Localizing subcategory

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

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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 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 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

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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 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   (with   a suitable ring) modulo a localizing subcategory.

See also

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References

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  • Nicolae Popescu; 1973; Abelian Categories with Applications to Rings and Modules; Academic Press, Inc.; out of print.