The classification of endotrivial complexes

Abstract

Let G be a finite group and k a field of prime characteristic p. We give a complete classification of endotrivial complexes, i.e. determine the Picard group $E_k\left(G\right)$ of the tensor-triangulated category $K^b\left({}_{kG}\mathrm{triv}\right)$, the bounded homotopy category of p-permutation modules, which Balmer and Gallauer recently considered in [8]. For p-groups, we identify $E_k(-)$ with the rational p-biset functor $C{F}_b(-)$ of Borel-Smith functions and recover a short exact sequence of rational p-biset functors constructed by Bouc and Yalçin. As a consequence, we prove that every p-permutation autoequivalence of a p-group arises from a splendid Rickard autoequivalence. Additionally, we give a positive answer to a question of Gelvin and Yalçin in [22], showing the kernel of the Bouc homomorphism for an arbitrary finite group G is described by superclass functions $f:s_p\left(G\right)\to Z$ satisfying the oriented Artin-Borel-Smith conditions.