is in a certain sense the most general abelian category with this property.
Forming Serre quotients of abelian categories is thus formally akin to forming quotients of groups.
Serre quotients are somewhat similar to quotient categories, the difference being that with Serre quotients all involved categories are abelian and all functors are exact.
Serre quotients also often have the character of localizations of categories, especially if the Serre subcategory is localizing.
and whose morphisms from X to Y are given by the direct limit (of abelian groups)
denote quotient objects computed in
is induced by the universal property of the direct limit.
An alternative, equivalent construction of the quotient category uses what is called a "calculus of fractions" to define the morphisms of
This is a multiplicative system in the sense of Gabriel-Zisman, and one can localize the category
of finite-dimensional vector spaces is a Serre-subcategory of
This has the effect of identifying all finite-dimensional vector spaces with 0, and of identifying two linear maps whenever their difference has finite-dimensional image.
The Serre quotient here is equivalent to the category
of all vector spaces over the rationals, with the canonical functor
Similarly, the Serre quotient of the category of finitely generated abelian groups by the subcategory of finitely generated torsion groups is equivalent to the category of finite-dimensional vectorspaces over
[2] Here, the Serre quotient behaves like a localization.
is an abelian category, and the canonical functor
The Serre quotient and canonical functor are characterized by the following universal property: if
, we have if and only if According to a theorem by Jean-Pierre Serre, the category
of coherent sheaves on a projective scheme
is a commutative noetherian graded ring, graded by the non-negative integers and generated by degree-0 and finitely many degree-1 elements, and
refers to the Proj construction) can be described as the Serre quotient
denotes the category of finitely-generated graded modules over
is the Serre subcategory consisting of all those graded modules
[4][5] A similar description exists for the category of quasi-coherent sheaves on
The Gabriel–Popescu theorem states that any Grothendieck category
is equivalent to a Serre quotient of the form
denotes the abelian category of right modules over some unital ring
[6] Daniel Quillen's algebraic K-theory assigns to each exact category
is a Serre subcategory of the abelian category
, there is a long exact sequence of the form[7]