Serre subcategory

A locally small full subcategory of an Abelian category such that for every exact sequence

in it is the case that if and only if and . 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 .

Given a Serre subcategory, one can define the quotient category , whose objects are the objects of and whose morphisms are defined by

The quotient category is Abelian.

A Serre subcategory is called localizing if the canonical functor has a right adjoint , called the section functor. If 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

Comments

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)