On endotrivial complexes and the generalized Dade group

Let p be a prime, G a p-group, and k a field of characteristic p. We give a complete description of the group Eₖ (G) of endotrivial complexes, identifying Eₖ (G) with the additive subgroup of integer-valued class functions satisfying the Borel-Smith conditions. This is done by constructing a short exact sequence isomorphic to a short exact sequence of rational p-biset functors constructed by Bouc and Yalcin, endowing Eₖ with rational p-biset structure. We also determine all the possible h-marks which arise in Eₖ (G) when G is a finite group via the induced restriction map to a Sylow p-subgroup. As a consequence, we prove that every p-permutation autoequivalence of a p-group extends to a splendid Rickard autoequivalence. Additionally, we give a positive answer to a question of Gelvin and Yalcin, showing the kernel of the Bouc homomorphism for an arbitrary finite group G is described by integer-valued class functions on p-subgroups satisfying the oriented Artin-Borel-Smith conditions.