Modular fixed points in equivariant homotopy theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912451751051264 |
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| author | Fuhrmann, Yorick |
| author_facet | Fuhrmann, Yorick |
| contents | We show that the derived $\infty$-category of permutation modules is equivalent to the category of modules over the Eilenberg-MacLane spectrum associated to a constant Mackey functor in the $\infty$-category of equivariant spectra. On such module categories we define a modular fixed point functor using geometric fixed points followed by an extension of scalars and identify it with the modular fixed point functor on derived permutation modules introduced by Balmer-Gallauer. As an application, we show that the Picard group of such a module category for a $p$-group is given by the group of class functions satisfying the Borel-Smith conditions. In the language of representation theory, this result was first obtained by Miller. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21413 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modular fixed points in equivariant homotopy theory Fuhrmann, Yorick Algebraic Topology Representation Theory We show that the derived $\infty$-category of permutation modules is equivalent to the category of modules over the Eilenberg-MacLane spectrum associated to a constant Mackey functor in the $\infty$-category of equivariant spectra. On such module categories we define a modular fixed point functor using geometric fixed points followed by an extension of scalars and identify it with the modular fixed point functor on derived permutation modules introduced by Balmer-Gallauer. As an application, we show that the Picard group of such a module category for a $p$-group is given by the group of class functions satisfying the Borel-Smith conditions. In the language of representation theory, this result was first obtained by Miller. |
| title | Modular fixed points in equivariant homotopy theory |
| topic | Algebraic Topology Representation Theory |
| url | https://arxiv.org/abs/2506.21413 |