Four lemma - Knowlpedia

Let

\begin{array}{ccccccccc} A_1 &\to& A_2 &\to& A_3 &\to& A_4 &\to& A_5\\ \downarrow f_1 && \downarrow f_2 && \downarrow f_3 && \downarrow f_4 && \downarrow f_5\\ B_1 &\to& B_2 &\to& B_3 &\to& B_4 &\to& B_5 \end{array}

be a commutative diagram of $R$ -modules with exact rows.

The four lemma has two forms:

If $f_1$ is surjective and $f_2,f_4$ are injective, then $f_3$ is injective.
If $f_2,f_4$ are surjective and $f_5$ is injective, then $f_3$ is surjective.

Remarks

For modules, the assertions follow by diagram chasing. Analogous categorical formulations replace injective and surjective maps by suitable monomorphisms and epimorphisms. The result is closely related to the five lemma.