localization of an abelian category

Abstract localization functors among abelian categories have several descriptions. Additional descriptions exist if in addition the category is Grothendieck.

A nonempty subcategory of an abelian category is thick (in the sense of Pierre Gabriel) if it is closed under subobjects, quotients and extensions. In the special case of the abelian categories of RR-modules (where RR is a ring) this agrees with the notion of a Serre subcategory (in general the latter is a stronger notion).

Following Jean-Pierre Serre, given a thick subcategory TT, define the quotient 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. There is a canonical quotient functor Q:AA/TQ: A\to A/T which is the identity on objects. The quotient category A/TA/T is abelian.

Created on June 8, 2011 at 16:01:50. See the history of this page for a list of all contributions to it.