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
Let $$\mathcal{A}$$ be an abelian category. A non-empty full subcategory $$\mathcal{C}$$ is called a Serre subcategory (or also a dense subcategory), if for every short exact sequence $$0\rightarrow A' \rightarrow A\rightarrow A''\rightarrow 0$$ in $$\mathcal{A}$$ the object $$A$$ is in $$\mathcal{C}$$ if and only if the objects $$A'$$ and $$A''$$ belong to $$\mathcal{C}$$. In words: $$\mathcal{C}$$ is closed under subobjects, quotient objects and extensions.

Each Serre subcategory $$\mathcal{C}$$ of $$\mathcal{A}$$ is itself an abelian category, and the inclusion functor $$\mathcal{C}\to\mathcal{A}$$ 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 $$\mathcal{A}$$) the quotient category (in the sense of Gabriel, Grothendieck, Serre) $$\mathcal{A}/\mathcal{C}$$, which has the same objects as $$\mathcal{A}$$, is abelian, and comes with an exact functor (called the quotient functor) $$T\colon\mathcal{A}\rightarrow\mathcal{A}/\mathcal{C}$$ whose kernel is $$\mathcal{C}$$.

Localizing subcategories
Let $$\mathcal{A}$$ be locally small. The Serre subcategory $$\mathcal{C}$$ is called localizing if the quotient functor $$T\colon\mathcal{A}\rightarrow\mathcal{A}/\mathcal{C}$$ has a right adjoint $$S\colon\mathcal{A}/\mathcal{C}\rightarrow\mathcal{A}$$. Since then $$T$$, as a left adjoint, preserves colimits, each localizing subcategory is closed under colimits. The functor $$T$$ (or sometimes $$ST$$) is also called the localization functor, and $$S$$ the section functor. The section functor is left-exact and fully faithful.

If the abelian category $$\mathcal{A}$$ is moreover cocomplete and has injective hulls (e.g. if it is a Grothendieck category), then a Serre subcategory $$\mathcal{C}$$ is localizing if and only if $$\mathcal{C}$$ 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 $$\mathcal{A}$$ is a Grothendieck category and $$\mathcal{C}$$ a localizing subcategory, then $$\mathcal{C}$$ and the quotient category $$\mathcal{A}/\mathcal{C}$$ are again Grothendieck categories.

The Gabriel-Popescu theorem implies that every Grothendieck category is the quotient category of a module category $$\operatorname{Mod}(R)$$ (with $$R$$ a suitable ring) modulo a localizing subcategory.