower A C M] [IsScalarTower B L M]
    [IsScalarTower A K L] [IsScalarTower K L M]
    [IsScalarTower A B M] [IsScalarTower A K M]
    [IsFractionRing A K] [IsFractionRing B L] [IsFractionRing C M]
    (p : ℕ) [Fact p.Prime] [FiniteDimensional L M] [IsPurelyInseparable L M]
    (hdegree : Module.finrank L M = p)
    (π : A) (hπ : Irreducible π)
    (hNagata : IsNagataRing B)
    (hweak : Ideal.map (algebraMap A B) (IsLocalRing.maximalIdeal A) =
      IsLocalRing.maximalIdeal B) :
    Ideal.map (algebraMap A C) (IsLocalRing.maximalIdeal A) =
        IsLocalRing.maximalIdeal C ∨
      (∃ x : C, x ^ p = algebraMap A C π ∧ Algebra.adjoin B {x} = ⊤) ∨
      ∃ (K₁ : Type u) (_ : Field K₁) (_ : Algebra K K₁) (_ : Algebra A K₁)
        (_ : IsScalarTower A K K₁) (_ : FiniteDimensional K K₁)
        (_ : Algebra.IsSeparable K K₁),
        Module.finrank K K₁ = p ∧
          IsDiscreteValuationRing.IsTotallyRamifiedWithRespectTo A K K₁ ∧
          let L₁ := L ⊗[K] K₁
          let M₁ := M ⊗[K] K₁
          let A₁ := integralClosure A K₁
          let B₁ := integralClosure B L₁
          let C₁ := integralClosure C M₁
          IsField L₁ ∧ IsField M₁ ∧
            ∃ hA₁ : IsDiscreteValuationRing A₁,
              letI := hA₁
              ∃ hB₁dom : IsDomain B₁,
                letI := hB₁dom
                ∃ hB₁ : IsDiscreteValuationRing B₁,
                  letI := hB₁
                  ∃ hC₁dom : IsDomain C₁,
                    letI := hC₁dom
                    ∃ hC₁ : IsDiscreteValuationRing C₁,
                      letI := hC₁
                      ∃ (g : L₁ →+* M₁) (f : A₁ →+* B₁) (f' : B₁ →+* C₁),
                        Function.Injective f ∧ Function.Injective f' ∧
                          (∀ l : L, g
                            ((Algebra.TensorProduct.includeLeft : L →ₐ[K] L₁) l) =
                            (Algebra.TensorProduct.includeLeft : M →ₐ[K] M₁)
                              (algebraMap L M l)) ∧
                          (∀ k : K₁, g
                            ((Algebra.TensorProduct.includeRight : K₁ →ₐ[K] L₁) k) =
                            (Algebra.TensorProduct.includeRight : K₁ →ₐ[K] M₁) k) ∧
                          (∀ a : A₁, ((f a : B₁) : L₁) =
                            (Algebra.TensorProduct.includeRight : K₁ →ₐ[K] L₁) (a : K₁)) ∧
                          (∀ b : B₁, ((f' b : C₁) : M₁) = g (b : L₁)) ∧
                          Ideal.map f (IsLocalRing.maximalIdeal A₁) =
                            IsLocalRing.maximalIdeal B₁ ∧
                          Ideal.map f' (IsLocalRing.maximalIdeal B₁) =
                            IsLocalRing.maximalIdeal C₁

Tag 09EZ, a lemma

Tag 09EZ in the Stacks Project.

IsLocalRing.exists_residueField_section_compatible, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/LocalRing/ResidueSection.lean, line 51.

A local ring of prime characteristic with nilpotent maximal ideal has a coefficient field, and it can be extended compatibly across a local homomorphism whose target also has nilpotent maximal ideal and whose residue field extension is separable.

theorem exists_residueField_section_compatible
    {p : ℕ} [Fact p.Prime]
    (A : Type u) [CommRing A] [IsLocalRing A] [CharP A p]
    (hA : IsNilpotent (IsLocalRing.maximalIdeal A)) :
    ∃ σ : IsLocalRing.ResidueField A →+* A,
      (IsLocalRing.residue A).comp σ = RingHom.id _ ∧
      ∀ (A' : Type v) [CommRing A'] [IsLocalRing A'] [Algebra A A']
        [IsLocalHom (algebraMap A A')],
        IsNilpotent (IsLocalRing.maximalIdeal A') →
        Algebra.IsSeparableExtension (IsLocalRing.ResidueField A)
          (IsLocalRing.ResidueField A') →
        ∃ σ' : IsLocalRing.ResidueField A' →+* A',
          (IsLocalRing.residue A').comp σ' = RingHom.id _ ∧
          (algebraMap A A').comp σ =
            σ'.comp (algebraMap (IsLocalRing.ResidueField A)
              (IsLocalRing.ResidueField A'))

Tag 09F0, a lemma

Tag 09F0 in the Stacks Project.

IsDiscreteValuationRing.isGalois_and_ramification_cases_of_adjoin_root_artinSchreier, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/ArtinSchreier/RamificationCases.lean, line 689.

Let A be a discrete valuation ring with fraction field K of characteristic p > 0. If L = K[z] and z ^ p - z = ξ, then L / K is Galois and is either trivial, unramified of degree p, totally ramified of ramification index p, or weakly unramified with purely inseparable residue extension of degree p.

Moreover, for a uniformizer π of A, an integral ξ gives one of the first two cases; a pole of order prime to p gives the totally ramified case; and a pole whose positive order is divisible by p and whose leading coefficient is not a p-th power in the residue field gives the weakly unramified, residually inseparable case.

theorem isGalois_and_ramification_cases_of_adjoin_root_artinSchreier
    (A : Type uA) (K : Type uK) (L : Type uL)
    [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [Field K] [Algebra A K] [IsFractionRing A K]
    [Field L] [Algebra K L] [Algebra A L] [IsScalarTower A K L]
    (p : ℕ) (hp : p.Prime) [CharP K p]
    (ξ : K) (z : L)
    (hz : z ^ p - z = algebraMap K L ξ)
    (hgen : Algebra.adjoin K {z} = ⊤)
    (π : A) (hπ : Irreducible π) :
    ∃ hfin : Module.Finite K L,
      letI := hfin
      let B := integralClosure A L
      let mA := IsLocalRing.maximalIdeal A
      let trivial : Prop := Function.Surjective (algebraMap K L)
      let unramified : Prop :=
        Module.finrank K L = p ∧
          ∀ m : Ideal B, m.IsMaximal → ∃ _ : m.LiesOver mA,
            Ideal.ramificationIdx' mA m = 1 ∧
              @Algebra.IsSeparable (A ⧸ mA) (B ⧸ m) _ _
                (Ideal.Quotient.algebraOfLiesOver m mA)
      let totallyRamified : Prop :=
        Stacks.KummerExtension.IsTotallyRamified A K L ∧
          ∀ m : Ideal B, m.IsMaximal → m.LiesOver mA →
            Ideal.ramificationIdx' mA m = p
      let residuallyInseparable : Prop :=
        ∃ hDVR : IsDiscreteValuationRing B,
          letI := hDVR
          let mB := IsLocalRing.maximalIdeal B
          ∃ _ : mB.LiesOver mA,
            Ideal.ramificationIdx' mA mB = 1 ∧
              letI := Ideal.Quotient.algebraOfLiesOver mB mA
              IsPurelyInseparable (A ⧸ mA) (B ⧸ mB) ∧
                Module.finrank (A ⧸ mA) (B ⧸ mB) = p
      IsGalois K L ∧
        (trivial ∨ unramified ∨ totallyRamified ∨ residuallyInseparable) ∧
        ((∃ a : A, ξ = algebraMap A K a) → trivial ∨ unramified) ∧
        (∀ (n : ℕ) (a : A), 0 < n → ¬p ∣ n → IsUnit a →
          ξ = (algebraMap A K π)⁻¹ ^ n * algebraMap A K a → totallyRamified) ∧
        (∀ (n : ℕ) (a : A), 0 < n → p ∣ n →
          (¬∃ b : IsLocalRing.ResidueField A, IsLocalRing.residue A a = b ^ p) →
          ξ = (algebraMap A K π)⁻¹ ^ n * algebraMap A K a →
            residuallyInseparable)

Tag 09F1, a lemma

Tag 09F1 in the Stacks Project.

IsDiscreteValuationRing.exists_isGalois_totallyRamified_isWeakSolution_of_charP, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/ArtinSchreier/CharPSolution.lean, line 690.

In equal characteristic p, a weakly unramified extension A ⊆ B followed by a degree-p Galois extension of fraction fields admits a finite Galois base extension which is totally ramified over A and is a weak solution for A → C, provided the image of the residue field of A in that of B is the intersection of all sets of p ^ n-th powers for n ≥ 1.

This is Tag 09F1 as stated. The two consequences of hresidue that the proof consumes are isolated in exists_isGalois_totallyRamified_isWeakSolution_of_charP_of_perfectCore_le, which is strictly more general: it drops the inclusion image ⊆ perfectCore entirely, keeping only perfectness of κ_A and perfectCore κ_B p ⊆ image.

theorem exists_isGalois_totallyRamified_isWeakSolution_of_charP
    (A : Type uA) (B : Type uB) (C : Type uC)
    (K : Type uK) (L : Type uL) (M : Type uM)
    [CommRing A] [CommRing B] [CommRing C]
    [IsDomain A] [IsDomain B] [IsDomain C]
    [IsDiscreteValuationRing A] [IsDiscreteValuationRing B]
    [IsDiscreteValuationRing C]
    [Field K] [Field L] [Field M]
    [Algebra A B] [Algebra B C] [Algebra A C]
    [Algebra A K] [Algebra B L] [Algebra C M]
    [Algebra K L] [Algebra L M] [Algebra K M]
    [Algebra A L] [Algebra B M] [Algebra A M]
    [IsScalarTower A B C] [IsScalarTower A B L] [IsScalarTower A K L]
    [IsScalarTower B C M] [IsScalarTower B L M]
    [IsScalarTower A C M] [IsScalarTower A K M] [IsScalarTower K L M]
    [IsExtension A B] [IsExtension B C]
    [IsExtension A C]
    [IsFractionRing A K] [IsFractionRing B L] [IsFractionRing C M]
    (p : ℕ) (hp : p.Prime) [CharP K p]
    [FiniteDimensional L M] [IsGalois L M]
    (hdegree : Module.finrank L M = p)
    (hweak : IsWeaklyUnramified A B)
    (hresidue : ∀ x : IsLocalRing.ResidueField B,
      (∃ a : IsLocalRing.ResidueField A,
          algebraMap (IsLocalRing.ResidueField A) (IsLocalRing.ResidueField B) a = x) ↔
        ∀ n : ℕ, 1 ≤ n → ∃ y : IsLocalRing.ResidueField B, y ^ (p ^ n) = x) :
    ∃ (K₁ : Type uK) (_ : Field K₁) (_ : Algebra K K₁) (_ : Algebra A K₁)
      (_ : IsScalarTower A K K₁) (_ : Module.Finite K K₁) (_ : IsGalois K K₁),
      IsTotallyRamifiedWithRespectTo A K K₁ ∧
        ∃ D : IntegralClosureBaseChange.{uA, uC, uK, uM, uK,
            max uM uK, uK, max uM uK} A C K M K₁,
          D.IsWeakSolution

Tag 09F2, a lemma

Tag 09F2 in the Stacks Project.

main_statement, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/NumberTheory/Cyclotomic/OneAddPow/QuotientPolynomial.lean, line 768.

