Exact sequence

Sequence of homomorphisms such that each kernel equals the preceding image

The sequence of groups and homomorphisms may be either finite or infinite.

To understand the definition, it is helpful to consider relatively simple cases where the sequence is finite and begins or ends with the trivial group. Traditionally, this, along with the single identity element, is denoted 0 (additive notation, usually when the groups are abelian), or denoted 1 (multiplicative notation).

A long exact sequence is equivalent to a family of short exact sequences in the following sense: Given a long sequence

The first sequence may also be written without using special symbols for monomorphism and epimorphism:

These homomorphisms are restrictions of similarly defined homomorphisms that form the short exact sequence

so the image of the curl is a subset of the kernel of the divergence. The converse is somewhat involved:

This short exact sequence also permits a much shorter proof of the validity of the Helmholtz decomposition that does not rely on brute-force vector calculus. Consider the subsequence

The importance of short exact sequences is underlined by the fact that every exact sequence results from "weaving together" several overlapping short exact sequences. Consider for instance the exact sequence

Suppose in addition that the cokernel of each morphism exists, and is isomorphic to the image of the next morphism in the sequence:

Conversely, given any list of overlapping short exact sequences, their middle terms form an exact sequence in the same manner.

In the theory of abelian categories, short exact sequences are often used as a convenient language to talk about sub- and factor objects.

Given any chain complex, its homology can therefore be thought of as a measure of the degree to which it fails to be exact.