Theorems · Theorem · commutative algebra

AdicCompletion.map_injective

∀ {R : Type u} [inst : CommRing R] (I : Ideal R) {M : Type u} [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  {N : Type u} [inst_3 : AddCommGroup N] [inst_4 : Module R N] [IsNoetherianRing R] [Module.Finite R N] {f : M →ₗ[R] N},
  Function.Injective ⇑f → Function.Injective ⇑(AdicCompletion.map I f)

Adic completion preserves injectivity of finite modules over a Noetherian ring.

Defined in

Mathlib.RingTheory.AdicCompletion.Exactness

Cited by

1 results in Mathlib

Foundations

Depth 125 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleIsNoetherianRingModule.Finite