Theorems · Theorem · commutative algebra

Module.free_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))) → Module.Free R M

If M is a finitely presented module over a local ring (R, 𝔪) such that m ⊗ M → M is injective, then M is free.

Defined in

Mathlib.RingTheory.LocalRing.Module

Cited by

1 results in Mathlib

Foundations

Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleIsLocalRingModule.FinitePresentation