Lemma 15.117.13 (Stacks tag 09F2). Let p be prime, A a commutative ring containing a primitive p-th root of unity ζ (encoded by f : ℤ[ζ_p] → A), and w = 1 - ζ. Then there is ONE common polynomial P ∈ A[z] such that:

• (Statement 1, integrality) P is the quotient ((1 + w z)^p - 1)/w^p over A: w^p · P = (1 + wz)^p - 1 in A[z], P = ∑_{0 < i ≤ p} c_i z^i, and w^{p-i} · c_i = (p choose i) for 0 < i ≤ p (predicate intProp). In particular P is monic of degree p. The corresponding statement in (A[w⁻¹])[z] (predicate quotChar) is recorded too, but it is weaker — vacuous when w is nilpotent, see quotChar_of_isNilpotent — so it is intProp that carries the assertion "P ∈ A[z]";
• (Statement 1, shape) P(z) = z^p - z + ∑_{0 < i < p} a_i z^i with a_i ∈ (w) (predicate shapeProp);
• (Statement 2, functional equation) P(z_1 + z_2 + w z_1 z_2) = P(z_1) + P(z_2) + w^p P(z_1) P(z_2) in A[z_1, z_2] (predicate funcEq).

Bundling all of them into a single existential guarantees they hold for the same P.

theorem main_statement :
    ∃ P : A[X], intProp f P ∧ quotChar f P ∧ shapeProp f P ∧ funcEq f P

Tag 09F3, a lemma

Tag 09F3 in the Stacks Project.

isGalois_and_ramification_cases_of_adjoin_root_cyclotomic_quotient, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/NumberTheory/Cyclotomic/OneAddPow/RamificationCases.lean, line 511.

Let A be a mixed-characteristic discrete valuation ring containing a primitive p-th root of unity, and let P be the cyclotomic quotient polynomial of Tag 09F2. If L = K[z] with P(z) = ξ, then L / K is Galois and is either trivial, unramified of degree p, totally ramified with ramification index p, or weakly unramified with purely inseparable residue extension of degree p.

Moreover, for a uniformizer π of A, an integral ξ gives one of the first two cases; a pole of positive order not divisible by p gives the totally ramified case; and a pole whose order is divisible by p and whose leading coefficient is not a p-th power in the residue field gives the weakly unramified, residually inseparable case.

theorem isGalois_and_ramification_cases_of_adjoin_root_cyclotomic_quotient
    (A : Type uA) (K : Type uK) (L : Type uL)
    [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [Field K] [Algebra A K] [IsFractionRing A K] [CharZero K]
    [Field L] [Algebra K L] [Algebra A L] [IsScalarTower A K L]
    (p : ℕ) [Fact p.Prime] [CharP (IsLocalRing.ResidueField A) p]
    (f : CyclotomicRing p ℤ ℚ →+* A) (P : Polynomial A)
    (hP : intProp f P ∧ quotChar f P ∧ shapeProp f P ∧ funcEq f P)
    (ξ : K) (z : L)
    (hz : Polynomial.aeval z P = algebraMap K L ξ)
    (hgen : Algebra.adjoin K {z} = ⊤)
    (π : A) (hπ : Irreducible π) :
    ∃ hfin : Module.Finite K L,
      letI := hfin
      let B := integralClosure A L
      let mA := IsLocalRing.maximalIdeal A
      let trivial : Prop := Function.Surjective (algebraMap K L)
      let unramified : Prop :=
        Module.finrank K L = p ∧
          ∀ m : Ideal B, m.IsMaximal → ∃ _ : m.LiesOver mA,
            Ideal.ramificationIdx' mA m = 1 ∧
              @Algebra.IsSeparable (A ⧸ mA) (B ⧸ m) _ _
                (Ideal.Quotient.algebraOfLiesOver m mA)
      let totallyRamified : Prop :=
        Stacks.KummerExtension.IsTotallyRamified A K L ∧
          ∀ m : Ideal B, m.IsMaximal → m.LiesOver mA →
            Ideal.ramificationIdx' mA m = p
      let residuallyInseparable : Prop :=
        ∃ hDVR : IsDiscreteValuationRing B,
          letI := hDVR
          let mB := IsLocalRing.maximalIdeal B
          ∃ _ : mB.LiesOver mA,
            Ideal.ramificationIdx' mA mB = 1 ∧
              letI := Ideal.Quotient.algebraOfLiesOver mB mA
              IsPurelyInseparable (A ⧸ mA) (B ⧸ mB) ∧
                Module.finrank (A ⧸ mA) (B ⧸ mB) = p
      IsGalois K L ∧
        (trivial ∨ unramified ∨ totallyRamified ∨ residuallyInseparable) ∧
        ((∃ a : A, ξ = algebraMap A K a) → trivial ∨ unramified) ∧
        (∀ (n : ℕ) (a : A), 0 < n → ¬p ∣ n → IsUnit a →
          ξ = (algebraMap A K π)⁻¹ ^ n * algebraMap A K a → totallyRamified) ∧
        (∀ (n : ℕ) (a : A), 0 < n → p ∣ n →
          (¬∃ b : IsLocalRing.ResidueField A, IsLocalRing.residue A a = b ^ p) →
          ξ = (algebraMap A K π)⁻¹ ^ n * algebraMap A K a →
            residuallyInseparable)

Tag 09F4, a lemma

Tag 09F4 in the Stacks Project.

Stacks.DegreePExtensionLevel.exists_totallyRamified_galoisExtension_isWeakSolution_or_finiteLevel, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/ExtensionLevel/FiniteLevel.lean, line 859.

In mixed characteristic, a weakly unramified tower followed by a Galois extension of prime degree admits a finite, totally ramified base extension. This base extension is either a weak solution for the full tower or makes the canonical tensor-product extension a degree-p extension of finite level.

theorem exists_totallyRamified_galoisExtension_isWeakSolution_or_finiteLevel
    (A B C K L M : Type u)
    [CommRing A] [CommRing B] [CommRing C]
    [IsDomain A] [IsDomain B] [IsDomain C]
    [IsDiscreteValuationRing A] [IsDiscreteValuationRing B]
    [IsDiscreteValuationRing C]
    [Field K] [Field L] [Field M]
    [Algebra A B] [Algebra B C] [Algebra A C]
    [Algebra A K] [Algebra B L] [Algebra C M]
    [Algebra K L] [Algebra L M] [Algebra K M]
    [Algebra A L] [Algebra A M] [Algebra B M]
    [IsScalarTower A B C] [IsScalarTower A B L]
    [IsScalarTower B C M] [IsScalarTower B L M]
    [IsScalarTower A C M] [IsScalarTower A K L]
    [IsScalarTower A K M] [IsScalarTower K L M]
    [IsDiscreteValuationRing.IsExtension A B] [IsDiscreteValuationRing.IsExtension B C]
    [IsDiscreteValuationRing.IsExtension A C]
    [IsFractionRing A K] [IsFractionRing B L] [IsFractionRing C M]
    (p : ℕ) [Fact p.Prime] [CharZero K]
    [CharP (IsLocalRing.ResidueField A) p]
    (hweak : IsDiscreteValuationRing.IsWeaklyUnramified A B)
    (f : CyclotomicRing p ℤ ℚ →+* B)
    [FiniteDimensional L M] [IsGalois L M]
    (hdegree : Module.finrank L M = p) :
    ∃ (K₁ : Type u) (_ : Field K₁) (_ : Algebra K K₁) (_ : Algebra A K₁)
      (_ : IsScalarTower A K K₁) (_ : Module.Finite K K₁),
      IsDiscreteValuationRing.IsTotallyRamifiedWithRespectTo A K K₁ ∧
        ((∃ D : IsDiscreteValuationRing.IntegralClosureBaseChange.{u, u, u, u, u, u, u, u} A C K M
          K₁,
            D.IsWeakSolution) ∨
          let L₁ := L ⊗[K] K₁
          let M₁ := M ⊗[K] K₁
          ∃ (hL₁ : IsField L₁) (hM₁ : IsField M₁),
            letI := hL₁.toField
            letI := hM₁.toField
            let ι : L₁ →ₐ[K] M₁ :=
              Algebra.TensorProduct.map (IsScalarTower.toAlgHom K L M) (AlgHom.id K K₁)
            letI : Algebra L₁ M₁ := ι.toRingHom.toAlgebra
            let B₁ := integralClosure B L₁
            ∃ (hB₁ : IsDiscreteValuationRing B₁) (hfrac : IsFractionRing B₁ L₁)
              (hfin : Module.Finite L₁ M₁),
              letI := hB₁
              letI := hfrac
              letI := hfin
              ∃ P : Polynomial B,
                intProp f P ∧ quotChar f P ∧ shapeProp f P ∧ funcEq f P ∧
                  IsDegreePExtensionOfFiniteLevel B₁ L₁ M₁ p
                    ((algebraMap B B₁).comp f) (P.map (algebraMap B B₁)))

Tag 09F5, a lemma

Tag 09F5 in the Stacks Project.

Stacks.DiscreteValuationRing.EppLevelInduction.exists_totallyRamified_isWeakSolution_or_extensionLevel_le, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/ExtensionLevel/LevelLowering.lean, line 850.

