# Split exact sequence

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.

## Equivalent characterizations

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

$0\to A\ {\stackrel {a}{\to }}\ B\ {\stackrel {b}{\to }}\ C\to 0$ is called split exact if it is isomorphic to the sequence where the middle term is the direct sum of the outer ones:

$0\to A\ {\stackrel {i}{\to }}\ A\oplus C\ {\stackrel {p}{\to }}\ C\to 0$ The requirement that the sequence is isomorphic means that there is an isomorphism $f:B\to A\oplus C$ such that the composite $f\circ a$ is the natural inclusion $i:A\to A\oplus C$ and such that the composite $p\circ f$ equals b.

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

## Examples

Any short exact sequence of vector spaces is split exact. This is a rephrasing of the fact that any set of linearly independent vectors in a vector space can be extended to a basis.

The exact sequence $0\to \mathbf {Z} {\stackrel {2}{\to }}\mathbf {Z} \to \mathbf {Z} /2\to 0$ (where the first map is multiplication by 2) is not split exact.

Pure exact sequences can be characterized as the filtered colimits of split exact sequences.