The 5-lemma
Suppose we have a commutative diagram of abelian groups and homomorphisms
such that both rows are exact and the vertical arrows labeled “” are isomorphisms. Then the middle mapping is also an isomorphism.
Proof. Diagram chasing. See Hatcher Section 2.1, p. 129.
2-out-of-3 property
Given a diagram of chain complexes of the form
, if any two vertical maps induce isomorphisms on homology of all degrees, then so does the third.