Stacks of group representations.
Summary: 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 \textit{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 \(T(G)\to T(H)\), where \(T(-)\) denotes the group of endotrivial representations.
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