nLab
localizing subcategory

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Notions of subcategory

Homotopy theory

Contents

Definition

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

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

Properties

In general abelian categories

A localizing subcategory KerQKer Q determines Q:ABQ:A\to B up to an equivalence of categories commuting with the localization functors; it is the quotient functor Q T:AA/TQ_T : A\to A/T to the Serre quotient category A/TA/T. The right adjoint S T:A/TAS_T : A/T\to A to Q TQ_T is usually called the section functor. Denote the unit of the adjunction η:Id AS TQ T\eta : Id_A\to S_T Q_T. Then for XObAX\in Ob A, Kerη XXKer \eta_X\subset X is the maximal subobject of XX contained in XX, called the TT-torsion part of XX. An object XX is TT-torsionfree if the TT-torsion part of XX is 00, i.e. η X\eta_X is isomorphism, and XX is TT-closed (local object with respect to morphisms inverting under QQ) if η X\eta_X is an isomorphism. The section functor S TS_T realizes the equivalence of categories between A/TA/T and the full subcategory of AA generated by TT-closed objects.

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

Localizing subcategories are precisely those which are topologizing, closed under extensions and closed under all colimits which exist in AA. In other words, AA and AA'' are in TT iff any given extension AA' of AA by AA'' is in TT; and it is closed under colimits existing in AA.

A strictly full subcategory TAT\subset A is localizing iff the class Σ T\Sigma_T of all fMorAf\in Mor A for which KerfObTKer f\in Ob T and CokerfObTCoker f\in Ob T is precisely the class of all morphisms inverted by some left exact localization admiting right adjoint.

A reflective (strongly) thick subcategory TT is always localizing and the converse holds if AA has injective envelopes.

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

(i) TT is localizing

(ii) TT is closed under coproducts

(iii) TT is cocomplete (closed under arbitrary colimits)

(iv) any colimit of objects in TT in AA belongs to TT

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

In RMod_R Mod.

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

Literature

The notion is introduced by Gabriel:

A comprehensive (and very reliable) source is

  • N. Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375

Revised on March 7, 2014 05:35:25 by Zoran Škoda (161.53.130.104)