Compactly generated relative stable categories.
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 category of \(H\)-projective \(kG\)-modules is additively generated by a single module \(x\) which is the induced module of a \(kH\)-module \(y\) which is an additive generator of \(\mathrm{Mod}(kH)\). By the usual projectivising process, the homotopy category of complexes of \(H\)-projective \(kG\)-modules \(K(H\mathrm{-Proj})\) is equivalent to the homotopy category of complexes of projective \(\Gamma^{op}\)-modules where \(\Gamma=\mathrm{End}_{kG}(x)\) is the endomorphism ring of \(x\). So \(K(H\mathrm{-Proj})\) is compactly generated, by a result of the second author [Adv. Math. 193, No. 1, 223-232 (2005; Zbl 1068.18012)]. In the third step, the authors then realize \(\mathrm{StMod}_H(kG)\) as the right orthogonal of \(x\) in \(K(H\mathrm{-Proj})\). So we can apply a result of \textit{S. Iyengar} and \textit{H. Krause} [Doc. Math., J. DMV 11, 207-240 (2006; Zbl 1119.13014)], and deduce that \(\mathrm{StMod}_H(kG)\) is compactly generated. It would be interesting to find a set of compact generators for \(\mathrm{StMod}_H(kG)\) in the case considered in this paper, and more generally to investigate when \(\mathrm{StMod}_H(kG)\) is compactly generated for a subgroup \(H\).
compactly generated triangulated categories
scientific article; zbMATH DE number 5997605