Theorems · Theorem · commutative algebra

IsLocalRing.split_injective_iff_lTensor_residueField_injective

∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : AddCommGroup N]
  [inst_3 : Module R M] [inst_4 : Module R N] [inst_5 : IsLocalRing R] [Module.Finite R M] [Module.Finite R N]
  [Module.Free R N] (l : M →ₗ[R] N),
  (∃ l', l' ∘ₗ l = LinearMap.id) ↔ Function.Injective ⇑(LinearMap.lTensor (IsLocalRing.ResidueField R) l)

Given a linear map l : M → N over a local ring (R, 𝔪, k) with M finite and N finite free, l is a split injection if and only if k ⊗ l is a (split) injection.

Defined in

Mathlib.RingTheory.LocalRing.Module

Cited by

1 results in Mathlib

Foundations

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

Assumes

CommRingAddCommGroupAddCommGroupModuleModuleIsLocalRingModule.FiniteModule.FiniteModule.Free