Editing Localization of a category (section)

==Examples==

===Serre's ''C''-theory===

[[Jean-Pierre Serre|Serre]] introduced the idea of working in [[homotopy theory]] ''[[Ideal (ring theory)|modulo]]'' some class ''C'' of [[abelian group]]s. This meant that groups ''A'' and ''B'' were treated as isomorphic, if for example ''A/B'' lay in ''C''. 

===Module theory===

In the theory of [[module (mathematics)|module]]s over a [[commutative ring]] ''R'', when ''R'' has [[Krull dimension]] ≥ 2, it can be useful to treat modules ''M'' and ''N'' as ''pseudo-isomorphic'' if ''M/N'' has [[support of a module|support]] of codimension at least two. This idea is much used in [[Iwasawa theory]].

===Derived categories===

The [[derived category]] of an [[abelian category]] is much used in [[homological algebra]]. It is the localization of the category of chain complexes (up to homotopy) with respect to the [[quasi-isomorphism]]s.

=== Quotients of abelian categories ===
Given an [[abelian category]] ''A'' and a [[Serre subcategory]] ''B,'' one can define the [[Quotient of an abelian category|quotient category]] ''A/B,'' which is an abelian category equipped with an [[exact functor]] from ''A'' to ''A/B'' that is [[Essentially surjective functor|essentially surjective]] and has kernel ''B.'' This quotient category can be constructed as a localization of ''A'' by the class of morphisms whose kernel and cokernel are both in ''B.''

===Abelian varieties up to isogeny===<!-- This section is linked from [[Elliptic curve]] -->

An [[isogeny]] from an [[abelian variety]] ''A'' to another one ''B'' is a surjective morphism with finite [[Kernel (category theory)|kernel]]. Some theorems on abelian varieties require the idea of ''abelian variety up to isogeny'' for their convenient statement. For example, given an abelian subvariety ''A<sub>1</sub>'' of ''A'', there is another subvariety ''A<sub>2</sub>'' of ''A'' such that

:''A<sub>1</sub>'' &times; ''A<sub>2</sub>''

is ''isogenous'' to ''A'' (Poincaré's reducibility  theorem: see for example ''Abelian Varieties'' by [[David Mumford]]). To call this a [[direct sum]] decomposition, we should work in the category of abelian varieties up to isogeny.