Serre subcategory

A locally small full subcategory $ \mathfrak S $ of an Abelian category $ \mathfrak A $ such that for every exact sequence

$$ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 $$

in $ \mathfrak A $ it is the case that $ B \in \mathfrak S $ if and only if $ A \in \mathfrak S $ and $ C \in \mathfrak S $. In this context, local smallness of a category is the condition: A collection of representatives of the isomorphism classes of subobjects of any object forms a set. Serre subcategories can be characterized as kernels of functors defined on $ \mathfrak A $.

Given a Serre subcategory, one can define the quotient category $ \mathfrak A / \mathfrak S $, whose objects are the objects of $ \mathfrak A $ and whose morphisms are defined by

$$ \mathop{\rm Mor} _ {\mathfrak A / \mathfrak S } ( X, Y) = \ \lim\limits _ { {Y ^ \prime , X/X ^ \prime \in \mathfrak S } vec } \ \mathop{\rm Mor} _ {\mathfrak A } ( X ^ \prime , Y/Y ^ \prime ) . $$

The quotient category $ \mathfrak A / \mathfrak S $ is Abelian.

A Serre subcategory is called localizing if the canonical functor $ T: \mathfrak A \rightarrow \mathfrak A / \mathfrak S $ has a right adjoint $ S: \mathfrak A / \mathfrak S \rightarrow \mathfrak A $, called the section functor. If $ \mathfrak A $ is a Grothendieck category with coproducts, then a Serre subcategory is localizing if and only if it is closed under coproducts. Thus one obtains a generalization of the classical theory of localization of modules over a commutative ring. This method embraces many constructions of rings of fractions and torsion theories (radicals) of modules over associative rings.

The concept of a Serre subcategory was introduced by J.-P. Serre [1], who called them classes. By using this concept he obtained a far-reaching generalization of a theorem of Hurewicz (see Homotopy group).

Comments

Serre subcategories are also called thick subcategories or dense subcategories. See also Localization in categories.

References

[1]	J.-.P. Serre, "Groupes d'homotopie et classes de groupes abéliens" Ann. of Math. , 58 : 2 (1953) pp. 258–294 Zbl 0052.19303
[2]	C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973)
[3]	N. Popesco, P. Gabriel, "Caractérisations des catégories abéliennes avec générateurs et limites inductives exactes" C.R. Acad. Sci. Paris , 258 : 17 (1964) pp. 4188–4190
[a1]	N. Popescu, "Abelian categories with applications to rings and modules" , Acad. Press (1973)