In
mathematics, the quotient (also called Serre quotient or Gabriel quotient) of an
abelian category by a
Serre subcategory is the abelian category which, intuitively, is obtained from by ignoring (i.e. treating as
zero) all
objects from . There is a canonical
exact
functor whose kernel is , and 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.
Formally, is the
category whose objects are those of and whose
morphisms from X to Y are given by the
direct limit (of
abelian groups)
where the limit is taken over
subobjects and such that and . (Here, and denote
quotient objects computed in .) These pairs of subobjects are ordered by .
Composition of morphisms in is induced by the
universal property of the direct limit.
The canonical functor sends an object X to itself and a morphism to the corresponding element of the direct limit with X′ = X and Y′ = 0.
An alternative, equivalent construction of the quotient category uses what is called a "
calculus of fractions" to define the morphisms of . Here, one starts with the class of those morphisms in whose kernel and cokernel both belong to . This is a multiplicative system in the sense of Gabriel-Zisman, and one can localize the category at the system to obtain .
[1]
Let be a
field and consider the abelian category of all
vector spaces over . Then the full subcategory of finite-
dimensional vector spaces is a Serre-subcategory of . The Serre quotient has as objects the -vector spaces, and the set of morphisms from to in is (which is a
quotient of vector spaces). 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. This example shows that the Serre quotient can behave like a
quotient category.
For another example, take the
abelian category Ab of all abelian groups and the Serre subcategory of all
torsion abelian groups. The Serre quotient here is
equivalent to the category of all vector spaces over the rationals, with the canonical functor given by tensoring with . 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.
The Serre quotient is an abelian category, and the canonical functor is
exact and surjective on objects. The kernel of is , i.e., is
zero in if and only if belongs to .
The Serre quotient and canonical functor are characterized by the following
universal property: if is any abelian category and is an exact functor such that is a zero in for each object , then there is a unique exact functor such that .
[3]
Given three abelian categories , , , we have
if and only if
- there exists an exact and
essentially surjective functor whose kernel is and such that for every morphism in there exist morphisms and in so that is an isomorphism and .
Theorems involving Serre quotients
Serre's description of coherent sheaves on a projective scheme
According to a theorem by
Jean-Pierre Serre, the category of
coherent sheaves on a projective scheme (where 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
where denotes the category of finitely-generated graded modules over and is the Serre subcategory consisting of all those graded modules which are 0 in all degrees that are high enough, i.e. for which there exists such that for all .
[4]
[5]
A similar description exists for the category of
quasi-coherent sheaves on , even if is not noetherian.
The
Gabriel–Popescu theorem states that any
Grothendieck category is
equivalent to a Serre quotient of the form , where denotes the abelian category of right
modules over some
unital ring , and is some
localizing subcategory of .
[6]
Quillen's localization theorem
Daniel Quillen's
algebraic K-theory assigns to each
exact category a sequence of abelian groups , and this assignment is functorial in . Quillen proved that, if is a Serre subcategory of the abelian category , there is a
long exact sequence of the form
[7]