In mathematics, Serre and localizing subcategories form important classes of subcategories of an abelian category.
They are strongly linked to the notion of a quotient category.
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
is closed under subobjects, quotient objects and extensions.
is itself an abelian category, and the inclusion functor
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 (for locally small
) the quotient category (in the sense of Gabriel, Grothendieck, Serre)
, is abelian, and comes with an exact functor (called the quotient functor)
is called localizing if the quotient functor
, as a left adjoint, preserves colimits, each localizing subcategory is closed under colimits.
) is also called the localization functor, and
The section functor is left-exact and fully faithful.
is moreover cocomplete and has injective hulls (e.g. if it is a Grothendieck category), then a Serre subcategory
is closed under arbitrary coproducts (a.k.a.
Hence the notion of a localizing subcategory is equivalent to the notion of a hereditary torsion class.
is a Grothendieck category and
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
a suitable ring) modulo a localizing subcategory.