nLab thick subcategory

this entry needs attention

Definition

A non-empty full (pre-)triangulated subcategory is called thick (or épaisse) if it is “closed under direct summands” (e.g. Murayama, Def. 11, see Stacks Project, Def. 13.6.1).

Sometimes this is considered in the generality of abelian categories, where a thick subcategory is something like an additive full subcategory which is closed under direct summands, kernels of epimorphisms, cokernels of monomorphisms, and extensions (e.g. Dichev 2009).

The following needs harmonizing.

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 $T$ of an abelian category $A$ is thick (in the strong sense) iff for every exact sequence

$$0 \longrightarrow M\longrightarrow M''\longrightarrow M'\longrightarrow 0$$

in $A$, the object $M''$ is in $T$ iff $M$ and $M'$ are in $T$.

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 $A$ one denotes by $\bar{T}$ the full subcategory of $A$ generated by all objects $N$ for which any (nonzero) subquotient of $N$ in $T$ has a (nonzero) subobject from $T$. This becomes an idempotent operation on the class of subcategories of $A$ where $T\subset \bar{T}$ iff $T$ is topologizing. Moreover $\bar{T}$ is always thick in the stronger sense. Serre subcategories in the strong sense are those (nonempty) subcategories which are stable under the operation $T\mapsto\bar{T}$.

Serre quotient category

Following the extensions of an early work of Serre by Grothendieck and Gabriel, for a thick subcategory $T$ in an abelian category $A$, one defines the (Serre) quotient category $A/T$ as the one having the same objects as $A$ and hom-sets given by

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

where the colimit runs through all subobjects $X'\subset X$, $Y'\subset Y$ such that $X/X' \in Ob T$, $Y'\in Ob T$. The quotient functor $Q \colon A\to A/T$ is obvious.

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

A basic example is the quotient of the category of abelian groups modulo the torsion groups. This category is equivalent to the category of $\mathbb{Q}$-vector spaces, by the functor which maps an abelian group $M$ to the scalar extension $M \otimes_{\mathbb{Z}} \mathbb{Q}$. (See the Stacks Project, Tag 0B0J for a proof.)

Localizing subcategories

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

$$F \coloneqq \{X\in Ob A\,|\,A(T,X) = 0\}$$

Thick subcategories and saturation

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

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

Conversely, let $\Sigma$ be a saturated class of morphisms of $A$ 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 $p:A\to A_\Sigma$ 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).

Related concepts

• thick subcategory theorem

References

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

• The Stacks Project, Quotients of triangulated categories

• eom, Localization of categories

• 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.

• Takumi Murayama, The classification of thick subcategories and Balmer’s reconstruction theorem (pdf)

Discussion for quiver representations:

• Nikolay Dimitrov Dichev, Thick subcategories for quiver representations 2009 (pdf)