Theorems · Theorem · commutative algebra

Module.Finite.of_exact

∀ {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},
  Function.Exact ⇑f ⇑g → Function.Surjective ⇑g → ∀ [Module.Finite R M] [Module.Finite R P], Module.Finite R N

If $M → N → P → 0$ is exact and $M$ and $P$ are finitely generated then so is $N$. This is the Module.Finite version of Submodule.fg_of_fg_map_of_fg_inf_ker.

Defined in

Mathlib.RingTheory.Finiteness.Finsupp

Cited by

1 results in Mathlib

Foundations

Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddCommGroupModuleAddCommGroupModuleModule.FiniteModule.Finite