# 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).

#### References

 [1] J.-.P. Serre, "Groupes d'homotopie et classes de groupes abéliens" Ann. of Math. , 58 : 2 (1953) pp. 258–294 [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

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

#### References

 [a1] N. Popescu, "Abelian categories with applications to rings and modules" , Acad. Press (1973)
How to Cite This Entry:
Serre subcategory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Serre_subcategory&oldid=48679
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article