this entry needs attention
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 of an abelian category is thick (in the strong sense) iff for every exact sequence
in , the object is in iff and are in .
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 one denotes by the full subcategory of generated by all objects for which any (nonzero) subquotient of in has a (nonzero) subobject from . This becomes an idempotent operation on the class of subcategories of where iff is topologizing. Moreover is always thick in the stronger sense. Serre subcategories in the strong sense are those (nonempty) subcategories which are stable under the operation .
Following the extensions of an early work of Serre by Grothendieck and Gabriel, for a thick subcategory in an abelian category , one defines the (Serre) quotient category as the one having the same objects as and hom-sets given by
where the colimit runs through all subobjects , such that , . The quotient functor 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 of all morphisms whose kernel and cokernel are in . Although 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 -vector spaces, by the functor which maps an abelian group to the scalar extension . (See the Stacks Project, Tag 0B0J for a proof.)
A thick subcategory (here always in the strong sense) is said to be localizing if is thick and the canonical functor admits a right adjoint , often called the section functor. In other words is a reflective subcategory of . Every coreflective thick subcategory admits a section functor, and the converse holds if has injective envelopes. A thick subcategory is a coreflective iff is a torsion theory where
Recall that a class of morphisms with category of fractions is called saturated if coincides with the class of morphisms inverted by the canonical functor .
Let be an abelian category, be a thick subcategory in the strong sense and let be the class of morphisms in such that their kernel and cokernel are in . Then has a right and a left calculus of fractions and is saturated for the canonical functor . Furthermore, maps to zero objects precisely the objects in .
Conversely, let be a saturated class of morphisms of 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 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).
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.
Takumi Murayama, The classification of thick subcategories and Balmer’s reconstruction theorem (pdf)
Discussion for quiver representations:
Last revised on August 3, 2021 at 10:29:35. See the history of this page for a list of all contributions to it.