Localizing subcategory
In mathematics, Serre and localizing subcategories form important classes of subcategories of an abelian category. Localizing subcategories are certain Serre subcategories. They are strongly linked to the notion of a quotient category.
Serre subcategories
Let be an abelian category. A non-empty full subcategory is called a Serre subcategory, if for every short exact sequence in the object is in if and only if the objectsand belong to. In words: is closed under subobjects, quotient objects
and extensions.
Each Serre subcategory of is itself an abelian category, and the inclusion functor is exact. The importance of this notion stems from the fact that kernels of exact functors between abelian categories are Serre subcategories, and that one can build the quotient category , which has the same objects as, is abelian, and comes with an exact functor whose kernel is.
Localizing subcategories
Let be locally small. The Serre subcategory is called localizing if the quotient functorhas a
right adjoint
. Since then, as a left adjoint, preserves colimits, each localizing subcategory is closed under colimits. The functor is also called the localization functor, and the section functor. The section functor is left-exact and fully faithful.
If the abelian category is moreover
cocomplete and has injective hulls, then a Serre
subcategory is localizing if and only if
is closed under arbitrary coproducts. Hence the notion of a localizing subcategory is
equivalent to the notion of a hereditary torsion class.
If is a Grothendieck category and a localizing subcategory, then and the quotient category
are again Grothendieck categories.
The Gabriel-Popescu theorem implies that every Grothendieck category is the quotient category of a module category modulo a localizing subcategory.