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 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 as the one having 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 the strong sense) is said to be localizing if TT is thick and the canonical functor 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\}

Thick subcategories and saturation

Recall that a class Σ\Sigma of morphisms with category of fractions 𝒞 Σ\mathcal{C}_\Sigma is called saturated if Σ\Sigma coincides with the class of morphisms inverted by the canonical functor 𝒞𝒞 Σ\mathcal{C}\to \mathcal{C}_\Sigma.

Let AA be an abelian category, TT be a thick subcategory in the strong sense and let Σ T\Sigma_T be the class of morphisms in AA such that their kernel and cokernel are in TT. Then Σ T\Sigma_T has a right and a left calculus of fractions and is saturated for the canonical functor p:AA/Tp:A\to A/T. Furthermore, pp maps to zero objects precisely the objects in TT.

Conversely, let Σ\Sigma be a saturated class of morphisms of AA with a calculus of fractions on the right and on the left. Then the full subcategory on the objects that are mapped to zero by the canonical p:AA Σp:A\to A_\Sigma is thick. In other words, there is a bijection between thick subcategories in the strong sense and saturated classes of morphisms with a calculus of fractions on the right and on the left.

(For this material see Schubert 1970, pp.105-107).

References

  • Pierre Gabriel, Des catégories abéliennes, Bulletin de la Société Mathématique de France, 90 (1962), p. 323-448 (numdam)

  • Jacob Lurie, Chromatic Homotopy Theory, lecture 26, Thick subcategories (pdf)

  • A. L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, MIA 330, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9

  • H. Schubert, Kategorien II , Springer Heidelberg 1970.

Revised on September 10, 2017 14:46:23 by Thomas Holder (89.204.130.80)