Theorems · Theorem · commutative algebra

Module.length_eq_add_of_exact

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

Length is additive in exact sequences.

Defined in

Mathlib.RingTheory.Length

Cited by

6 results in Mathlib

Foundations

Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddCommGroupAddCommGroupModuleModule