Relatively endotrivial complexes
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915604814888960 |
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| author | Miller, Sam K. |
| author_facet | Miller, Sam K. |
| contents | Let $G$ be a finite group and $k$ be a field of characteristic $p > 0$. In prior work, we studied endotrivial complexes, the invertible objects of the bounded homotopy category $K^b({}_{kG}\mathbf{triv})$ of $p$-permutation $kG$-modules. Using the notion of projectivity relative to a $kG$-module, we expand on this study by defining notions of "relatively" endotrivial chain complexes, analogous to Lassueur's construction of relatively endotrivial $kG$-modules. We obtain equivalent characterizations of relative endotriviality and find corresponding local homological data which almost completely determine the isomorphism class of a relatively endotrivial complex. We show this local data must partially satisfy the Borel-Smith conditions, and consider the behavior of restriction to subgroups containing Sylow $p$-subgroups $S$ of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08042 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relatively endotrivial complexes Miller, Sam K. Group Theory Representation Theory 20J05, 20C05, 20C20 Let $G$ be a finite group and $k$ be a field of characteristic $p > 0$. In prior work, we studied endotrivial complexes, the invertible objects of the bounded homotopy category $K^b({}_{kG}\mathbf{triv})$ of $p$-permutation $kG$-modules. Using the notion of projectivity relative to a $kG$-module, we expand on this study by defining notions of "relatively" endotrivial chain complexes, analogous to Lassueur's construction of relatively endotrivial $kG$-modules. We obtain equivalent characterizations of relative endotriviality and find corresponding local homological data which almost completely determine the isomorphism class of a relatively endotrivial complex. We show this local data must partially satisfy the Borel-Smith conditions, and consider the behavior of restriction to subgroups containing Sylow $p$-subgroups $S$ of $G$. |
| title | Relatively endotrivial complexes |
| topic | Group Theory Representation Theory 20J05, 20C05, 20C20 |
| url | https://arxiv.org/abs/2402.08042 |