Editing Exact sequence (section)

==Applications of exact sequences==
In the theory of abelian categories, short exact sequences are often used as a convenient language to talk about [[subobject | subobjects]] and factor objects.

The [[extension problem]] is essentially the question "Given the end terms ''A'' and ''C'' of a short exact sequence, what possibilities exist for the middle term ''B''?" In the category of groups, this is equivalent to the question, what groups ''B'' have ''A'' as a normal subgroup and ''C'' as the corresponding factor group?  This problem is important in the [[classification of finite simple groups|classification of groups]]. See also [[Outer automorphism group]].

Notice that in an exact sequence, the composition ''f''<sub>''i''+1</sub> ∘ ''f''<sub>''i''</sub> maps ''A''<sub>''i''</sub> to 0 in ''A''<sub>''i''+2</sub>, so every exact sequence is a [[chain complex]]. Furthermore, only ''f''<sub>''i''</sub>-images of elements of ''A''<sub>''i''</sub> are mapped to 0 by ''f''<sub>''i''+1</sub>, so the [[homology (mathematics)|homology]] of this chain complex is trivial. More succinctly:
:Exact sequences are precisely those chain complexes which are [[acyclic complex|acyclic]].
Given any chain complex, its homology can therefore be thought of as a measure of the degree to which it fails to be exact.

If we take a series of short exact sequences linked by chain complexes (that is, a short exact sequence of chain complexes, or from another point of view, a chain complex of short exact sequences), then we can derive from this a '''long exact sequence''' (that is, an exact sequence indexed by the natural numbers) on homology by application of the [[zig-zag lemma]]. It comes up in [[algebraic topology]] in the study of [[relative homology]]; the [[Mayer–Vietoris sequence]] is another example. Long exact sequences induced by short exact sequences are also characteristic of [[derived functor]]s.

[[Exact functor]]s are [[functor]]s that transform exact sequences into exact sequences.