LocalRing.lean`

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Unramified lifts of separable residue field extensions

Material for Mathlib.RingTheory.Unramified.LocalRing, whose characterization of unramified extensions of local rings (Algebra.FormallyUnramified.iff_map_maximalIdeal_eq) this complements with an existence statement.

A finite separable extension k' of the residue field k of a discrete valuation ring R (with fraction field K) lifts to an unramified extension L/K of the same degree: writing k' = k[X]/(p̄) via a power basis (primitive element), we lift p̄ to a monic P ∈ R[X] and set L = K[X]/(P). The integral extension S = R[X]/(P) is again a discrete valuation ring, with S/πS ≅ k[X]/(p̄) = k' — so its maximal ideal is generated by a uniformizer π of R (ramification index 1) and its residue field is k'.

The ingredients — AdjoinRoot.isDiscreteValuationRing_of_irreducible_map_residue (R[X]/(P) is an unramified discrete valuation ring for monic P with irreducible reduction, in FLT.Mathlib.RingTheory.AdjoinRoot) and the separability transfer Polynomial.Monic.separable_map_algebraMap_of_separable_map_residue (in FLT.Mathlib.RingTheory.DiscreteValuationRing.Basic) — are assembled here into

exists_unramified_extension_of_residueField : the lifting theorem.

The quadratic case of the lifting theorem is the local-field input to WeierstrassCurve.exists_quadraticTwist_hasSplitMultiplicativeReduction.

Imports (5)

Imported by 2 files

theorem exists_unramified_extension_of_residueField Unramified lifting, arbitrary degree line 56 source
Source documentation
Unramified lifting, arbitrary degree. A finite separable extension k' of the residue field of a discrete valuation ring R lifts to an unramified extension L of its fraction field K: there is a finite separable extension L/K with [L : K] = [k' : k] and a discrete valuation ring S with fraction field L, containing R via a local homomorphism, whose residue field is k'. (The residue degree being [L : K] forces ramification index 1.)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDomain, IsDiscreteValuationRing, Field, Algebra, IsFractionRing, IsLocalRing.ResidueField, FiniteDimensional, Algebra.IsSeparable, Exists, IsScalarTower, Module.Finite, IsLocalHom, RingHom, Algebra.algebraMap, And, Eq, Nat, Module.finrank, Nonempty and 1 more
Proof uses: Polynomial, Polynomial.Monic, Polynomial.Separable, Irreducible, Polynomial.natDegree, AdjoinRoot, Field.exists_monic_irreducible_adjoinRoot_algEquiv, LT.lt, OfNat.ofNat, Eq.rec, Module.finrank_pos, Eq.symm, Polynomial.map, IsLocalRing.residue, IsLocalRing.exists_monic_map_residue_eq, Eq.mpr, id, congrArg, Polynomial.Monic.irreducible_of_irreducible_map, Ne and 62 more
117 instances, coercions and auxiliary constants left out.

Tactic trace: 32 steps
Open to load the checked before-and-after goal states.
```
theorem exists_unramified_extension_of_residueField
    (k' : Type u) [Field k'] [Algebra (ResidueField R) k']
    [FiniteDimensional (ResidueField R) k'] [Algebra.IsSeparable (ResidueField R) k'] :
    ∃ (L : Type u) (_ : Field L) (_ : Algebra K L) (_ : FiniteDimensional K L)
      (_ : Algebra.IsSeparable K L) (_ : Algebra R L) (_ : IsScalarTower R K L) (S : Type u)
      (_ : CommRing S) (_ : IsDomain S) (_ : IsDiscreteValuationRing S) (_ : Algebra R S)
      (_ : Module.Finite R S) (_ : Algebra S L) (_ : IsScalarTower R S L) (_ : IsFractionRing S L)
      (_ : IsLocalHom (algebraMap R S)),
      Module.finrank K L = Module.finrank (ResidueField R) k'
        ∧ Nonempty (ResidueField S ≃ₐ[ResidueField R] k') := by
…
```