Self-equivalences of stable module categories
type: Item
instance of: scholarly article
title: Self-equivalences of stable module categories
zbMATH Open document ID: 0949.20001
DOI: 10.1007/PL00004789
author: Raphaël Rouquier
author: Jon F. Carlson
published in: Mathematische Zeitschrift
publication date: 2000-09-19T00:00:00Z
review text: Let \(G\) and \(H\) be finite groups and \(k\) a field of characteristic \(p>0\). The stable module category of \(kG\)-modules \(\text{stmod}(kG)\) is the category with objects being finitely generated \(kG\)-modules and morphisms being module homomorphisms modulo those factoring through a projective module. Denote by \({\mathcal T}(G)\) the triangulated subcategory of \(\text{stmod}(kG)\) generated by the trivial module. The main result of the paper is that the image of an endotrivial module \(M\) by an equivalence \(F\colon{\mathcal T}(G)\to{\mathcal T}(H)\) of triangulated categories is an endotrivial module. The proof uses a tricky argument considering the dimension of the group cohomology of \(G\) and \(H\). Various applications are given. It is proven for example that any such equivalence \(F\) may be modified so that it fixes the trivial module. Furthermore, it is proven that if \(G\) and \(H\) are two \(p\)-groups, then \(kG\) and \(kH\) are stably equivalent of Morita type if and only if \(kG\) and \(kH\) are isomorphic as algebras. A rather detailed analysis of the case of an Abelian \(p\)-group acted upon by a cyclic \(p'\)-group gives that there are no non-trivial autoequivalences of Morita type of the stable module category of this group. Stable autoequivalences of standard type were first studied in a systematic way by \textit{R. Rouquier} and the reviewer for derived module categories [see CMS Conf. Proc. 18, 721-749 (1996; Zbl 0855.16015)]. There the case of a stable equivalence is sketched as well. As a byproduct the authors get in that situation that having a splendid equivalence between the derived categories of the principal blocks of \(kG\) and \(kH\) is equivalent to having a stable equivalence between these blocks.
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 1406454
zbMATH Keywords: finitely generated modules
zbMATH Keywords: \(p\)-groups
zbMATH Keywords: finite groups
zbMATH Keywords: stable module categories
zbMATH Keywords: endotrivial modules
zbMATH Keywords: triangulated categories
zbMATH Keywords: autoequivalences of Morita type
zbMATH Keywords: derived module categories
zbMATH Keywords: stable equivalences
zbMATH Keywords: splendid equivalences
zbMATH Keywords: derived categories
zbMATH Keywords: principal blocks
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1406454