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Stacks of group representations. (English) Zbl 1351.20004

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 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.

MSC:

20C20 Modular representations and characters
14A20 Generalizations (algebraic spaces, stacks)
18F10 Grothendieck topologies and Grothendieck topoi
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18E30 Derived categories, triangulated categories (MSC2010)
14F20 Étale and other Grothendieck topologies and (co)homologies
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References:

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