Editing Localization of a category (section)

===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.