nLab
thick subcategory

Thick subcategories and Serre quotient categories

Definition

A full triangulated subcategory is thick (or épaisse) if it is closed under extensions.

Sometimes the same definition is used in abelian categories as well. However, for many authors, including Pierre Gabriel, in abelian categories, this term denotes the stronger notion of a topologizing subcategory closed under extensions; in other words, a nonempty full subcategory TT of an abelian category AA is thick (in the strong sense) iff with all objects contains all its subquotients and all extensions, i.e. for every exact sequence

0MMM0 0 \longrightarrow M\longrightarrow M''\longrightarrow M'\longrightarrow 0

in AA, the object MM'' is in TT iff MM and MM' are in TT.

For some authors the thick subcategory (strong version) is called a Serre subcategory (in a weak sense), the term which we reserve for (generally) a stronger notion.

For any subcategory of an 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 where TT¯T\subset \bar{T} iff TT is topologizing. Moreover T¯\bar{T} is always thick in the stronger sense. Serre subcategories in the strong sense are those (nonempty) subcategories which are stable under the operation TT¯T\mapsto\bar{T}.

Serre quotient category

Following the extensions of an early work of Serre by Grothendieck and Gabriel, for a thick subcategory TT in an abelian category AA, one defines the (Serre) quotient category A/TA/T has the same objects as TT and

(A/T)(X,Y):=colimA(X,Y/Y) (A/T)(X,Y) := colim A(X',Y/Y')

where the colimit runs through all subobjects XXX'\subset X, YYY'\subset Y such that X/XObTX/X' \in Ob T, YObTY'\in Ob T. The quotient functor Q:AA/TQ: A\to A/T is obvious.

Notice that the set of morphisms is small, so that the Serre quotient category exists. On the other hand, one can construct an equivalent localization by the Gabriel-Zisman localizing at the class Σ\Sigma of all morphisms whose kernel and cokernel are in TT. Although Σ\Sigma admits the calculus of fractions, this method does not guarantee the existence in general.

Localizing subcategories

A thick subcategory (here always in strong sense) is said to be localizing if and QQ admits a right adjoint A/TAA/T\to A, often called the section functor. In other words A/TA/T is a reflective subcategory of AA. Every coreflective thick subcategory TT admits a section functor, and the converse holds if AA has injective envelopes. A thick subcategory TAT\subset A is a coreflective iff (T,F)(T,F) is a torsion theory where

F:={XObA|A(T,X)=0} F := \{X\in Ob A\,|\,A(T,X) = 0\}

References

Revised on March 24, 2014 06:45:53 by Urs Schreiber (89.204.138.56)