Localizing subcategory

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

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

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

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

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

• Giraud subcategory

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