In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way.

A short exact sequence of abelian groups or of modules over a fixed ring, or more generally of objects in an abelian category

is called split exact if it is isomorphic to the exact sequence where the middle term is the direct sum of the outer ones:

with ${\displaystyle i:A\to A\oplus C}$ being the natural inclusion of A into the direct sum, and ${\displaystyle p:A\oplus C\to C}$ denoting the natural projection of the direct sum onto the second summand. The requirement that the sequence is isomorphic means that there is an isomorphism ${\displaystyle f:B\to A\oplus C}$ such that the composite ${\displaystyle f\circ a}$ is the natural inclusion ${\displaystyle i:A\to A\oplus C}$ and such that the composite ${\displaystyle p\circ f}$ equals b. This can be summarized by a commutative diagram as:

The splitting lemma provides further equivalent characterizations of split exact sequences. The sequence

is split exact if and only if there exists ${\displaystyle r:C\to B}$ such that ${\displaystyle b\circ r=1_{C}}$, which is the case if and only if there exists ${\displaystyle s:B\to A}$ such that ${\displaystyle s\circ a=1_{A}}$.

A trivial example of a split short exact sequence is

where ${\displaystyle M_{1},M_{2}}$ are R-modules, ${\displaystyle q}$ is the canonical injection and ${\displaystyle p}$ is the canonical projection.