The traditional Tag 09F5 interface. The global residue-field hypothesis identifies the perfect core with the image of the base residue field, so the finite-support alternative in exists_totallyRamified_isWeakSolution_or_extensionLevel_le_or_finiteSupport_of_uniformizer is impossible.

theorem exists_totallyRamified_isWeakSolution_or_extensionLevel_le
    (A B C K L M : Type u)
    [CommRing A] [CommRing B] [CommRing C]
    [IsDomain A] [IsDomain B] [IsDomain C]
    [IsDiscreteValuationRing A] [IsDiscreteValuationRing B]
    [IsDiscreteValuationRing C]
    [Field K] [Field L] [Field M]
    [Algebra A B] [Algebra B C] [Algebra A C]
    [Algebra A K] [Algebra B L] [Algebra C M]
    [Algebra K L] [Algebra L M] [Algebra K M]
    [Algebra A L] [Algebra A M] [Algebra B M]
    [IsScalarTower A B C] [IsScalarTower A B L]
    [IsScalarTower B C M] [IsScalarTower B L M]
    [IsScalarTower A C M] [IsScalarTower A K L]
    [IsScalarTower A K M] [IsScalarTower K L M]
    [IsDiscreteValuationRing.IsExtension A B] [IsDiscreteValuationRing.IsExtension B C]
    [IsDiscreteValuationRing.IsExtension A C]
    [IsFractionRing A K] [IsFractionRing B L] [IsFractionRing C M]
    (p : ℕ) [Fact p.Prime] [CharZero K]
    [CharP (IsLocalRing.ResidueField A) p]
    (hweak : IsDiscreteValuationRing.IsWeaklyUnramified A B)
    (f : CyclotomicRing p ℤ ℚ →+* B) (P : Polynomial B)
    (hP : intProp f P ∧ quotChar f P ∧ shapeProp f P ∧ funcEq f P)
    [Module.Finite L M]
    (hfinite : Stacks.DegreePExtensionLevel.IsDegreePExtensionOfFiniteLevel B L M p f P)
    (l : ℚ)
    (hlevel : ∃ (π : B) (e₁ : ℕ) (he₁ : 0 < e₁)
        (hne : (Stacks.DegreePExtensionLevel.extensionLevelNumerators B L M P π).Nonempty),
        Irreducible π ∧ Associated (wElt f) (π ^ e₁) ∧
          l = Stacks.DegreePExtensionLevel.extensionLevel B L M P π e₁ he₁ hne)
    (hl : 0 < l)
    (hresidue : ∀ x : IsLocalRing.ResidueField B,
      (∃ a : IsLocalRing.ResidueField A,
          algebraMap (IsLocalRing.ResidueField A) (IsLocalRing.ResidueField B) a = x) ↔
        ∀ n : ℕ, 1 ≤ n → ∃ y : IsLocalRing.ResidueField B, y ^ (p ^ n) = x) :
    ∃ (K₁ : Type u) (_ : Field K₁) (_ : Algebra K K₁) (_ : Algebra A K₁)
      (_ : IsScalarTower A K K₁) (_ : Module.Finite K K₁)
      (_ : Algebra.IsSeparable K K₁),
      IsDiscreteValuationRing.IsTotallyRamifiedWithRespectTo A K K₁ ∧
        ((∃ D : IsDiscreteValuationRing.IntegralClosureBaseChange.{u, u, u, u, u, u, u, u}
            A C K M K₁, D.IsWeakSolution) ∨
          let L₁ := L ⊗[K] K₁
          let M₁ := M ⊗[K] K₁
          ∃ (hL₁ : IsField L₁) (hM₁ : IsField M₁),
            letI := hL₁.toField
            letI := hM₁.toField
            let ι : L₁ →ₐ[K] M₁ :=
              Algebra.TensorProduct.map (IsScalarTower.toAlgHom K L M) (AlgHom.id K K₁)
            letI : Algebra L₁ M₁ := ι.toRingHom.toAlgebra
            let B₁ := integralClosure B L₁
            ∃ (hB₁ : IsDiscreteValuationRing B₁) (hfrac : IsFractionRing B₁ L₁)
              (hfin : Module.Finite L₁ M₁),
              letI := hB₁
              letI := hfrac
              letI := hfin
              let f₁ := (algebraMap B B₁).comp f
              let P₁ := P.map (algebraMap B B₁)
              Stacks.DegreePExtensionLevel.IsDegreePExtensionOfFiniteLevel B₁ L₁ M₁ p f₁ P₁ ∧
                ∃ (π₁ : B₁) (e₁ : ℕ) (he₁ : 0 < e₁)
                  (hne : (Stacks.DegreePExtensionLevel.extensionLevelNumerators B₁ L₁ M₁ P₁
                    π₁).Nonempty),
                  Irreducible π₁ ∧ Associated (wElt f₁) (π₁ ^ e₁) ∧
                    Stacks.DegreePExtensionLevel.extensionLevel B₁ L₁ M₁ P₁ π₁ e₁ he₁ hne ≤
                      max 0 (max (l - 1) (2 * l - (p : ℚ))))

Tag 09F6, an equation

Tag 09F6 in the Stacks Project.

IsDiscreteValuationRing.eval_sub_pow_sub_mem_uniformizer_zpow, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/EliminateRamification.lean, line 163.

For the Kummer polynomial P associated to a primitive p-th root of unity, if w = 1 - ζ has order e₁ and π ^ n P(z) is integral, then P(z) ≡ z ^ p - z modulo π ^ (e₁ - n) B.

theorem eval_sub_pow_sub_mem_uniformizer_zpow
    (B : Type uB) (K : Type uK)
    [CommRing B] [IsDomain B] [IsDiscreteValuationRing B]
    [Field K] [Algebra B K] [IsFractionRing B K]
    (p : ℕ) [Fact p.Prime]
    (f : CyclotomicRing p ℤ ℚ →+* B) (P : Polynomial B)
    (hP : shapeProp f P)
    (π : B) (hπ : Irreducible π) (e₁ n : ℕ)
    (hw : Associated (wElt f) (π ^ e₁)) (z : K)
    (hz : ∃ b : B,
      (algebraMap B K π) ^ n * P.eval₂ (algebraMap B K) z = algebraMap B K b) :
    ∃ b : B,
      P.eval₂ (algebraMap B K) z - (z ^ p - z) =
        (algebraMap B K π) ^ ((e₁ : ℤ) - n) * algebraMap B K b

Tag 09F7, an equation

Tag 09F7 in the Stacks Project.

IsDiscreteValuationRing.add_mul_correction_sModEq, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/EliminateRamification.lean, line 219.

Stacks 09F7. The correction term in the finite-level group law vanishes modulo π ^ (-2n + p e₁) B.

theorem add_mul_correction_sModEq
    {B K : Type*} [CommRing B] [IsDomain B] [Field K]
    [Algebra B K] [IsFractionRing B K]
    {p n e₁ : ℕ} {π w : B} {ξ₁ ξ₂ : K}
    (hw : Associated w (π ^ e₁))
    (hξ₁ : ξ₁ ∈ FractionalIdeal.spanSingleton B⁰
      ((algebraMap B K π) ^ (-(n : ℤ))))
    (hξ₂ : ξ₂ ∈ FractionalIdeal.spanSingleton B⁰
      ((algebraMap B K π) ^ (-(n : ℤ)))) :
    SModEq (FractionalIdeal.spanSingleton B⁰
        ((algebraMap B K π) ^ (-(2 : ℤ) * n + p * e₁ : ℤ)) : Submodule B K)
      (ξ₁ + ξ₂ + (algebraMap B K w) ^ p * ξ₁ * ξ₂) (ξ₁ + ξ₂)

Tag 09F8, a lemma

Tag 09F8 in the Stacks Project.

IsDiscreteValuationRing.exists_finite_extension_isWeakSolution, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/EppAlgClosedResidue/IntermediateFieldInduction.lean, line 681.

Epp's theorem in the complete case: under the residue-field intersection hypothesis, every finite extension at the top of a weakly unramified tower admits a finite base extension which is a weak solution for the full tower.

theorem exists_finite_extension_isWeakSolution
    (A : Type uA) (B : Type uB) (C : Type uC)
    (K : Type uK) (L : Type uL) (M : Type uM)
    [CommRing A] [CommRing B] [CommRing C]
    [IsDomain A] [IsDomain B] [IsDomain C]
    [IsDiscreteValuationRing A] [IsDiscreteValuationRing B]
    [IsDiscreteValuationRing C]
    [Field K] [Field L] [Field M]
    [Algebra A B] [Algebra B C] [Algebra A C]
    [Algebra A K] [Algebra B L] [Algebra C M]
    [Algebra K L] [Algebra L M] [Algebra K M]
    [Algebra A L] [Algebra B M] [Algebra A M]
    [IsScalarTower A B C] [IsScalarTower A B L] [IsScalarTower A K L]
    [IsScalarTower B C M] [IsScalarTower B L M]
    [IsScalarTower A C M] [IsScalarTower A K M] [IsScalarTower K L M]
    [IsDiscreteValuationRing.IsExtension A B] [IsDiscreteValuationRing.IsExtension B C]
    [IsDiscreteValuationRing.IsExtension A C]
    [IsFractionRing A K] [IsFractionRing B L] [IsFractionRing C M]
    (p : ℕ) [Fact p.Prime] [CharP (IsLocalRing.ResidueField A) p]
    [IsAlgClosed (IsLocalRing.ResidueField A)]
    [IsAdicComplete (IsLocalRing.maximalIdeal A) A]
    [IsAdicComplete (IsLocalRing.maximalIdeal B) B]
    (hweak : IsDiscreteValuationRing.IsWeaklyUnramified A B)
    [Module.Finite L M]
    (hresidue : ∀ x : IsLocalRing.ResidueField B,
      (∃ a : IsLocalRing.ResidueField A,
          algebraMap (IsLocalRing.ResidueField A) (IsLocalRing.ResidueField B) a = x) ↔
        ∀ n : ℕ, 1 ≤ n → ∃ y : IsLocalRing.ResidueField B, y ^ (p ^ n) = x) :
    ∃ (K₁ : Type uK) (_ : Field K₁) (_ : Algebra K K₁) (_ : Algebra A K₁)
      (_ : IsScalarTower A K K₁) (_ : Module.Finite K K₁),
      ∃ D : IsDiscreteValuationRing.IntegralClosureBaseChange.{uA, uC, uK, uM, uK,
          max uM uK, uK, max uM uK} A C K M K₁,
        D.IsWeakSolution

Tag 09F9, a theorem

Tag 09F9 in the Stacks Project.

Stacks.EppTheorem.exists_finite_extension_isWeakSolution, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/EppTheorem.lean, line 26.

