JournalsjemsVol. 17, No. 1pp. 189–228

Stacks of group representations

  • Paul Balmer

    UCLA, Los Angeles, USA
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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 GG, the derived and the stable categories of representations of a subgroup HH can be constructed out of the corresponding category for GG 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 HH can be extended to GG. We show that the presheaves of plain, derived and stable representations all form stacks on the category of finite GG-sets (or the orbit category of GG), with respect to a suitable Grothendieck topology that we call the sipp topology.

When HH contains a Sylow subgroup of~GG, we use sipp Čech cohomology to describe the kernel and the image of the homomorphism T(G)T(H)T(G)\to T(H), where T()T(-) denotes the group of endotrivial representations.

Cite this article

Paul Balmer, Stacks of group representations. J. Eur. Math. Soc. 17 (2015), no. 1, pp. 189–228

DOI 10.4171/JEMS/501