Theorems · Theorem · category theory

Function.Exact.exact_mapQ_iff

∀ {R : Type u_1} {M : Type u_2} {N : Type u_4} {P : Type u_6} [inst : Ring R] [inst_1 : AddCommGroup M]
  [inst_2 : AddCommGroup N] [inst_3 : AddCommGroup P] [inst_4 : Module R M] [inst_5 : Module R N] [inst_6 : Module R P]
  {f : M →ₗ[R] N} {g : N →ₗ[R] P},
  Function.Exact ⇑f ⇑g →
    ∀ {p : Submodule R M} {q : Submodule R N} {r : Submodule R P} (hpq : p ≤ Submodule.comap f q)
      (hqr : q ≤ Submodule.comap g r),
      Function.Exact ⇑(p.mapQ q f hpq) ⇑(q.mapQ r g hqr) ↔ g.range ⊓ r ≤ Submodule.map g q

A necessary and sufficient condition for an exact sequence to descend to a quotient.

Defined in

Mathlib.Algebra.Exact.Basic

Cited by

1 results in Mathlib

Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupAddCommGroupAddCommGroupModuleModuleModule