Serre subcategory


Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Homology theories


Notions of subcategory



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

MMMM\to M'\to M''

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


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, 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) := \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}.