Epp's theorem when all input types lie in one universe.

theorem exists_finite_extension_isWeakSolution
    (A B K L : Type u)
    [CommRing A] [CommRing B] [IsDomain A] [IsDomain B]
    [IsDiscreteValuationRing A] [IsDiscreteValuationRing B]
    [Field K] [Field L]
    [Algebra A B] [Algebra A K] [Algebra A L] [Algebra B L] [Algebra K L]
    [IsScalarTower A B L] [IsScalarTower A K L]
    [IsExtension A B] [IsFractionRing A K] [IsFractionRing B L]
    (hEpp : ∀ p : ℕ, p.Prime → CharP (ResidueField A) p →
      ∀ [ExpChar (ResidueField B) p], EppCondition A B p) :
    ∃ (K₁ : Type u) (_ : Field K₁) (_ : Algebra K K₁) (_ : Algebra A K₁)
      (_ : IsScalarTower A K K₁) (_ : Module.Finite K K₁),
      ∃ D : IntegralClosureBaseChange.{u, u, u, u, u, u, u, u} A B K L K₁,
        D.IsWeakSolution

Sources

• Section 15.117, Tag 09EL
• AxiomMath/Stacks at 5ed82d14

LeanIsDiscreteValuationRing.IntegralClosureBaseChange.IsWeakSolution · IsDiscreteValuationRing.IntegralClosureBaseChange.IsSolution · IsDiscreteValuationRing.IntegralClosureBaseChange.IsSeparableSolution · IsDiscreteValuationRing.weakSolution_isInseparable_and_inseparable_isSolution · IsDiscreteValuationRing.isField_tensorProduct_and_isDVR_baseChange_of_totallyRamified · IsDiscreteValuationRing.weakSolutions_and_solutions_go_down_and_compose · IsDiscreteValuationRing.normal_extension_acts_transitively_on_maximalSpectrum_integralClosure · Stacks.EisensteinExtension.exists_totallyRamifiedExtension_with_uniformizer_eq

Section15.118 Eliminating ramification, II#

Section 15.118 of the Stacks Project, Tag 0GLQ. 9 tags here are carried by 9 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0GLR, a lemma

Tag 0GLR in the Stacks Project.

IsDiscreteValuationRing.exists_isSolution_of_tower, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/SolutionTower.lean, line 590.

A finite extension of a solution for an extension of discrete valuation rings is again a solution.

theorem exists_isSolution_of_tower
    (A : Type uA) (B : Type uB) (K : Type uK) (L : Type uL)
    (K₁ : Type uK₁) (K₂ : Type uK₂)
    [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [CommRing B] [IsDomain B] [IsDiscreteValuationRing B]
    [Field K] [Field L] [Field K₁] [Field K₂]
    [Algebra A B] [Algebra A K] [Algebra A L] [Algebra B L] [Algebra K L]
    [Algebra K K₁] [Algebra K₁ K₂] [Algebra K K₂]
    [Algebra A K₁] [Algebra A K₂]
    [IsScalarTower A B L] [IsScalarTower A K L]
    [IsScalarTower A K K₁] [IsScalarTower A K K₂]
    [IsScalarTower K K₁ K₂] [IsScalarTower A K₁ K₂]
    [IsExtension A B]
    [IsFractionRing A K] [IsFractionRing B L]
    [Module.Finite K K₁] [Module.Finite K₁ K₂]
    (hsolution :
      ∃ D₁ : IntegralClosureBaseChange.{uA, uB, uK, uL, uK₁,
          max uL uK₁, uK₁, max uL uK₁} A B K L K₁,
        D₁.IsSolution.{uA, uB, uK, uL, uK₁, max uL uK₁, uK₁, max uL uK₁,
          max uL uK₁}) :
    letI : Module.Finite K K₂ := Module.Finite.trans K₁ K₂
    ∃ D₂ : IntegralClosureBaseChange.{uA, uB, uK, uL, uK₂,
        max uL uK₂, uK₂, max uL uK₂} A B K L K₂,
      D₂.IsSolution.{uA, uB, uK, uL, uK₂, max uL uK₂, uK₂, max uL uK₂,
        max uL uK₂}

Tag 0GLS, a lemma

Tag 0GLS in the Stacks Project.

IsNagataRing.of_isSeparable_fractionRing, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/Nagata/Descent.lean, line 463.

Let A → B be an extension of discrete valuation rings, and let K → L be the induced extension of their fraction fields. If B is Nagata and L/K is separable, then A is Nagata.

theorem of_isSeparable_fractionRing
    (A B K L : Type u) [CommRing A] [IsDomain A] [CommRing B] [IsDomain B]
    [IsDiscreteValuationRing A] [IsDiscreteValuationRing B] [Algebra A B]
    [IsDiscreteValuationRing.IsExtension A B]
    [Field K] [Field L] [Algebra A K] [IsFractionRing A K]
    [Algebra B L] [IsFractionRing B L] [Algebra A L] [Algebra K L]
    [IsScalarTower A B L] [IsScalarTower A K L]
    (hB : IsNagataRing B) (hsep : Algebra.IsSeparableExtension K L) :
    IsNagataRing A

Tag 0GLT, a lemma

Tag 0GLT in the Stacks Project.

exists_bound_finite_extension_ringEquiv_mod, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/FiniteExtension/ModPow/QuotientEquiv.lean, line 241.

Stacks 0GLT. Let A' ⊂ A be an extension of rings and f ∈ A', with A finite over A', f a nonzerodivisor on A, and A'_f = A_f. Then there is n₀ > 0 such that for all n ≥ n₀: for every ring B', every nonzerodivisor g ∈ B', and every ring isomorphism φ' : A'/f^n A' ≃ B'/g^n B' sending the class of f to the class of g, there exist a finite ring extension B' ⊂ B and a ring isomorphism φ : A/fA ≃ B/gB compatible with φ', i.e. making the square of reduction maps commute.

theorem exists_bound_finite_extension_ringEquiv_mod
    (A' : Type u) (A : Type v) [CommRing A'] [CommRing A] [Algebra A' A]
    (hinj : Function.Injective (algebraMap A' A))
    (f : A')
    [Module.Finite A' A]
    (hnzd : algebraMap A' A f ∈ nonZeroDivisors A)
    (hloc : LocalizationEq A' A f) :
    ∃ n₀ : ℕ, 0 < n₀ ∧ ∀ n : ℕ, n₀ ≤ n →
      ∀ (B' : Type w) [CommRing B'] (g : B') (_ : g ∈ nonZeroDivisors B')
        (hn : 1 ≤ n)
        (φ' : (A' ⧸ Ideal.span {f ^ n}) ≃+* (B' ⧸ Ideal.span {g ^ n})),
        φ' (Ideal.Quotient.mk _ f) = Ideal.Quotient.mk _ g →
        ∃ (B : Type w) (_ : CommRing B) (_ : Algebra B' B),
          Function.Injective (algebraMap B' B) ∧
          Module.Finite B' B ∧
          ∃ φ : (A ⧸ Ideal.span {algebraMap A' A f}) ≃+*
              (B ⧸ Ideal.span {algebraMap B' B g}),
            (φ : (A ⧸ Ideal.span {algebraMap A' A f}) →+* _).comp
                (reductionMap A' A f n hn)
              = (reductionMap B' B g n hn).comp
                (φ' : (A' ⧸ Ideal.span {f ^ n}) →+* _)

Tag 0GLU, a remark

Tag 0GLU in the Stacks Project.

exists_compatible_finite_extensions_of_quotient_equiv, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/FiniteExtension/ModPow/QuotientEquiv.lean, line 469.

Stacks 0GLU. The construction of exists_bound_finite_extension_ringEquiv_mod is functorial. Given a commutative square of finite injective ring extensions satisfying its hypotheses, and a compatible square of isomorphisms modulo a sufficiently large common power, the two constructed finite extensions admit a map extending the given map of base rings. Both the square of extensions and the square of the resulting isomorphisms modulo the nonzerodivisors commute. As in the construction B = g^{-t} N ⊆ B'_g of bSubalg, each extension becomes an isomorphism after inverting the corresponding image of g, and this image is a nonzerodivisor in the constructed ring.

theorem exists_compatible_finite_extensions_of_quotient_equiv
    (A₁' : Type u₁) (A₁ : Type v₁) (A₂' : Type u₂) (A₂ : Type v₂)
    [CommRing A₁'] [CommRing A₁] [CommRing A₂'] [CommRing A₂]
    [Algebra A₁' A₁] [Algebra A₂' A₂]
    (iA₁ : Function.Injective (algebraMap A₁' A₁))
    (iA₂ : Function.Injective (algebraMap A₂' A₂))
    (a' : A₁' →+* A₂') (a : A₁ →+* A₂)
    (ha : a.comp (algebraMap A₁' A₁) = (algebraMap A₂' A₂).comp a')
    (f : A₁')
    (hfin₁ : Module.Finite A₁' A₁) (hfin₂ : Module.Finite A₂' A₂)
    (hnzd₁ : algebraMap A₁' A₁ f ∈ nonZeroDivisors A₁)
    (hnzd₂ : algebraMap A₂' A₂ (a' f) ∈ nonZeroDivisors A₂)
    (hloc₁ : LocalizationEq A₁' A₁ f)
    (hloc₂ : LocalizationEq A₂' A₂ (a' f)) :
    ∃ (n₀₁ : ℕ) (_ : 0 < n₀₁) (n₀₂ : ℕ) (_ : 0 < n₀₂),
      ∀ (n : ℕ) (hnmax : max n₀₁ n₀₂ ≤ n),
      ∀ (B₁' : Type w₁) (B₂' : Type w₂) [CommRing B₁'] [CommRing B₂']
        (b' : B₁' →+* B₂') (g : B₁')
        (_ : g ∈ nonZeroDivisors B₁') (_ : b' g ∈ nonZeroDivisors B₂')
        (φ₁' : (A₁' ⧸ Ideal.span {f ^ n}) ≃+* (B₁' ⧸ Ideal.span {g ^ n}))
        (φ₂' : (A₂' ⧸ Ideal.span {(a' f) ^ n}) ≃+*
          (B₂' ⧸ Ideal.span {(b' g) ^ n})),
        φ₂'.toRingHom.comp
            (Ideal.quotientMap (Ideal.span {(a' f) ^ n}) a' (by
              apply Ideal.span_le.2
              rintro _ rfl
              change a' (f ^ n) ∈ Ideal.span {(a' f) ^ n}
              rw [map_pow]
              exact Ideal.subset_span (Set.mem_singleton _))) =
          (Ideal.quotientMap (Ideal.span {(b' g) ^ n}) b' (by
            apply Ideal.span_le.2
            rintro _ rfl
            change b' (g ^ n) ∈ Ideal.span {(b' g) ^ n}
            rw [map_pow]
            exact Ideal.subset_span (Set.mem_singleton _))).comp φ₁'.toRingHom →
        φ₁' (Ideal.Quotient.mk _ f) = Ideal.Quotient.mk _ g →
        ∃ (B₁ : Type w₁) (_ : CommRing B₁) (_ : Algebra B₁' B₁)
          (B₂ : Type w₂) (_ : CommRing B₂) (_ : Algebra B₂' B₂),
          Function.Injective (algebraMap B₁' B₁) ∧ Module.Finite B₁' B₁ ∧
          Function.Injective (algebraMap B₂' B₂) ∧ Module.Finite B₂' B₂ ∧
          LocalizationEq B₁' B₁ g ∧ LocalizationEq B₂' B₂ (b' g) ∧
          algebraMap B₁' B₁ g ∈ nonZeroDivisors B₁ ∧
          algebraMap B₂' B₂ (b' g) ∈ nonZeroDivisors B₂ ∧
          ∃ (b : B₁ →+* B₂)
            (hb : b.comp (algebraMap B₁' B₁) = (algebraMap B₂' B₂).comp b')
            (φ₁ : (A₁ ⧸ Ideal.span {algebraMap A₁' A₁ f}) ≃+*
              (B₁ ⧸ Ideal.span {algebraMap B₁' B₁ g}))
            (φ₂ : (A₂ ⧸ Ideal.span {algebraMap A₂' A₂ (a' f)}) ≃+*
              (B₂ ⧸ Ideal.span {algebraMap B₂' B₂ (b' g)})),
            φ₂.toRingHom.comp
                (Ideal.quotientMap (Ideal.span {algebraMap A₂' A₂ (a' f)}) a (by
                  apply Ideal.span_le.2
                  rintro _ rfl
                  change a (algebraMap A₁' A₁ f) ∈
                    Ideal.span {algebraMap A₂' A₂ (a' f)}
                  rw [show a (algebraMap A₁' A₁ f) =
                    algebraMap A₂' A₂ (a' f) from DFunLike.congr_fun ha f]
                  exact Ideal.subset_span (Set.mem_singleton _))) =
              (Ideal.quotientMap (Ideal.span {algebraMap B₂' B₂ (b' g)}) b (by
                apply Ideal.span_le.2
                rintro _ rfl
                change b (algebraMap B₁' B₁ g) ∈
                  Ideal.span {algebraMap B₂' B₂ (b' g)}
                rw [show b (algebraMap B₁' B₁ g) =
                  algebraMap B₂' B₂ (b' g) from DFunLike.congr_fun hb g]
                exact Ideal.subset_span (Set.mem_singleton _))).comp
                φ₁.toRingHom ∧
            φ₁.toRingHom.comp (reductionMap A₁' A₁ f n (by omega)) =
              (reductionMap B₁' B₁ g n (by omega)).comp φ₁'.toRingHom ∧
            φ₂.toRingHom.comp (reductionMap A₂' A₂ (a' f) n (by omega)) =
              (reductionMap B₂' B₂ (b' g) n (by omega)).comp φ₂'.toRingHom

