Derived induction theory for the K-theory of modular group algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911239654866944 |
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| author | Vogeli, Chase |
| author_facet | Vogeli, Chase |
| contents | We prove an induction theorem for the higher algebraic K-groups of group algebras $kG$ of finite groups $G$ over characteristic $p$ finite fields $k$. For a certain class of finite groups, which we call $p$-isolated, this reduces calculations to calculations for their $p$-subgroups. We do so by showing that the stable module categories of $kH$ as $H$ ranges over subgroups of $G$ assemble into a categorical Green functor, which results in a spectral Green functor structure on K-theory. By general induction theory, this reduces proving a spectrum-level induction statement to proving an induction statement on $π_0$ Green functors, which we accomplish using modular representation theory. For $p$-isolated groups with Sylow $p$-subgroups of order $p$, we produce explicit new calculations of K-groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25763 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Derived induction theory for the K-theory of modular group algebras Vogeli, Chase K-Theory and Homology Algebraic Topology 19D50, 55P91, 19A22, 20C20 We prove an induction theorem for the higher algebraic K-groups of group algebras $kG$ of finite groups $G$ over characteristic $p$ finite fields $k$. For a certain class of finite groups, which we call $p$-isolated, this reduces calculations to calculations for their $p$-subgroups. We do so by showing that the stable module categories of $kH$ as $H$ ranges over subgroups of $G$ assemble into a categorical Green functor, which results in a spectral Green functor structure on K-theory. By general induction theory, this reduces proving a spectrum-level induction statement to proving an induction statement on $π_0$ Green functors, which we accomplish using modular representation theory. For $p$-isolated groups with Sylow $p$-subgroups of order $p$, we produce explicit new calculations of K-groups. |
| title | Derived induction theory for the K-theory of modular group algebras |
| topic | K-Theory and Homology Algebraic Topology 19D50, 55P91, 19A22, 20C20 |
| url | https://arxiv.org/abs/2510.25763 |