Five lemma

This is a lemma involving commutative diagrams that can be proved by means of diagram chasing

Let $A_1\to A_2\to A_3\to A_4\to A_5$ and $B_1\to B_2\to B_2\to B_4\to B_5$ be exact sequences of homomorphisms. Suppose there are maps $f_i:A_i\to B_i$ such that the diagram of all these maps commutes. Then the following are true:

• If $f_2$ and $f_4$ are injective and $f_1$ is surjective, then $f_3$ is injective
• If $f_2$ and $f_4</math/aresurjcetiveand<math>f_5$ is injective then $f_3$ is surjective
• if $f_1,f_2,f_4,f_5$ are isomorphisms, then $f_3$ is also an isomorphism