nLab
localizing subcategory

Definition

A subcategory $T$ of an abelian category $A$ is a localizing subcategory (French: sous-catégorie localisante) if there exists an exact localization functor $Q:A\to B$ having a right adjoint $B\hookrightarrow A$ (which is automatically then fully faithful) and for which $T=\mathrm{Ker}Q$ i.e. the full subcategory of $A$ generated by objects $a\in \mathrm{Ob}(A)$ such that $Q(a)=0$.

One sometimes says that $T$ is the localizing subcategory associated with quotient (or localized) category $B$ (which is then equivalent to $A/T$).

Properties

In general abelian categories

A localizing subcategory $\mathrm{Ker}Q$ determines $Q:A\to B$ up to an equivalence of categories commuting with the localization functors; it is the quotient functor $Q_T:A\to A/T$ to the Serre quotient category $A/T$. The right adjoint $S_T:A/T\to A$ to $Q_T$ is usually called the section functor. Denote the unit of the adjunction $\eta :{\mathrm{Id}}_A\to S_T{Q}_T$. Then for $X\in \mathrm{Ob}A$, $\mathrm{Ker}{\eta }_X\subset X$ is the maximal subobject of $X$ contained in $X$, called the $T$-torsion part of $X$. An object $X$ is $T$-torsionfree if the $T$-torsion part of $X$ is $0$, i.e. ${\eta }_X$ is isomorphism, and $X$ is $T$-closed if ${\eta }_X$ is an isomorphism. The section functor $S_T$ realizes the equivalence of categories between $A/T$ and the full subcategory of $A$ generated by $T$-closed objects.

${\eta }_X$ is isomorphism, i.e. the $T$-torsion part of $X$ is $0$.

A thick subcategory $T\subset A$ (in strong sense) is localizing iff every object $M$ in $A$ has the largest subobject among the subobjects from $T$ and if the only subobject from $T$ is a zero object then there is a monomorphism from $M$ to a $T$-closed object.

Localizing subcategories are precisely those which are topologizing, closed under extensions and closed under all colimits which exist in $A$. In other words, $A$ and $A″$ are in $T$ iff any given extension $A\prime$ of $A$ by $A″$ is in $T$; and it is closed under colimits existing in $A$.

A strictly full subcategory $T\subset A$ is localizing iff the class ${\Sigma }_T$ of all $f\in \mathrm{Mor}A$ for which $\mathrm{Ker}f\in \mathrm{Ob}T$ and $\mathrm{Coker}f\in \mathrm{Ob}T$ is precisely the class of all morphisms inverted by some left exact localization admiting right adjoint.

A reflective? (strongly) thick subcategory $T$ is always localizing and the converse holds if $A$ has injective envelopes.

If $A$ admit colimits and has a set of generators, then any localizing subcategory $T\subset A$,and the Serre quotient $A/T$, admit colimits and has a set of generators (Gabriel, Prop. 9) and the quotient functor $Q_T:A\to A/T$ preserves colimits (in the same Grothendieck universe if we work with universes). The generators of $A/T$ are the images of the generators in $A$ under the quotient functor $Q_T$. If $A$ is locally noetherian abelian category then any localizing subcategory $T\subset A$ and the quotient category $A/T$ are locally noetherian (Gabriel, Cor. 1). (If $A$ is locally finitely presented, $A$ and $A/T$ are locally finitely presented.?) If $A$ is locally noetherian and $A_{\mathrm{Noether}}\subset A$ is the full subcategory of noetherian objects in $A$, then the assignment which to any localizing subcategory $T\subset A$ assigns the full subcategory $T_{\mathrm{Noether}}\subset T$ of noetherian objects in $T$ is the bijection between the localizing subcategories in $A$ and (strongly) thick subcategories in $A_{\mathrm{Noether}}$ (Gabriel Prop. 10).

In locally finitely presentable abelian categories

In this setup, there is a bijective correspondence between hereditary torsion theories, localizing subcategories and exact localizations having right adjoint.

In Grothendieck categories

For a strongly thick subcategory (i.e. weakly Serre subcategory) $T$ in a Grothendieck category $A$ the following are equivalent:

(i) $T$ is localizing

(ii) $T$ is closed under coproducts

(iii) $T$ is cocomplete (closed under arbitrary colimits)

(iv) any colimit of objects in $T$ in $A$ belongs to $T$

(v) the corresponding localizing functor $F:A\to A/T$ preserves colimits

In ${}_R\mathrm{Mod}$.

There is a canonical correspondence between topologizing filters of a unital ring and localizing subcategories in the category $R$Mod of (say left) unital modules of the ring.

Literature

• Pierre Gabriel, Des catégories abéliennes, Bulletin de la Société Mathématique de France 90 (1962), 323-448 (numdam)

• Francis Borceux, Handbook of categorical algebra, II.1

• Henning Krause, The spectrum of a locally coherent category, J. Pure Appl. Algebra 114 (1997), 259-271, pdf

• Ryo Takahashi, On localizing subcategories of derived categories (2000) (pdf)