Split exact sequence

Summary

A split exact sequence is a short exact sequence $0 \to A \to B \to C \to 0$ in which the middle object is the biproduct $A \oplus C$, compatibly with the two maps. These are the trivial extensions: $B$ is just $A$ and $C$ placed side by side. By the splitting lemma, a short exact sequence splits as soon as one of its two maps has a one-sided inverse.