Self-equivalences of stable module categories
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.
scientific article; zbMATH DE number 1406454