Theorems · Theorem · commutative algebra

Submodule.fg_ker_comp

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

The kernel of the composition of two linear maps is finitely generated if both kernels are and the first morphism is surjective.

Defined in

Mathlib.RingTheory.Finiteness.Finsupp

Cited by

1 results in Mathlib

Foundations

Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddCommGroupModuleAddCommGroupModule