Split exact sequence

A short exact sequence

(see short exact sequence ) splits if there exists a homomorphism $s:C\to B$ with $p\circ s=\mathrm{id}_C$ (a section), or equivalently a homomorphism $r:B\to A$ with $r\circ i=\mathrm{id}_A$ (a retraction), where $\mathrm{id}$ is the identity map. In this case one has an isomorphism $B\cong A\oplus C$ as in direct sums ; a standard proof is the splitting lemma .

Split exactness is the precise condition that the extension carries no “twisting”: $B$ is just a direct sum of the ends.

Examples:

• The sequence $0\to A \to A\oplus C \to C\to 0$ splits (take $s(c)=(0,c)$).
• For an $R$-linear projection $p:B\to C$ with a right-inverse $s$, the sequence $0\to \ker(p)\to B\to C\to 0$ splits.
• (Nonexample) For $n>1$, the sequence $0\to \mathbb Z \xrightarrow{\times n} \mathbb Z \to \mathbb Z/n\mathbb Z \to 0$ does not split.