# Stacks of group representations

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 1, page 189-228
- ISSN: 1435-9855

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topBalmer, Paul. "Stacks of group representations." Journal of the European Mathematical Society 017.1 (2015): 189-228. <http://eudml.org/doc/277691>.

@article{Balmer2015,

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 $T(G)\rightarrow T(H)$, where $T(-)$ denotes the group of endotrivial representations.},

author = {Balmer, Paul},

journal = {Journal of the European Mathematical Society},

keywords = {restriction; extension; monad; stack; modular representations; finite group; ring object; descent; endotrivial representation; restriction; extension; monad; stack; modular representations; finite group; ring object; descent; endotrivial representation},

language = {eng},

number = {1},

pages = {189-228},

publisher = {European Mathematical Society Publishing House},

title = {Stacks of group representations},

url = {http://eudml.org/doc/277691},

volume = {017},

year = {2015},

}

TY - JOUR

AU - Balmer, Paul

TI - Stacks of group representations

JO - Journal of the European Mathematical Society

PY - 2015

PB - European Mathematical Society Publishing House

VL - 017

IS - 1

SP - 189

EP - 228

AB - 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 $T(G)\rightarrow T(H)$, where $T(-)$ denotes the group of endotrivial representations.

LA - eng

KW - restriction; extension; monad; stack; modular representations; finite group; ring object; descent; endotrivial representation; restriction; extension; monad; stack; modular representations; finite group; ring object; descent; endotrivial representation

UR - http://eudml.org/doc/277691

ER -

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