Stacks of group representations

Abstract

We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group G, the derived and the stable categories of representations of a subgroup H can be constructed out of the corresponding category for G by a purely triangulated-categorical construction, analogous to étale extension in algebraic geometry. In the case of finite groups, we then use descent methods to investigate when modular representations of the subgroup H can be extended to G. We show that the presheaves of plain, derived and stable representations all form stacks on the category of finite G-sets (or the orbit category of G), with respect to a suitable Grothendieck topology that we call the sipp topology. When H contains a Sylow subgroup of G, we use sipp Čech cohomology to describe the kernel and the image of the homomorphism T (G) → T (H), where T (-) denotes the group of endotrivial representations.