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.
Let be an abelian category. A non-empty full subcategory is called a Serre subcategory (or also a dense subcategory), if for every short exact sequence in the object is in if and only if the objects and belong to . In words: is closed under subobjects, quotient objects and extensions.
The importance of this notion stems from the fact that kernels of exact functors between abelian categories have this property, and that one can build (for locally small ) the quotient category (in the sense of Gabriel, Grothendieck, Serre) , which has the same objects as , is abelian, and comes with an exact functor (called the quotient functor) whose kernel is .
Let be locally small. The Serre subcategory is called localizing, if the quotient functor has a right adjoint . Since then , as a left adjoint, preserves colimits, each localizing subcategory is closed under colimits. The functor (or sometimes ) 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 (e.g. if it is a Grothendieck category), then a Serre subcategory is localizing if and only if is closed under arbitrary coproducts (a.k.a. direct sums). 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 the quotient category is again a Grothendieck category.
- Nicolae Popescu; 1973; Abelian Categories with Applications to Rings and Modules; Academic Press, Inc.; out of print.