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exact sequence

An exact sequence is a chain complex with vanishing homology.

This is sometimes called a long exact sequence, where a short exact sequence is an exact sequence of the form

$$0\to A\to B\to C\to 0.$$0 \to A \to B \to C \to 0 .

Of course, by adding $0$ to either end (or both!), one turns a short exact sequence into a chain complex and in particular into a long exact sequence.

A short exact sequence may also be defined in more elementary terms as a sequence

$$A\stackrel{i}{\to }B\stackrel{p}{\to }C$$A \stackrel{i}\to B \stackrel{p}\to C

such that $i$ is a monomorphism, $p$ is an epimorphism, and the image of $i$ equals the kernel of $p$ (equivalently, the coimage of $p$ equals the cokernel of $i$).

A split exact sequence is a short exact sequence in which $i$ is a split monomorphism, or (equivalently) in which $p$ is a split epimorphism. In this case, $B$ may be decomposed as the biproduct $A\oplus C$ (with $i$ and $p$ the usual biproduct inclusion and projection); this sense in which $B$ is ‘split’ into $A$ and $C$ is the origin of the general terms ‘split (mono/epi)morphism’.

The above works in any abelian category, and possible more generally. See also exact sequence of Hopf algebras.