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 formula_1 be an abelian category. A non-empty full subcategory formula_2 is called a "Serre subcategory" (or also a "dense subcategory"), if for every short exact sequence formula_3 in formula_1 the object formula_5 is in formula_2 if and only if the objects formula_7
and formula_8 belong to formula_2. In words: formula_2 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 formula_1) the quotient category (in the sense of Gabriel, Grothendieck,
Serre) formula_12, which has the same objects as formula_1, is abelian, and comes with an exact functor (called the quotient functor) formula_14 whose kernel is formula_2.
Localizing subcategories.
Let formula_1 be locally small. The Serre subcategory formula_2 is called "localizing", if the quotient functor
formula_14 has a
right adjoint
formula_19. Since then formula_20, as a left adjoint, preserves colimits, each localizing subcategory is closed under colimits. The functor formula_20 (or sometimes formula_22) is also called the "localization functor", and formula_23 the "section functor". The section functor is left-exact and fully faithful.
If the abelian category formula_1 is moreover
cocomplete and has injective hulls (e.g. if it is a Grothendieck category), then a Serre
subcategory formula_2 is localizing if and only if
formula_2 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 formula_1 is a Grothendieck category and
formula_2 a localizing subcategory, then the quotient category
formula_12 is again a Grothendieck category.
The Gabriel-Popescu theorem implies that every Grothendieck category is the quotient category of a module category formula_30 (with formula_31 a suitable ring) modulo a localizing subcategory.