Compactly generated relative stable categories.
type: Item
instance of: scholarly article
title: Compactly generated relative stable categories.
zbMATH Open document ID: 1238.20006
author: Matthew Grime
author: Peter Jørgensen
published in: Algebras and Representation Theory
publication date: 2012-01-13T00:00:00Z
review text: Let \(k\) be a field, \(G\) a finite group and \(H\) a subgroup of \(G\). Let \(\mathrm{Mod}(kG)\) be the category of all left \(kG\)-modules (which are not necessarily finitely generated). A \(kG\)-module is called \(H\)-projective if it is a direct summand of a module induced from \(H\) to \(G\). The relative stable category \(\mathrm{StMod}_H(kG)\) is defined as \(\mathrm{Mod}(kG)\) modulo morphisms factoring through \(H\)-projective \(kG\)-modules. As in the case of the usual stable category, this relative stable category is also a triangulated category. While the usual stable category is known to be compactly generated and is generated by finitely generated modules which are compact, it is not known whether this is true for \(\mathrm{StMod}_H(kG)\). Of course finitely generated modules are still compact objects in \(\mathrm{StMod}_H(kG)\), but they are NOT enough to compactly generate \(\mathrm{StMod}_H(kG)\), we should find more modules to do the task. The main result of the paper under review is that if \(kH\) has finite representation type, then the relative stable category \(\mathrm{StMod}_H(kG)\) is compactly generated, although the method could not give an explicit set of compact generators. The proof is as follows. The first step is to realize \(\mathrm{StMod}_H(kG)\) as the homotopy category of exact complexes of \(H\)-projective \(kG\)-modules, denoted by \(\mathrm{Tate}_H(kG)\) in this paper. This is a variant of some known results, for example, \textit{H. Krause}, [in Compos. Math. 141, No. 5, 1128-1162 (2005; Zbl 1090.18006)], showed that the homotopy category of exact complexes of injective modules over a Noetherian ring \(A\), denoted by \(K_{ac}(\text{Inj\,}A)\), is the infinite completion of the singular category, [see also Theorem 9.6.4 of \textit{M. Hovey, J. H. Palmieri} and \textit{N. P. Strickland}, Mem. Am. Math. Soc. 610 (1997; Zbl 0881.55001)]. The second step is the following remark. If \(H\) has finite representation type, then the cat…
reviewed by: Guodong Zhou
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 16E05
zbMATH DE Number: 5997605
zbMATH Keywords: finite groups
zbMATH Keywords: categories of left modules
zbMATH Keywords: compactly generated triangulated categories
zbMATH Keywords: finite representation type
zbMATH Keywords: relative projective modules
zbMATH Keywords: relative stable categories
zbMATH Keywords: right orthogonal categories
MaRDI profile type: Publication
full work available at URL: s10468-009-9187-9
OpenAlex ID: W1992966050
cites work: Representation Theory of Artin Algebras II
cites work: Q4318498
cites work: Localization and duality in topology and modular representation theory.
cites work: Transfer maps and virtual projectivity
cites work: On the derived category of a finite-dimensional algebra
cites work: Axiomatic stable homotopy theory
cites work: Acyclicity versus total acyclicity for complexes over noetherian rings
cites work: The homotopy category of complexes of projective modules
cites work: The connection between the $K$-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel
cites work: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability
cites work: Idempotent Modules in the Stable Category
DOI: 10.1007/S10468-009-9187-9
description: scientific article; zbMATH DE number 5997605