Split exact sequence

The term split exact sequence is used in two different ways by different people. Some people mean a short exact sequence that right-splits (thus corresponding to a semidirect product) and some people mean a short exact sequence that left-splits (which implies it right-splits, and corresponds to a direct product). This article takes the latter approach, but both are in common use. When reading a book or paper, it is important to note precisely which of the two meanings is in use.

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 exact 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. This can be summarized by a commutative diagram as:

File:Commutative diagram for split exact sequence - fixed.svg

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

Examples

A trivial example of a split short exact sequence is

$0\to M_1\stackrel{q}{\to }{M}_1\oplus M_2\stackrel{p}{\to }{M}_2\to 0$

where $M_1,M_2$ are R-modules, $q$ is the canonical injection and $p$ is the canonical projection.

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 $0\to \mathbf{𝐙}\stackrel{2}{\to }\mathbf{𝐙}\to \mathbf{𝐙}/2\mathbf{𝐙}\to 0$ (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.