On the Picard group of the stable module category for infinite groups
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916258133311488 |
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| author | Gómez, Juan Omar |
| author_facet | Gómez, Juan Omar |
| contents | We introduce the stable module $\infty$-category for groups of type $Φ$ as an enhancement of the stable category defined by N. Mazza and P. Symonds. For groups of type $Φ$ which act on a tree, we show that the stable module $\infty$-category decomposes in terms of the associated graph of groups. For groups which admit a finite-dimensional cocompact model for the classifying space for proper actions, we exhibit a decomposition in terms of the stable module $\infty$-categories of their finite subgroups. We use these decompositions to provide methods to compute the Picard group of the stable module category. In particular, we provide a description of the Picard group for countable locally finite $p$-groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_08260 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Picard group of the stable module category for infinite groups Gómez, Juan Omar Representation Theory Algebraic Topology 20C07 (Primary) 18G65, 18N60 (Secondary) We introduce the stable module $\infty$-category for groups of type $Φ$ as an enhancement of the stable category defined by N. Mazza and P. Symonds. For groups of type $Φ$ which act on a tree, we show that the stable module $\infty$-category decomposes in terms of the associated graph of groups. For groups which admit a finite-dimensional cocompact model for the classifying space for proper actions, we exhibit a decomposition in terms of the stable module $\infty$-categories of their finite subgroups. We use these decompositions to provide methods to compute the Picard group of the stable module category. In particular, we provide a description of the Picard group for countable locally finite $p$-groups. |
| title | On the Picard group of the stable module category for infinite groups |
| topic | Representation Theory Algebraic Topology 20C07 (Primary) 18G65, 18N60 (Secondary) |
| url | https://arxiv.org/abs/2303.08260 |