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

${\displaystyle 0\to A\ {\stackrel {a}{\to }}\ B\ {\stackrel {b}{\to }}\ C\to 0}$

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

${\displaystyle 0\to A\ {\stackrel {i}{\to }}\ A\oplus C\ {\stackrel {p}{\to }}\ C\to 0}$

The requirement that the sequence is isomorphic means that there is an isomorphism ${\displaystyle f:B\to A\oplus C}$ 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.

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

Related notions

Pure exact sequences can be characterized as the filtered colimits of split exact sequences.[1]

References

1. Fuchs (2015, Ch. 5, Thm. 3.4)

Sources

• Fuchs, László (2015), Abelian Groups, Springer Monographs in Mathematics, Springer, ISBN 9783319194226