Stacks of group representations
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2013
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| _version_ | 1866917770847846400 |
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| author | Balmer, Paul |
| author_facet | Balmer, Paul |
| contents | 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 Cech cohomology to describe the kernel and the image of the homomorphism $T(G)\to T(H)$, where $T(-)$ denotes the group of endotrivial representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1302_6290 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | Stacks of group representations Balmer, Paul Representation Theory Category Theory 20C20, 14A20, 18F10, 18E30, 18C20 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 Cech cohomology to describe the kernel and the image of the homomorphism $T(G)\to T(H)$, where $T(-)$ denotes the group of endotrivial representations. |
| title | Stacks of group representations |
| topic | Representation Theory Category Theory 20C20, 14A20, 18F10, 18E30, 18C20 |
| url | https://arxiv.org/abs/1302.6290 |