Stacks of group representations

Balmer, 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 -