UC Santa Cruz

Let G be a finite group and k a field of prime characteristic p. In this dissertation, we explore endotrivial complexes, the invertible objects of the homotopy category of p-permutation modules Kb (kG triv), and their applications towards other questions of interest in representation theory. We develop methods for characterizing endotrivial complexes in terms of numerical invariants and local data, and provide a classification of these objects for all finite groups. As a corollary, we show that every p-permutation autoequivalence of a p-group lifts to a splendid Rickard autoequivalence, and deduce the kernel of the Bouc homomorphism for any finite group. We also develop a notion of relative endotriviality, analogous to Lassueur’s construction of relatively endotrivial modules. This notion is critical for the classification of endotrivial complexes, and also allows us to give a complete characterization of endosplit p-permutation resolutions, chain complexes which play a critical role in Brou´e’s abelian defect group conjecture. Finally, we develop a theory of Galois descent for these complexes, showing that the question of Galois descent of an endosplit p-permutation resolution reduces to the question of Galois descent for the module it resolves.