nLab Serre subcategory

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Localization theory

Notions of subcategory

Contents

Definition

A full subcategory TT of an abelian category AA is a Serre subcategory if it is nonempty and for any exact sequence

MMM M\to M'\to M''

MM' is in TT if MM and MM'' are in TT.

Remark

(Terminology) This notion in general is called a thick subcategory in Gabriel’s thesis, Des Catégories Abéliennes. The terminology is a minefield as there are variants that occur in the literature, see thick subcategory, and also the Stack’s Project section on this. Some of these variants are mentioned below.

Serre quotient

Following Serre 1953, one then defines the category A/TA/T whose objects are the objects of AA and where the morphisms in A/TA/T are defined by

Hom A/T(X,Y)colimHom A(X,Y/Y),\mathrm{Hom}_{A/T}(X,Y) \coloneqq \mathrm{colim}\, \mathrm{Hom}_A(X',Y/Y'),

where the colimit is over all X,YX',Y' in AA such that YY' and X/XX/X' are in TT. The quotient category A/TA/T is abelian.

Very often the same definition of Serre subcategory is used in an arbitrary abelian category AA (we will say in that case weakly Serre subcategory); but in fact, at least when the abelian category is not a Grothendieck category, it is more appropriate to ask for an additional condition in the definition of Serre subcategory, so that the standard theorems on correspondences with other canonical data in localization theory remain valid.

To this aim, for any subcategory of an arbitrary abelian category AA one denotes by T¯\bar{T} the full subcategory of AA generated by all objects NN for which any (nonzero) subquotient of NN in TT has a (nonzero) subobject from TT. This becomes an idempotent operation on the class of subcategories of AA with TT¯T\subset \bar{T} iff TT is topologizing. Moreover T¯\bar{T} is always thick in the stronger sense (that is, thick and topologizing).

Serre subcategories in the strong sense are those nonempty full subcategories which are stable under the operation TT¯T\mapsto\bar{T}.

References

Serre considered such a class in the case of the category of abelian groups

Last revised on August 22, 2024 at 10:16:01. See the history of this page for a list of all contributions to it.