Tag 0GLV, a lemma

Tag 0GLV in the Stacks Project.

IsDiscreteValuationRing.exists_isSeparable_solution_of_isPurelyInseparable_finrank_eq_prime, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/SeparableSolution/InseparableStep.lean, line 45.

Let A → B be an extension of discrete valuation rings with separable fraction-field extension L/K, where K has prime characteristic p, and suppose that B is Nagata. If K → K₁ → K₂ is a tower of finite extensions, K₂ is a solution for A → B, and K₂/K₁ is purely inseparable of degree p, then some separable extension K₃/K₁ is a solution for A → B.

theorem exists_isSeparable_solution_of_isPurelyInseparable_finrank_eq_prime
    (A : Type uA) (B : Type uB) (K : Type uK) (L : Type uL)
    (K₁ : Type uK₁) (K₂ : Type uK₂)
    [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [CommRing B] [IsDomain B] [IsDiscreteValuationRing B]
    [Field K] [Field L] [Field K₁] [Field K₂]
    [Algebra A B] [Algebra A K] [Algebra A L] [Algebra B L] [Algebra K L]
    [Algebra K K₁] [Algebra K₁ K₂] [Algebra K K₂]
    [Algebra A K₁] [Algebra A K₂]
    [IsScalarTower A B L] [IsScalarTower A K L]
    [IsScalarTower A K K₁] [IsScalarTower A K K₂]
    [IsScalarTower K K₁ K₂] [IsScalarTower A K₁ K₂]
    [IsExtension A B]
    [IsFractionRing A K] [IsFractionRing B L]
    [Module.Finite K K₁] [Module.Finite K₁ K₂]
    (p : ℕ) (hp : p.Prime) [CharP K p]
    (hsep : Algebra.IsSeparableExtension K L)
    (hB : IsNagataRing B)
    (hsolution :
      letI : Module.Finite K K₂ := Module.Finite.trans K₁ K₂
      ∃ D₂ : IntegralClosureBaseChange.{uA, uB, uK, uL, uK₂,
          max uL uK₂, uK₂, max uL uK₂} A B K L K₂,
        D₂.IsSolution.{uA, uB, uK, uL, uK₂, max uL uK₂, uK₂, max uL uK₂,
          max uL uK₂})
    (hpi : IsPurelyInseparable K₁ K₂)
    (hdegree : Module.finrank K₁ K₂ = p) :
    ∃ (K₃ : Type uK₁) (_ : Field K₃) (_ : Algebra K₁ K₃) (_ : Algebra K K₃)
      (_ : Algebra A K₃) (_ : IsScalarTower K K₁ K₃) (_ : IsScalarTower A K K₃)
      (_ : IsScalarTower A K₁ K₃) (_ : Module.Finite K K₃),
      Algebra.IsSeparable K₁ K₃ ∧
        ∃ D₃ : IntegralClosureBaseChange.{uA, uB, uK, uL, uK₁,
            max uL uK₁, uK₁, max uL uK₁} A B K L K₃,
          D₃.IsSolution.{uA, uB, uK, uL, uK₁, max uL uK₁, uK₁, max uL uK₁,
            max uL uK₁}

Tag 0BRN, a lemma

Tag 0BRN in the Stacks Project.

IsDiscreteValuationRing.exists_isSeparableSolution_of_exists_isSolution, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/SeparableSolution/InseparableStep.lean, line 784.

Let A → B be an extension of discrete valuation rings with separable fraction-field extension L/K. If B is Nagata and there is a solution for A → B, then there is a separable solution for A → B.

theorem exists_isSeparableSolution_of_exists_isSolution
    (A : Type uA) (B : Type uB) (K : Type uK) (L : Type uL)
    [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [CommRing B] [IsDomain B] [IsDiscreteValuationRing B]
    [Field K] [Field L]
    [Algebra A B] [Algebra A K] [Algebra A L] [Algebra B L] [Algebra K L]
    [IsScalarTower A B L] [IsScalarTower A K L]
    [IsExtension A B]
    [IsFractionRing A K] [IsFractionRing B L]
    (hsep : Algebra.IsSeparableExtension K L)
    (hB : IsNagataRing B)
    (hsolution :
      ∃ (K₂ : Type uE) (_ : Field K₂) (_ : Algebra K K₂) (_ : Algebra A K₂)
      (_ : IsScalarTower A K K₂) (_ : Module.Finite K K₂),
      ∃ D₂ : IntegralClosureBaseChange.{uA, uB, uK, uL, uE,
          max uL uE, uE, max uL uE} A B K L K₂,
        D₂.IsSolution.{uA, uB, uK, uL, uE,
          max uL uE, uE, max uL uE, max uL uE}) :
    ∃ (K₁ : Type uE) (_ : Field K₁) (_ : Algebra K K₁) (_ : Algebra A K₁)
      (_ : IsScalarTower A K K₁) (_ : Module.Finite K K₁),
      ∃ D₁ : IntegralClosureBaseChange.{uA, uB, uK, uL, uE,
          max uL uE, uE, max uL uE} A B K L K₁,
        D₁.IsSeparableSolution.{uA, uB, uK, uL, uE,
          max uL uE, uE, max uL uE, max uL uE}

Tag 09IH, a lemma

Tag 09IH in the Stacks Project.

IsDiscreteValuationRing.integralClosure_reducedTensorProduct_isNoetherian_and_spec_surjective, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/EliminateRamificationII.lean, line 430.

Let A → B be an essentially finite type extension of discrete valuation rings with fraction fields K → L. If K'/K is algebraic and the integral closure A' of A in K' is Noetherian, then the integral closure B' of B in the reduction of L ⊗[K] K' is Noetherian. The canonical map Spec(B') → Spec(A') is surjective and all corresponding residue field extensions are finitely generated.

theorem integralClosure_reducedTensorProduct_isNoetherian_and_spec_surjective
    (A : Type uA) (B : Type uB) (K : Type uK) (L : Type uL)
    [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [CommRing B] [IsDomain B] [IsDiscreteValuationRing B]
    [Algebra A B] [FaithfulSMul A B] [IsLocalHom (algebraMap A B)]
    [Field K] [Algebra A K] [IsFractionRing A K]
    [Field L] [Algebra B L] [IsFractionRing B L]
    [Algebra A L] [Algebra K L]
    [IsScalarTower A B L] [IsScalarTower A K L]
    [Algebra.EssFiniteType A B]
    (K' : Type v) [Field K'] [Algebra K K'] [Algebra.IsAlgebraic K K']
    [Algebra A K'] [IsScalarTower A K K']
    [IsNoetherianRing (integralClosure A K')] :
    let L' := (L ⊗[K] K') ⧸ nilradical (L ⊗[K] K')
    let A' := integralClosure A K'
    let B' := integralClosure B L'
    IsNoetherianRing B' ∧
      ∃ f : A' →+* B',
        (∀ x : A',
          ((B').val (f x) : L') =
            algebraMap (L ⊗[K] K') L'
              (Algebra.TensorProduct.includeRight (x : K'))) ∧
        Function.Surjective (PrimeSpectrum.comap f) ∧
        ∀ q : PrimeSpectrum B',
          (Ideal.ResidueField.map (q.asIdeal.comap f) q.asIdeal f rfl).EssFiniteType

Tag 09II, a proposition

Tag 09II in the Stacks Project.

Stacks.EliminateRamification.exists_finite_extension_isSolution, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/EliminateRamificationTheorem.lean, line 90.

Tag 09II. An essentially finite type extension of discrete valuation rings admits a finite base-field extension which is a solution.

theorem exists_finite_extension_isSolution
    (A : Type uA) (B : Type uB) (K : Type uK) (L : Type uL)
    [CommRing A] [CommRing B] [IsDomain A] [IsDomain B]
    [IsDiscreteValuationRing A] [IsDiscreteValuationRing B] [Field K] [Field L]
    [Algebra A B] [Algebra A K] [Algebra A L] [Algebra B L] [Algebra K L]
    [IsScalarTower A B L] [IsScalarTower A K L]
    [IsExtension A B] [IsFractionRing A K] [IsFractionRing B L]
    [Algebra.EssFiniteType A B] :
    ∃ (K₁ : Type uK) (_ : Field K₁) (_ : Algebra K K₁) (_ : Algebra A K₁)
      (_ : IsScalarTower A K K₁) (_ : Module.Finite K K₁),
      ∃ D : IntegralClosureBaseChange.{uA, uB, uK, uL, uK, max uL uK, uK, max uL uK}
        A B K L K₁,
        D.IsSolution.{uA, uB, uK, uL, uK, max uL uK, uK, max uL uK, v}

Tag 0BRP, a lemma

Tag 0BRP in the Stacks Project.

