Quotient of an abelian category explained
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.
Definition
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
(X',Y')\preccurlyeq(X'',Y'')\LongleftrightarrowX''\subseteqX'andY'\subseteqY''
.
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] Examples
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
\cal{C}={\rmMod}(k)/{\rmmod}(k)
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
\operatorname{Mod}(\Bbb{Q})
of all vector spaces over the rationals, with the canonical functor
Ab\to\operatorname{Mod}(\Bbb{Q})
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.
Properties
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
\overline{F}\colonlA/lB\tolC
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
\operatorname{mod}\Bbb{Z}(R)
denotes the category of finitely-generated graded modules over
and
| \Bbb{Z}(R) |
\operatorname{mod} | |
| tor |
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.
Gabriel–Popescu theorem
is
equivalent to a Serre quotient of the form
\operatorname{Mod}(R)/\cal{B}
, where
denotes the abelian category of right
modules over some
unital ring
, and
is some
localizing subcategory of
.
[6] Quillen's localization theorem
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]
Notes and References
- https://stacks.math.columbia.edu/tag/02MN Section 12.10
- Web site: 109.76 The category of modules modulo torsion modules . The Stacks Project.
- Gabriel, Pierre, Des categories abeliennes, Bull. Soc. Math. France 90 (1962), 323-448.
- Book: Görtz . Ulrich . Algebraic Geometry I: Schemes: With Examples and Exercises . Wedhorn . Torsten . Springer Nature . 2020 . 9783658307332. 2nd . 381 . en . Remark 13.21.
- Web site: Proposition 30.14.4 . The Stacks Project.
- N. Popesco . P. Gabriel . 1964. Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes . Comptes Rendus de l'Académie des Sciences . 258 . 4188–4190.
- Quillen . Daniel . 1973 . Higher algebraic K-theory: I . Higher K-Theories . Lecture Notes in Mathematics . 341 . en . Springer . 85–147 . 10.1007/BFb0067053 . 978-3-540-06434-3.