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 ${\mathcal{E}}_k(G)$ of the tensor-triangulated category $K^b({}_{kG}\mathbf{triv})$, the bounded homotopy category of $p$-permutation modules, which Balmer and Gallauer recently considered. For $p$-groups, we identify ${\mathcal{E}}_k(-)$ 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, showing the kernel of the Bouc homomorphism for an arbitrary finite group $G$ is described by superclass functions $f:s_p(G)\to \mathbb{Z}$ satisfying the oriented Artin-Borel-Smith conditions.

26 pages. v3: Added an observation by M. Gallauer that induction of endotrivial complexes arises from the topological norm map. Accepted version, to appear in Adv. Math