On endosplit $p$-permutation resolutions and Broué's conjecture for $p$-solvable groups

Abstract

Endosplit $p$-permutation resolutions play an instrumental role in verifying Broué's abelian defect group conjecture in numerous cases. We give a new characterization of all endosplit $p$-permutation resolutions and reduce the question of Galois descent of an endosplit $p$-permutation resolution to the Galois descent of the module it resolves. This is shown using techniques from the study of endotrivial complexes, the invertible objects of the bounded homotopy category of $p$-permutation modules. As an application, we show that a refinement of Broué's conjecture proposed by Kessar--Linckelmann holds for certain blocks of groups $G$ satisfying $G = O_{p',p,p'}(G)$ with abelian Sylow $p$-subgroup, the key reduction step in Harris--Linckelmann's verification of Broué's conjecture for all $p$-solvable groups.