LocalUniformization.lean`

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Centres of valuation subrings and local uniformization

Let F be a field over a field k and let 𝒪 be a valuation subring of F. For a k-subalgebra A of F contained in 𝒪, the centre of 𝒪 on A is the prime ideal 𝔪_𝒪 ∩ A of A: the point of the affine model Spec A whose local ring 𝒪 dominates.

𝒪 admits local uniformization over k if some finitely generated k-subalgebra A ⊆ 𝒪 with fraction field F is regular at the centre of 𝒪. Following Knaf and Kuhlmann, the condition is that the centre is a regular point, not a smooth one, which is the right notion over an imperfect ground field.

Main definitions

ValuationSubring.centerOn: the centre of a valuation subring on a subalgebra it contains.
ValuationSubring.HasLocalUniformization: some affine model is regular at the centre.

References

H. Knaf and F.-V. Kuhlmann, Every place admits local uniformization in a finite extension of the function field, Adv. Math. 221 (2009), 428--453.

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

def ValuationSubring.centerOn The centre of a valuation subring 𝒪 of F on a k-subalgebra A of F contained in 𝒪: the prime ideal 𝔪_𝒪 ∩ A of A line 56 source
Source documentation
The centre of a valuation subring 𝒪 of F on a k-subalgebra A of F contained in 𝒪: the prime ideal 𝔪_𝒪 ∩ A of A.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Algebra, ValuationSubring, Subalgebra, Subtype, Membership.mem, Ideal
Proof uses: Ideal.comap, RingHom, RingHom.codRestrict, AlgHom.toRingHom, Subalgebra.val, IsLocalRing.maximalIdeal
27 instances, coercions and auxiliary constants left out.
```
def centerOn (𝒪 : ValuationSubring F) (A : Subalgebra k F) (hA : ∀ a : A, (a : F) ∈ 𝒪) :
    Ideal A :=
…
```

instance ValuationSubring.(instance at line 61) line 61 source

Source documentation

The centre of 𝒪 on A is a prime ideal of A.

instance (𝒪 : ValuationSubring F) (A : Subalgebra k F) (hA : ∀ a : A, (a : F) ∈ 𝒪) :
    (𝒪.centerOn A hA).IsPrime :=
…

theorem ValuationSubring.mem_centerOn_iff The centre of 𝒪 on A consists of the elements of A of valuation less than one line 66 source
Source documentation
The centre of 𝒪 on A consists of the elements of A of valuation less than one.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Algebra, ValuationSubring, Subalgebra, Subtype, Membership.mem, Iff, Ideal, ValuationSubring.centerOn, LT.lt, ValuationSubring.ValueGroup, Valuation, ValuationSubring.valuation, OfNat.ofNat
Proof uses: Eq.mpr, Ideal.comap, RingHom, RingHom.codRestrict, AlgHom.toRingHom, Subalgebra.val, IsLocalRing.maximalIdeal, id, Eq, congrArg, ValuationSubring.centerOn.eq_1, propext, Ideal.mem_comap, ValuationSubring.valuation_lt_one_iff, Iff.rfl
78 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem mem_centerOn_iff (𝒪 : ValuationSubring F) (A : Subalgebra k F)
    (hA : ∀ a : A, (a : F) ∈ 𝒪) (a : A) :
    a ∈ 𝒪.centerOn A hA ↔ 𝒪.valuation (a : F) < 1 := by
…
```
def ValuationSubring.HasLocalUniformization Documented declaration: HasLocalUniformization line 78 source
Source documentation
A valuation subring 𝒪 of F admits local uniformization over k if some affine model of F over k inside 𝒪 is regular at the centre of 𝒪: there is a finitely generated k-subalgebra A of F with A ⊆ 𝒪 and fraction field F whose localisation at the centre of 𝒪 is a regular local ring.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, ValuationSubring, Algebra
Proof uses: Exists, Subalgebra, Subtype, Membership.mem, And, Algebra.FiniteType, IsFractionRing, IsRegularLocalRing, Localization.AtPrime, ValuationSubring.centerOn, Ideal.primeCompl
24 instances, coercions and auxiliary constants left out.
```
def HasLocalUniformization (𝒪 : ValuationSubring F) (k : Type*) [Field k] [Algebra k F] : Prop :=
…
```
theorem ValuationSubring.hasLocalUniformization_of_isRegularRing A regular affine model of F over k inside 𝒪 uniformizes 𝒪 line 84 source
Source documentation
A regular affine model of F over k inside 𝒪 uniformizes 𝒪.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Algebra, ValuationSubring, Subalgebra, Subtype, Membership.mem, Algebra.FiniteType, IsFractionRing, IsRegularRing, ValuationSubring.HasLocalUniformization
Proof uses: Exists.intro, Exists, And, IsRegularLocalRing, Localization.AtPrime, ValuationSubring.centerOn, Ideal.primeCompl, And.intro
37 instances, coercions and auxiliary constants left out.
```
theorem hasLocalUniformization_of_isRegularRing (𝒪 : ValuationSubring F) (A : Subalgebra k F)
    (hA : ∀ a : A, (a : F) ∈ 𝒪) [Algebra.FiniteType k A] [IsFractionRing A F]
    [IsRegularRing A] : 𝒪.HasLocalUniformization k :=
…
```