Editing Homological algebra (section)

====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).