Theorems · Theorem · number theory

Ideal.FinrankQuotientMap.linearIndependent_of_nontrivial

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] (K : Type u_1)
  [inst_3 : Field K] [inst_4 : Algebra R K] {V : Type u_3} {V' : Type u_4} {V'' : Type u_5} [inst_5 : AddCommGroup V]
  [inst_6 : Module R V] [inst_7 : Module K V] [IsScalarTower R K V] [inst_9 : AddCommGroup V'] [inst_10 : Module R V']
  [inst_11 : Module S V'] [IsScalarTower R S V'] [inst_13 : AddCommGroup V''] [inst_14 : Module R V'']
  [hRK : IsFractionRing R K] [IsDedekindDomain R],
  RingHom.ker (algebraMap R S) ≠ ⊤ →
    ∀ (F : V'' →ₗ[R] V),
      Function.Injective ⇑F →
        ∀ (f' : V'' →ₗ[R] V') {ι : Type u_6} {b : ι → V''}, LinearIndependent S (⇑f' ∘ b) → LinearIndependent K (⇑F ∘ b)

Let V be a vector space over K = Frac(R), S / R a ring extension and V' a module over S. If b, in the intersection V'' of V and V', is linear independent over S in V', then it is linear independent over R in V. The statement we prove is actually slightly more general: * it suffices that the inclusion algebraMap R S : R → S is nontrivial * the function f' : V'' → V' doesn't need to be injective

Defined in

Mathlib.NumberTheory.RamificationInertia.Basic

Cited by

1 results in Mathlib

Foundations

Depth 146 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraFieldAlgebraAddCommGroupModuleModuleIsScalarTowerAddCommGroupModuleModuleIsScalarTowerAddCommGroupModuleIsFractionRingIsDedekindDomain