In mathematics, Serre and localizing subcategories form important classes of
subcategories
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...
of an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
. Localizing subcategories are certain Serre subcategories. They are strongly linked to the notion of a
quotient category
In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but ...
.
Serre subcategories
Let
be an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
. A non-empty full
subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively ...
is called a ''Serre subcategory'' (or also a ''dense subcategory''), if for every short
exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the conte ...
in
the object
is in
if and only if the objects
and
belong to
. In words:
is closed under subobjects, quotient objects
and extensions.
Each Serre subcategory
of
is itself an abelian category, and the inclusion functor
is
exact. 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 mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but ...
(in the sense of
Gabriel
In Abrahamic religions (Judaism, Christianity and Islam), Gabriel (); Greek: grc, Γαβριήλ, translit=Gabriḗl, label=none; Latin: ''Gabriel''; Coptic: cop, Ⲅⲁⲃⲣⲓⲏⲗ, translit=Gabriêl, label=none; Amharic: am, ገብ� ...
,
Grothendieck,
Serre)
, which has the same objects as
, is abelian, and comes with an exact functor (called the quotient functor)
whose kernel is
.
Localizing subcategories
Let
be locally small. The Serre subcategory
is called ''localizing'' if the quotient functor
has a
right adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...
. Since then
, as a left adjoint, preserves
colimits
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions s ...
, each localizing subcategory is closed under colimits. The functor
(or sometimes
) is also called the ''localization functor'', and
the ''section functor''. The section functor is
left-exact and
fully faithful.
If the abelian category
is moreover
cocomplete In mathematics, a complete category is a category in which all small limits exist. That is, a category ''C'' is complete if every diagram ''F'' : ''J'' → ''C'' (where ''J'' is small) has a limit in ''C''. Dually, a cocomplete category is one in w ...
and has
injective hulls (e.g. if it is a
Grothendieck category In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957English translation in order to develop the machinery of homological algebra for modules and for sheaves i ...
), then a Serre
subcategory
is localizing if and only if
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
is a Grothendieck category and
a localizing subcategory, then
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
In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring ...
(with
a suitable
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
) modulo a localizing subcategory.
See also
*
Giraud subcategory In mathematics, Giraud subcategories form an important class of subcategories of Grothendieck categories. They are named after Jean Giraud.
Definition
Let \mathcal be a Grothendieck category. A full subcategory \mathcal is called ''reflective' ...
References
*
Nicolae Popescu
Nicolae Popescu (; 22 September 1937 – 29 July 2010) was a Romanian mathematician and professor at the University of Bucharest. He also held a research position at the Institute of Mathematics of the Romanian Academy, and was elected corresp ...
; 1973;
Abelian Categories with Applications to Rings and Modules; Academic Press, Inc.; out of print.
Category theory
Homological algebra