Thick subcategories of the bounded derived category of a finite group. (English) Zbl 1339.20044
From the introduction: A new proof of the classification for tensor ideal thick subcategories of the bounded derived category, and the stable category, of modular representations of a finite group is obtained. The arguments apply more generally to yield a classification of thick subcategories of the bounded derived category of an Artinian complete intersection ring. One of the salient features of this work is that it takes no recourse to infinite constructions, unlike previous proofs of these results.
In the paper [(*), Fundam. Math. 153, No. 1, 59-80 (1997; Zbl 0886.20007)], the first author, in collaboration with D. J. Benson and J. Rickard, established a classification for tensor ideal thick subcategories of the stable category of finitely generated \(kG\)-modules in the case that \(G\) is a finite group and \(k\) is a field of characteristic \(p>0\) dividing the order of \(G\). This result was inspired and influenced by an earlier classification of the thick subcategories of the perfect complexes over a commutative Noetherian ring by M. J. Hopkins [Lond. Math. Soc. Lect. Note Ser. 117, 73-96 (1987; Zbl 0657.55008)]. The statements are remarkably similar: In both cases a subcategory is determined by the support, suitably defined, of the objects in the subcategory.
However, the methods in the two settings were entirely different. At that time it did not seem possible to adapt Hopkins’ methods to modules over group algebras. Instead, the proofs in [(*), loc. cit.] used idempotent modules and idempotent functors developed by Rickard. Idempotent modules are, in general, infinitely generated, hence the proof had employed constructions from the stable category of all \(kG\)-modules to obtain a result that spoke only of finitely generated modules.
The main purpose of this paper is to show that the results in the two settings are directly related. To this end, we extend Hopkins’ arguments to obtain a classification of the thick subcategories of perfect Differential Graded modules over suitable DG algebras. The result about \(kG\)-modules is then deduced from it by a series of reductions, following the paradigm developed in the work of the second author with L. L. Avramov, R.-O. Buchweitz, and C. Miller [Adv. Math. 223, No. 5, 1731-1781 (2010; Zbl 1186.13006)], and with D. J. Benson and H. Krause [Ann. Math. (2) 174, No. 3, 1643-1684 (2011; Zbl 1261.20057)]. What is more, the arguments require no constructions involving infinitely generated modules, in contrast to the proof in [(*), loc. cit.].
In the paper [(*), Fundam. Math. 153, No. 1, 59-80 (1997; Zbl 0886.20007)], the first author, in collaboration with D. J. Benson and J. Rickard, established a classification for tensor ideal thick subcategories of the stable category of finitely generated \(kG\)-modules in the case that \(G\) is a finite group and \(k\) is a field of characteristic \(p>0\) dividing the order of \(G\). This result was inspired and influenced by an earlier classification of the thick subcategories of the perfect complexes over a commutative Noetherian ring by M. J. Hopkins [Lond. Math. Soc. Lect. Note Ser. 117, 73-96 (1987; Zbl 0657.55008)]. The statements are remarkably similar: In both cases a subcategory is determined by the support, suitably defined, of the objects in the subcategory.
However, the methods in the two settings were entirely different. At that time it did not seem possible to adapt Hopkins’ methods to modules over group algebras. Instead, the proofs in [(*), loc. cit.] used idempotent modules and idempotent functors developed by Rickard. Idempotent modules are, in general, infinitely generated, hence the proof had employed constructions from the stable category of all \(kG\)-modules to obtain a result that spoke only of finitely generated modules.
The main purpose of this paper is to show that the results in the two settings are directly related. To this end, we extend Hopkins’ arguments to obtain a classification of the thick subcategories of perfect Differential Graded modules over suitable DG algebras. The result about \(kG\)-modules is then deduced from it by a series of reductions, following the paradigm developed in the work of the second author with L. L. Avramov, R.-O. Buchweitz, and C. Miller [Adv. Math. 223, No. 5, 1731-1781 (2010; Zbl 1186.13006)], and with D. J. Benson and H. Krause [Ann. Math. (2) 174, No. 3, 1643-1684 (2011; Zbl 1261.20057)]. What is more, the arguments require no constructions involving infinitely generated modules, in contrast to the proof in [(*), loc. cit.].
MSC:
| 20J06 | Cohomology of groups |
| 20C20 | Modular representations and characters |
| 20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 16E45 | Differential graded algebras and applications (associative algebraic aspects) |
| 13D09 | Derived categories and commutative rings |
Keywords:
tensor ideal thick subcategories; bounded derived categories; stable categories; modular representations of finite groups; modules over group algebras; finitely generated modulesReferences:
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