IsDiscreteValuationRing.exists_isSeparableSolution_of_essFiniteType, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/RingTheory/DiscreteValuationRing/EppSeparableSolutionII.lean, line 120.

Let A → B be an extension of discrete valuation rings with fraction fields K ⊂ L such that B is essentially of finite type over A, A or B is a Nagata ring, and L / K is separable. Then there exists a separable solution for A → B.

theorem exists_isSeparableSolution_of_essFiniteType
    (A : Type uA) (B : Type uB) (K : Type uK) (L : Type uL)
    [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
    [CommRing B] [IsDomain B] [IsDiscreteValuationRing B]
    [Field K] [Field L]
    [Algebra A B] [Algebra A K] [Algebra A L] [Algebra B L] [Algebra K L]
    [IsScalarTower A B L] [IsScalarTower A K L]
    [IsExtension A B]
    [IsFractionRing A K] [IsFractionRing B L]
    [Algebra.EssFiniteType A B]
    (hNagata : IsNagataRing A ∨ IsNagataRing B)
    (hsep : Algebra.IsSeparableExtension K L) :
    ∃ (K₁ : Type uK) (_ : Field K₁) (_ : Algebra K K₁) (_ : Algebra A K₁)
      (_ : IsScalarTower A K K₁) (_ : Module.Finite K K₁),
      ∃ D₁ : IntegralClosureBaseChange.{uA, uB, uK, uL, uK,
          max uL uK, uK, max uL uK} A B K L K₁,
        D₁.IsSeparableSolution.{uA, uB, uK, uL, uK,
          max uL uK, uK, max uL uK, max uL uK}

Sources

• Section 15.118, Tag 0GLQ
• AxiomMath/Stacks at 5ed82d14

LeanIsDiscreteValuationRing.exists_isSolution_of_tower · IsNagataRing.of_isSeparable_fractionRing · exists_bound_finite_extension_ringEquiv_mod · exists_compatible_finite_extensions_of_quotient_equiv · IsDiscreteValuationRing.exists_isSeparable_solution_of_isPurelyInseparable_finrank_eq_prime · IsDiscreteValuationRing.exists_isSeparableSolution_of_exists_isSolution · IsDiscreteValuationRing.integralClosure_reducedTensorProduct_isNoetherian_and_spec_surjective · Stacks.EliminateRamification.exists_finite_extension_isSolution

Section15.119 Picard groups of rings#

Section 15.119 of the Stacks Project, Tag 0AFW. 2 tags here are carried by 4 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0B8H, a definition

Tag 0B8H in the Stacks Project.

Module.IsInvertibleModule, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Algebra/Module/Projective/Invertible.lean, line 62.

An R-module M is invertible when the functor M ⊗[R] - is an equivalence of the category of R-modules.

public abbrev Module.IsInvertibleModule
    (R : Type u) [CommRing R] (M : Type u) [AddCommGroup M] [Module R M] : Prop

Module.IsTrivialInvertibleModule, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Algebra/Module/Projective/Invertible.lean, line 68.

An invertible R-module is trivial if it is isomorphic to R as an R-module.

public abbrev Module.IsTrivialInvertibleModule
    (R : Type u) [CommRing R] (M : Type u) [AddCommGroup M] [Module R M] : Prop

Tag 0B8I, a lemma

Tag 0B8I in the Stacks Project.

Module.finite_projective_rankAtStalk_one_tfae_invertible_exists_tensor_inverse, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Algebra/Module/Projective/Invertible.lean, line 201.

Stacks 0B8I. For an R-module M, the following are equivalent: M is finite projective of rank one at every prime, M is invertible, and M admits a tensor inverse. Moreover, every tensor inverse of M is isomorphic to the linear dual of M.

public theorem Module.finite_projective_rankAtStalk_one_tfae_invertible_exists_tensor_inverse
    (R : Type u) [CommRing R] (M : Type v) [AddCommGroup M] [Module R M] :
    List.TFAE [
      Module.Finite R M ∧ Module.Projective R M ∧
        Module.rankAtStalk M = Function.const (PrimeSpectrum R) 1,
      Module.Invertible R M,
      ∃ (N : Type (max u v)) (_ : AddCommGroup N) (_ : Module R N),
        Nonempty (TensorProduct R M N ≃ₗ[R] R)] ∧
      ∀ (N : Type w) [AddCommGroup N] [Module R N],
        Nonempty (TensorProduct R M N ≃ₗ[R] R) →
          Nonempty (N ≃ₗ[R] Module.Dual R M)

Module.isInvertibleModule_iff_invertible, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Algebra/Module/Projective/Invertible.lean, line 290.

Stacks 0B8I, the description of invertibility the Stacks Project takes as its definition in Tag 0B8H: M ⊗[R] - is an equivalence of the category of R-modules exactly when M is invertible. Without this, Module.IsInvertibleModule and Module.Invertible are two unrelated predicates and the equivalence above is about the wrong condition (2).

public theorem Module.isInvertibleModule_iff_invertible :
    Module.IsInvertibleModule R M ↔ Module.Invertible R M

Sources

• Section 15.119, Tag 0AFW
• AxiomMath/Stacks at 5ed82d14

LeanModule.IsInvertibleModule · Module.IsTrivialInvertibleModule · Module.finite_projective_rankAtStalk_one_tfae_invertible_exists_tensor_inverse · Module.isInvertibleModule_iff_invertible

Section15.120 Determinants#

Section 15.120 of the Stacks Project, Tag 0FJ9. 7 tags here are carried by 7 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0FJA, a remark

Tag 0FJA in the Stacks Project.

ProjectiveDeterminant.determinant_submodule_eq_annihilator, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Algebra/Module/Projective/Determinant/Basic/Localization.lean, line 728.

Stacks 0FJA. The determinant line of a finite projective module is the annihilator of the degree-one elements in its exterior algebra.

public theorem determinant_submodule_eq_annihilator (R : Type u) (M : Type v)
    [CommRing R] [AddCommGroup M] [Module R M]
    [Module.Finite R M] [Module.Projective R M] :
    ProjectiveDeterminant.submodule (R := R) (M := M) =
      ⨅ m : M, LinearMap.ker (LinearMap.mulLeft R (ExteriorAlgebra.ι R m))

Tag 0FJB, a lemma

Tag 0FJB in the Stacks Project.

ProjectiveDeterminant.determinantTensorDeterminantEquiv_of_exact, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Algebra/Module/Projective/Determinant/ShortExact/Construction.lean, line 157. References: SGA 6 X (5.1), “affine case”.

A short exact sequence 0 → M' → M → M'' → 0 of finite projective modules has a canonical determinant isomorphism. Its value on x' ⊗ x'' is the wedge of the image of x' with any lift of x'' to the exterior algebra of M.

public noncomputable def determinantTensorDeterminantEquiv_of_exact
    {R : Type u} [CommRing R]
    {M' : Type v} {M : Type w} {M'' : Type x}
    [AddCommGroup M'] [Module R M'] [Module.Finite R M'] [Module.Projective R M']
    [AddCommGroup M] [Module R M] [Module.Finite R M] [Module.Projective R M]
    [AddCommGroup M''] [Module R M''] [Module.Finite R M''] [Module.Projective R M'']
    (f : M' →ₗ[R] M) (g : M →ₗ[R] M'')
    (hf : Function.Injective f) (hfg : Function.Exact f g) (hg : Function.Surjective g) :
    { γ : TensorProduct R
          (↥(⨅ m : M', LinearMap.ker
            (LinearMap.mulLeft R (ExteriorAlgebra.ι R m))))
          (↥(⨅ m : M'', LinearMap.ker
            (LinearMap.mulLeft R (ExteriorAlgebra.ι R m)))) ≃ₗ[R]
        ↥(⨅ m : M, LinearMap.ker
          (LinearMap.mulLeft R (ExteriorAlgebra.ι R m))) //
      ∀ (x' : ↥(⨅ m : M', LinearMap.ker
            (LinearMap.mulLeft R (ExteriorAlgebra.ι R m))))
        (x'' : ↥(⨅ m : M'', LinearMap.ker
            (LinearMap.mulLeft R (ExteriorAlgebra.ι R m))))
        (y'' : ExteriorAlgebra R M),
        ExteriorAlgebra.map g y'' = (x'' : ExteriorAlgebra R M'') →
          ((γ (x' ⊗ₜ[R] x'') : ↥(⨅ m : M, LinearMap.ker
              (LinearMap.mulLeft R (ExteriorAlgebra.ι R m)))) : ExteriorAlgebra R M) =
            ExteriorAlgebra.map f (x' : ExteriorAlgebra R M') * y'' }

Tag 0FJC, a lemma

Tag 0FJC in the Stacks Project.

ProjectiveDeterminant.determinantTensorDeterminantEquiv_naturality, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Algebra/Module/Projective/Determinant/ShortExact/Construction.lean, line 364.

