Editing Homological algebra (section)

===Exact sequences===
{{Main|Exact sequence}}
In the context of [[group theory]], a sequence
:<math>G_0 \;\xrightarrow{f_1}\; G_1 \;\xrightarrow{f_2}\; G_2 \;\xrightarrow{f_3}\; \cdots \;\xrightarrow{f_n}\; G_n</math>
of [[group (mathematics)|groups]] and [[group homomorphism]]s is called '''exact''' if the [[Image (mathematics)|image]] of each homomorphism is equal to the [[Kernel (algebra)|kernel]] of the next:
:<math>\mathrm{im}(f_k) = \mathrm{ker}(f_{k+1}).\!</math>
Note that the sequence of groups and homomorphisms may be either finite or infinite.

A similar definition can be made for certain other [[algebraic structure]]s.  For example, one could have an exact sequence of [[vector space]]s and [[linear map]]s, or of [[module (mathematics)|modules]] and [[module homomorphism]]s.  More generally, the notion of an exact sequence makes sense in any [[category (mathematics)|category]] with [[kernel (category theory)|kernel]]s and [[cokernel]]s.

====Short====
<!-- :<math>A \;\xrightarrow{f}\; B \;\twoheadrightarrow\; C</math> -->
The most common type of exact sequence is the '''short exact sequence'''. This is an exact sequence of the form
:<math>A \;\overset{f}{\hookrightarrow}\; B \;\overset{g}{\twoheadrightarrow}\; C</math>
where &fnof; is a [[monomorphism]] and ''g'' is an [[epimorphism]].  In this case, ''A'' is a [[subobject]] of ''B'', and the corresponding [[quotient]] is [[isomorphism|isomorphic]] to ''C'':
:<math>C \cong B/f(A).</math> 
(where  ''f(A)'' = im(''f'')).

A short exact sequence of abelian groups may also be written as an exact sequence with five terms:
:<math>0 \;\xrightarrow{}\; A \;\xrightarrow{f}\; B \;\xrightarrow{g}\; C \;\xrightarrow{}\; 0</math>
where 0 represents the [[Initial and terminal objects|zero object]], such as the [[trivial group]] or a zero-dimensional vector space.  The placement of the 0's forces &fnof; to be a monomorphism and ''g'' to be an epimorphism (see below).

====Long====
A long exact sequence is an exact sequence indexed by the [[natural number]]s.