Stacks of group representations.
type: Item
instance of: scholarly article
title: Stacks of group representations.
zbMATH Open document ID: 1351.20004
DOI: 10.4171/JEMS/501
author: Paul Balmer
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2015-03-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/1302.6290
review text: 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.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 18F10
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 14F20
zbMATH DE Number: 6419400
zbMATH Keywords: restrictions of representations
zbMATH Keywords: extensions of representations
zbMATH Keywords: stacks
zbMATH Keywords: modular representations
zbMATH Keywords: finite groups
zbMATH Keywords: ring objects
zbMATH Keywords: descent
zbMATH Keywords: endotrivial representations
zbMATH Keywords: categories of finite \(G\)-sets
zbMATH Keywords: derived categories
zbMATH Keywords: stable categories
MaRDI profile type: Publication
arXiv ID: 1302.6290
cites work: Seminar of algebraic geometry du Bois-Marie 1963--1964. Topos theory and étale cohomology of schemes (SGA 4). Vol. 1: Topos theory. Exp. I--IV
cites work: Q3096591
cites work: Separability and triangulated categories
cites work: Descent in triangulated categories
cites work: Modular representations of finite groups with trivial restriction to Sylow subgroups.
cites work: Q4413070
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description: scientific article