The canonical determinant isomorphism of a short exact sequence of finite projective modules is natural under an isomorphism of short exact sequences.

public theorem determinantTensorDeterminantEquiv_naturality
    {R : Type u} [CommRing R]
    {M' : Type v₁} {M : Type v₂} {M'' : Type v₃}
    {K' : Type w₁} {K : Type w₂} {K'' : Type w₃}
    [AddCommGroup M'] [Module R M'] [Module.Finite R M'] [Module.Projective R M']
    [AddCommGroup M] [Module R M] [Module.Finite R M] [Module.Projective R M]
    [AddCommGroup M''] [Module R M''] [Module.Finite R M''] [Module.Projective R M'']
    [AddCommGroup K'] [Module R K'] [Module.Finite R K'] [Module.Projective R K']
    [AddCommGroup K] [Module R K] [Module.Finite R K] [Module.Projective R K]
    [AddCommGroup K''] [Module R K''] [Module.Finite R K''] [Module.Projective R K'']
    (f : M' →ₗ[R] M) (g : M →ₗ[R] M'')
    (fK : K' →ₗ[R] K) (gK : K →ₗ[R] K'')
    (hf : Function.Injective f) (hfg : Function.Exact f g) (hg : Function.Surjective g)
    (hfK : Function.Injective fK) (hfgK : Function.Exact fK gK)
    (hgK : Function.Surjective gK)
    (a : M' ≃ₗ[R] K') (b : M ≃ₗ[R] K) (c : M'' ≃ₗ[R] K'')
    (hab : ∀ x : M', b (f x) = fK (a x))
    (hbc : ∀ x : M, c (g x) = gK (b x)) :
    let D_M' : Submodule R (ExteriorAlgebra R M') :=
      ⨅ x : M', LinearMap.ker (LinearMap.mulLeft R (ExteriorAlgebra.ι R x))
    let D_M : Submodule R (ExteriorAlgebra R M) :=
      ⨅ x : M, LinearMap.ker (LinearMap.mulLeft R (ExteriorAlgebra.ι R x))
    let D_M'' : Submodule R (ExteriorAlgebra R M'') :=
      ⨅ x : M'', LinearMap.ker (LinearMap.mulLeft R (ExteriorAlgebra.ι R x))
    let D_K' : Submodule R (ExteriorAlgebra R K') :=
      ⨅ x : K', LinearMap.ker (LinearMap.mulLeft R (ExteriorAlgebra.ι R x))
    let D_K : Submodule R (ExteriorAlgebra R K) :=
      ⨅ x : K, LinearMap.ker (LinearMap.mulLeft R (ExteriorAlgebra.ι R x))
    let D_K'' : Submodule R (ExteriorAlgebra R K'') :=
      ⨅ x : K'', LinearMap.ker (LinearMap.mulLeft R (ExteriorAlgebra.ι R x))
    let γM := (determinantTensorDeterminantEquiv_of_exact f g hf hfg hg).val
    let γK := (determinantTensorDeterminantEquiv_of_exact fK gK hfK hfgK hgK).val
    Set.BijOn (ExteriorAlgebra.map a)
        (D_M' : Set (ExteriorAlgebra R M')) (D_K' : Set (ExteriorAlgebra R K')) ∧
      Set.BijOn (ExteriorAlgebra.map b)
        (D_M : Set (ExteriorAlgebra R M)) (D_K : Set (ExteriorAlgebra R K)) ∧
      Set.BijOn (ExteriorAlgebra.map c)
        (D_M'' : Set (ExteriorAlgebra R M'')) (D_K'' : Set (ExteriorAlgebra R K'')) ∧
      ∀ (x' : D_M') (x'' : D_M'') (y' : D_K') (y'' : D_K''),
        ExteriorAlgebra.map a (x' : ExteriorAlgebra R M') =
          (y' : ExteriorAlgebra R K') →
        ExteriorAlgebra.map c (x'' : ExteriorAlgebra R M'') =
          (y'' : ExteriorAlgebra R K'') →
        ExteriorAlgebra.map b
            ((γM (x' ⊗ₜ[R] x'') : D_M) : ExteriorAlgebra R M) =
          ((γK (y' ⊗ₜ[R] y'') : D_K) : ExteriorAlgebra R K)

Tag 0FJD, a lemma

Tag 0FJD in the Stacks Project.

ProjectiveDeterminant.determinantEquiv_filtration, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Algebra/Module/Projective/Determinant/ShortExact/Filtration.lean, line 94.

For finite projective successive quotients of a filtration K ≤ L ≤ M, combining the canonical determinant equivalences first across K ≤ L or first across the induced sequence L / K → M / K → M / L gives the same equivalence to the determinant line of M.

public theorem determinantEquiv_filtration
    {R : Type u} [CommRing R] {M : Type v} [AddCommGroup M] [Module R M]
    (K L : Submodule R M) (hKL : K ≤ L)
    [Module.Finite R K] [Module.Projective R K]
    [Module.Finite R (L ⧸ K.submoduleOf L)]
    [Module.Projective R (L ⧸ K.submoduleOf L)]
    [Module.Finite R (M ⧸ L)] [Module.Projective R (M ⧸ L)] :
    let iQ : (L ⧸ K.submoduleOf L) →ₗ[R] (M ⧸ K) :=
      (K.submoduleOf L).mapQ K L.subtype (by
        intro x hx
        exact hx)
    let pQ : (M ⧸ K) →ₗ[R] (M ⧸ L) :=
      K.mapQ L LinearMap.id (by simpa using hKL)
    ∃ hLfin : Module.Finite R L,
      ∃ hLproj : Module.Projective R L,
        ∃ hMKfin : Module.Finite R (M ⧸ K),
          ∃ hMKproj : Module.Projective R (M ⧸ K),
            ∃ hMfin : Module.Finite R M,
              ∃ hMproj : Module.Projective R M,
                let _ : Module.Finite R L := hLfin
                let _ : Module.Projective R L := hLproj
                let _ : Module.Finite R (M ⧸ K) := hMKfin
                let _ : Module.Projective R (M ⧸ K) := hMKproj
                let _ : Module.Finite R M := hMfin
                let _ : Module.Projective R M := hMproj
                let hiQ : Function.Injective iQ
    := by
                  intro x y hxy
                  obtain ⟨x, rfl⟩

Tag 0FJE, a lemma

Tag 0FJE in the Stacks Project.

ProjectiveDeterminant.determinantTensorDeterminantEquiv_prodComm, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Algebra/Module/Projective/Determinant/ShortExact/ProdComm.lean, line 366.

Switching the summands in a direct sum is compatible with the canonical isomorphism from the tensor product of their determinant lines, up to the determinant of minus the identity on the tensor product of the summands.

public theorem determinantTensorDeterminantEquiv_prodComm
    {R : Type u} [CommRing R]
    (M' : Type v) (M'' : Type w)
    [AddCommGroup M'] [Module R M'] [Module.Finite R M'] [Module.Projective R M']
    [AddCommGroup M''] [Module R M''] [Module.Finite R M''] [Module.Projective R M''] :
    let D' : Submodule R (ExteriorAlgebra R M') :=
      ⨅ m : M', LinearMap.ker (LinearMap.mulLeft R (ExteriorAlgebra.ι R m))
    let D'' : Submodule R (ExteriorAlgebra R M'') :=
      ⨅ m : M'', LinearMap.ker (LinearMap.mulLeft R (ExteriorAlgebra.ι R m))
    let D'_prod_D'' : Submodule R (ExteriorAlgebra R (M' × M'')) :=
      ⨅ m : M' × M'', LinearMap.ker (LinearMap.mulLeft R (ExteriorAlgebra.ι R m))
    let D''_prod_D' : Submodule R (ExteriorAlgebra R (M'' × M')) :=
      ⨅ m : M'' × M', LinearMap.ker (LinearMap.mulLeft R (ExteriorAlgebra.ι R m))
    let γ' := (determinantTensorDeterminantEquiv_of_exact
      (LinearMap.inl R M' M'') (LinearMap.snd R M' M'')
      LinearMap.inl_injective Function.Exact.inl_snd LinearMap.snd_surjective).val
    let γ'' := (determinantTensorDeterminantEquiv_of_exact
      (LinearMap.inl R M'' M') (LinearMap.snd R M'' M')
      LinearMap.inl_injective Function.Exact.inl_snd LinearMap.snd_surjective).val
    ∃ ε : Rˣ,
      (ε : R) = ProjectiveDeterminant.det
        (-(LinearMap.id : TensorProduct R M' M'' →ₗ[R] TensorProduct R M' M'')) ∧
      ∀ x : TensorProduct R D' D'',
        ExteriorAlgebra.map (LinearEquiv.prodComm R M' M'')
            ((γ' x : D'_prod_D'') : ExteriorAlgebra R (M' × M'')) =
          ((γ'' ((ε : R) • TensorProduct.comm R D' D'' x) : D''_prod_D') :
            ExteriorAlgebra R (M'' × M'))

Tag 0FJF, a lemma

Tag 0FJF in the Stacks Project.

ProjectiveDeterminant.det_one_add_comp_comm, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Algebra/Module/Projective/Determinant/Basic/Localization.lean, line 782.

Stacks 0FJF. For maps in opposite directions between finite projective modules, the determinants of the identity plus the two possible composites agree.

public theorem det_one_add_comp_comm
    {R : Type u} [CommRing R]
    {M : Type v} [AddCommGroup M] [Module R M] [Module.Finite R M] [Module.Projective R M]
    {N : Type w} [AddCommGroup N] [Module R N] [Module.Finite R N] [Module.Projective R N]
    (a : M →ₗ[R] N) (b : N →ₗ[R] M) :
    ProjectiveDeterminant.det (LinearMap.id + a.comp b) =
      ProjectiveDeterminant.det (LinearMap.id + b.comp a)

Tag 0AFX, a lemma

Tag 0AFX in the Stacks Project.

K0.exists_determinant_hom, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/KTheory/Grothendieck/Determinant.lean, line 155. References: SGA 6 X (5.1), “affine case”.

Stacks 0AFX. The determinant gives a homomorphism from K₀(R) to the Picard group. For a finite locally free module of constant rank n, it sends its class to the class of its nth exterior power.

public theorem K0.exists_determinant_hom (R : Type u) [CommRing R] :
    ∃ det : K0 R →+ Additive (CommRing.Pic R),
      ∀ (M : FGProj R) (n : ℕ),
        Module.rankAtStalk M.carrier = Function.const (PrimeSpectrum R) n →
          ∃ h : Module.Invertible R (⋀[R]^n M.carrier),
            det (K0.mk M) =
              Additive.ofMul (@CommRing.Pic.mk R (⋀[R]^n M.carrier) inferInstance
                inferInstance inferInstance h)

Sources

• Section 15.120, Tag 0FJ9
• AxiomMath/Stacks at 5ed82d14

LeanProjectiveDeterminant.determinant_submodule_eq_annihilator · ProjectiveDeterminant.determinantTensorDeterminantEquiv_of_exact · ProjectiveDeterminant.determinantTensorDeterminantEquiv_naturality · ProjectiveDeterminant.determinantEquiv_filtration · ProjectiveDeterminant.determinantTensorDeterminantEquiv_prodComm · ProjectiveDeterminant.det_one_add_comp_comm · K0.exists_determinant_hom

Section15.121 Perfect complexes and K-groups#

Section 15.121 of the Stacks Project, Tag 0FJG. 2 tags here are carried by 2 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0AFY, a lemma

Tag 0AFY in the Stacks Project.

K0.exists_eulerChar_perfect, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/KTheory/Grothendieck/PerfectComplex.lean, line 675. References: SGA 6 0 (3.1), “special case”.

