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


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

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


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

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:



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

Examples
A trivial example of a split short exact sequence is
 * $$0 \to M_1 \mathrel{\stackrel{q}{\to}} M_1\oplus M_2 \mathrel{\stackrel{p}{\to}} M_2 \to 0$$

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