Localizing subcategory explained
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
be an
abelian category. A non-empty full
subcategory
is called a
Serre subcategory (or also a
dense subcategory), if for every short
exact sequence
in
the object
is in
if and only if the objects
and
belong to
. In words:
is closed under subobjects, quotient objectsand 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 the sense of
Gabriel,
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
. Since then
, as a left adjoint, preserves
colimits, 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 and has
injective hulls (e.g. if it is a
Grothendieck category), then a Serresubcategory
is localizing if and only if
is closed under arbitrary coproducts (a.k.a.direct sums). Hence the notion of a localizing subcategory isequivalent 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.
(with
a suitable
ring) modulo a localizing subcategory.
See also
References
- Nicolae Popescu; 1973; Abelian Categories with Applications to Rings and Modules; Academic Press, Inc.; out of print.