Stacks 0AFY. Perfect complexes over R have an Euler characteristic in K₀(R) which changes by (-1)ⁿ under a shift by n, is additive in distinguished triangles, and is the alternating sum of the classes of the terms of every bounded finite-projective representative.

public theorem K0.exists_eulerChar_perfect (R : Type u) [CommRing R]
    [HasDerivedCategory.{w} (ModuleCat.{u} R)] :
    ∃ c : {K : DerivedCategory (ModuleCat.{u} R) // StacksDraft.Perfect.IsPerfect K} → K0 R,
      (∀ (K : DerivedCategory (ModuleCat.{u} R)) (hK : StacksDraft.Perfect.IsPerfect K) (n : ℤ),
        ∃ hKn : StacksDraft.Perfect.IsPerfect (K⟦n⟧),
          c ⟨K⟦n⟧, hKn⟩ = (n.negOnePow : ℤ) • c ⟨K, hK⟩) ∧
      (∀ (T : Triangle (DerivedCategory (ModuleCat.{u} R))),
        T ∈ distTriang (DerivedCategory (ModuleCat.{u} R)) →
        ∀ (h₁ : StacksDraft.Perfect.IsPerfect T.obj₁) (h₂ : StacksDraft.Perfect.IsPerfect T.obj₂)
          (h₃ : StacksDraft.Perfect.IsPerfect T.obj₃),
          c ⟨T.obj₂, h₂⟩ = c ⟨T.obj₁, h₁⟩ + c ⟨T.obj₃, h₃⟩) ∧
      (∀ (K : DerivedCategory (ModuleCat.{u} R)) (hK : StacksDraft.Perfect.IsPerfect K)
        (P : CochainComplex (ModuleCat.{u} R) ℤ) (a b : ℤ),
        P.IsStrictlyGE a → P.IsStrictlyLE b →
        (hfinite : ∀ i : ℤ, Module.Finite R (P.X i)) →
        (hprojective : ∀ i : ℤ, Module.Projective R (P.X i)) →
        Nonempty (K ≅ DerivedCategory.Q.obj P) →
        c ⟨K, hK⟩ = ∑ i ∈ Finset.Icc a b,
          (i.negOnePow : ℤ) • K0.mk
            ({ carrier := P.X i, finite := hfinite i, projective := hprojective i } : FGProj R))

Tag 0FCU, a lemma

Tag 0FCU in the Stacks Project.

K0.exists_k0_perfectDerivedCategory_addEquiv, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/KTheory/Grothendieck/PerfectDerivedCategory.lean, line 505.

Stacks 0FCU. Euler characteristic induces an additive equivalence from the zeroth K-group of the perfect derived category of R to the zeroth K-group of finite projective R-modules. On a bounded finite-projective representative it is the alternating sum of the classes of its terms.

public theorem K0.exists_k0_perfectDerivedCategory_addEquiv (R : Type u) [CommRing R] :
    let P : ObjectProperty (DerivedCategory (ModuleCat.{u} R)) :=
      (ObjectProperty.singleton
        ((DerivedCategory.singleFunctor (ModuleCat.{u} R) 0).obj
          (ModuleCat.of R R))).triangEnvelope
    ∃ e : CategoryTheory.Pretriangulated.K0 P.FullSubcategory ≃+ K0 R,
      ∀ (K : P.FullSubcategory)
        (Q : CochainComplex (ModuleCat.{u} R) ℤ) (a b : ℤ),
        Q.IsStrictlyGE a → Q.IsStrictlyLE b →
        (hfinite : ∀ i : ℤ, Module.Finite R (Q.X i)) →
        (hprojective : ∀ i : ℤ, Module.Projective R (Q.X i)) →
        Nonempty (K.obj ≅ DerivedCategory.Q.obj Q) →
        e (CategoryTheory.Pretriangulated.K0.cls K) = ∑ i ∈ Finset.Icc a b,
          (i.negOnePow : ℤ) • K0.mk
            ({ carrier := Q.X i, finite := hfinite i, projective := hprojective i } : FGProj R)

Sources

• Section 15.121, Tag 0FJG
• AxiomMath/Stacks at 5ed82d14

LeanK0.exists_eulerChar_perfect · K0.exists_k0_perfectDerivedCategory_addEquiv

Section15.122 Determinants of endomorphisms of finite length modules#

Section 15.122 of the Stacks Project, Tag 0GSX. 3 tags here are carried by 5 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0GSY, a lemma

Tag 0GSY in the Stacks Project.

FiniteLengthEndomorphism.det_exact, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Algebra/Module/FiniteLength/Determinant.lean, line 724. The attribute's comment: “multiplicativity of the residue-field determinant”.

Determinant is multiplicative in an equivariant short exact sequence.

public theorem det_exact (M₁ : Type v₁) (M₂ : Type v₂) (M₃ : Type v₃)
    [AddCommGroup M₁] [Module R M₁] [AddCommGroup M₂] [Module R M₂]
    [AddCommGroup M₃] [Module R M₃]
    (h₁ : IsFiniteLength R M₁) (h₂ : IsFiniteLength R M₂) (h₃ : IsFiniteLength R M₃)
    (φ₁ : Module.End R M₁) (φ₂ : Module.End R M₂) (φ₃ : Module.End R M₃)
    (f : M₁ →ₗ[R] M₂) (g : M₂ →ₗ[R] M₃) (hf : Function.Injective f)
    (hexact : Function.Exact f g) (hg : Function.Surjective g)
    (hfφ : f.comp φ₁ = φ₂.comp f) (hgφ : g.comp φ₂ = φ₃.comp g) :
    det R h₂ φ₂ = det R h₁ φ₁ * det R h₃ φ₃

FiniteLengthEndomorphism.trace_exact, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Algebra/Module/FiniteLength/Determinant.lean, line 737. The attribute's comment: “additivity of the residue-field trace”.

Trace is additive in an equivariant short exact sequence.

public theorem trace_exact (M₁ : Type v₁) (M₂ : Type v₂) (M₃ : Type v₃)
    [AddCommGroup M₁] [Module R M₁] [AddCommGroup M₂] [Module R M₂]
    [AddCommGroup M₃] [Module R M₃]
    (h₁ : IsFiniteLength R M₁) (h₂ : IsFiniteLength R M₂) (h₃ : IsFiniteLength R M₃)
    (φ₁ : Module.End R M₁) (φ₂ : Module.End R M₂) (φ₃ : Module.End R M₃)
    (f : M₁ →ₗ[R] M₂) (g : M₂ →ₗ[R] M₃) (hf : Function.Injective f)
    (hexact : Function.Exact f g) (hg : Function.Surjective g)
    (hfφ : f.comp φ₁ = φ₂.comp f) (hgφ : g.comp φ₂ = φ₃.comp g) :
    trace R h₂ φ₂ = trace R h₁ φ₁ + trace R h₃ φ₃

FiniteLengthEndomorphism.charpoly_exact, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Algebra/Module/FiniteLength/Determinant.lean, line 750. The attribute's comment: “multiplicativity of the residue-field characteristic polynomial”.

Characteristic polynomial is multiplicative in an equivariant short exact sequence.

public theorem charpoly_exact (M₁ : Type v₁) (M₂ : Type v₂) (M₃ : Type v₃)
    [AddCommGroup M₁] [Module R M₁] [AddCommGroup M₂] [Module R M₂]
    [AddCommGroup M₃] [Module R M₃]
    (h₁ : IsFiniteLength R M₁) (h₂ : IsFiniteLength R M₂) (h₃ : IsFiniteLength R M₃)
    (φ₁ : Module.End R M₁) (φ₂ : Module.End R M₂) (φ₃ : Module.End R M₃)
    (f : M₁ →ₗ[R] M₂) (g : M₂ →ₗ[R] M₃) (hf : Function.Injective f)
    (hexact : Function.Exact f g) (hg : Function.Surjective g)
    (hfφ : f.comp φ₁ = φ₂.comp f) (hgφ : g.comp φ₂ = φ₃.comp g) :
    charpoly R h₂ φ₂ = charpoly R h₁ φ₁ * charpoly R h₃ φ₃

Tag 0GSZ, a lemma

Tag 0GSZ in the Stacks Project.

FiniteLengthEndomorphism.multiplication_det_eq_norm_pow_length, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Algebra/Module/FiniteLength/Determinant.lean, line 926.

For a local homomorphism of local rings with finite residue-field extension, the determinant over the source residue field of multiplication by u on a finite-length target module is the appropriate power of the residue-field norm of u.

public theorem multiplication_det_eq_norm_pow_length
    (R : Type u) [CommRing R] [IsLocalRing R]
    (R' : Type v) [CommRing R'] [IsLocalRing R'] [Algebra R R']
    [IsLocalHom (algebraMap R R')]
    [Module.Finite (IsLocalRing.ResidueField R) (IsLocalRing.ResidueField R')]
    (M' : Type w) [AddCommGroup M'] [Module R M'] [Module R' M'] [IsScalarTower R R' M']
    (hM' : IsFiniteLength R' M') (u : R') :
    let hM : IsFiniteLength R M' := Module.length_ne_top_iff.mp <| by
      rw [IsLocalRing.length_restrictScalars R R' M']
      exact WithTop.mul_ne_top (Module.length_ne_top_iff.mpr hM') (by
        rw [Module.length_eq_finrank]
        exact ENat.natCast_ne_top _)
    FiniteLengthEndomorphism.det R hM
        ((LinearMap.lsmul R' M' u).restrictScalars R) =
      Algebra.norm (IsLocalRing.ResidueField R) (IsLocalRing.residue R' u) ^
        (Module.length R' M').toNat

Tag 0GT0, a lemma

Tag 0GT0 in the Stacks Project.

FiniteLengthEndomorphism.map_det_pow_eq_det_baseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Algebra/Module/FiniteLength/Determinant.lean, line 1324.

Stacks 0GT0. Under a flat local homomorphism R → S whose closed fibre has finite length m, the determinant of a finite-length module endomorphism after scalar extension is the image in the residue field of S of the original determinant raised to the power m.

public theorem map_det_pow_eq_det_baseChange
    {R : Type u} {S : Type v} [CommRing R] [CommRing S]
    [IsLocalRing R] [IsLocalRing S] [Algebra R S]
    [IsLocalHom (algebraMap R S)] [Module.Flat R S]
    (m : ℕ)
    (hm : Module.length S
      (S ⧸ Ideal.map (algebraMap R S) (IsLocalRing.maximalIdeal R)) = m)
    {M : Type w} [AddCommGroup M] [Module R M]
    (hM : IsFiniteLength R M) (phi : Module.End R M) :
    ∃ hMS : IsFiniteLength S (S ⊗[R] M),
      IsLocalRing.ResidueField.map (algebraMap R S)
          (FiniteLengthEndomorphism.det R hM phi ^ m) =
        FiniteLengthEndomorphism.det S hMS (LinearMap.baseChange S phi)

Sources

• Section 15.122, Tag 0GSX
• AxiomMath/Stacks at 5ed82d14

LeanFiniteLengthEndomorphism.det_exact · FiniteLengthEndomorphism.trace_exact · FiniteLengthEndomorphism.charpoly_exact · FiniteLengthEndomorphism.multiplication_det_eq_norm_pow_length · FiniteLengthEndomorphism.map_det_pow_eq_det_baseChange

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