Theorems · Theorem · commutative algebra

Module.finitePresentation_of_split_exact

∀ {R : Type u_2} {M : Type u_4} {N : Type u_3} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  [inst_3 : AddCommGroup N] [inst_4 : Module R N] {P : Type u_1} [inst_5 : AddCommGroup P] [inst_6 : Module R P]
  [Module.FinitePresentation R N] (f : M →ₗ[R] N) (g : N →ₗ[R] P) (l : P →ₗ[R] N),
  g ∘ₗ l = LinearMap.id → Function.Injective ⇑f → Function.Exact ⇑f ⇑g → Module.FinitePresentation R M

Given a split exact sequence 0 → M → N → P → 0 with N finitely presented, then M is also finitely presented.

Defined in

Mathlib.Algebra.Module.FinitePresentation

Cited by

1 results in Mathlib

Foundations

Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddCommGroupModuleAddCommGroupModuleModule.FinitePresentation