Theorems · Theorem · number theory

Ideal.FinrankQuotientMap.span_eq_top

Deprecated since 2026-07-01Mathlib marks this declaration as deprecated.

∀ {R : Type u} [inst : CommRing R] {S : Type v} [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal R)
  {K : Type u_1} [inst_3 : Field K] [inst_4 : Algebra R K] {L : Type u_2} [inst_5 : Field L] [inst_6 : Algebra S L]
  [IsFractionRing S L] [IsDomain R] [IsDomain S] [inst_10 : Algebra K L] [Module.Finite R S] [inst_12 : Algebra R L]
  [IsScalarTower R S L] [IsScalarTower R K L] [Algebra.IsAlgebraic R S] [Module.IsTorsionFree R K],
  p ≠ ⊤ →
    ∀ (b : Set S),
      Submodule.span R b ⊔ Submodule.restrictScalars R (Ideal.map (algebraMap R S) p) = ⊤ →
        Submodule.span K (⇑(algebraMap S L) '' b) = ⊤

If b mod p spans S/p as R/p-space, then b itself spans Frac(S) as K-space. Here, * p is an ideal of R such that R / p is nontrivial * K is a field that has an embedding of R (in particular we can take K = Frac(R)) * L is a field extension of K * S is the integral closure of R in L More precisely, we avoid quotients in this statement and instead require that b ∪ pS spans S.

Defined in

Mathlib.NumberTheory.RamificationInertia.Basic

Cited by

1 results in Mathlib

Foundations

Depth 124 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraFieldAlgebraFieldAlgebraIsFractionRingIsDomainIsDomainAlgebraModule.FiniteAlgebraIsScalarTowerIsScalarTowerAlgebra.IsAlgebraicModule.IsTorsionFree