Theorems · Theorem · commutative algebra

Module.exists_basis_of_span_of_maximalIdeal_rTensor_injective

∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  [inst_3 : IsLocalRing R] [Module.FinitePresentation R M],
  Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype (IsLocalRing.maximalIdeal R))) →
    ∀ {ι : Type u} (v : ι → M), Submodule.span R (Set.range v) = ⊤ → ∃ κ a b, ∀ (i : κ), b i = v (a i)

If M is a finitely presented module over a local ring (R, 𝔪) such that m ⊗ M → M is injective, then every generating family contains a basis.

Defined in

Mathlib.RingTheory.LocalRing.Module

Cited by

2 results in Mathlib

Foundations

Depth 121 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleIsLocalRingModule.FinitePresentation