Projective smooth representations in natural characteristic
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916488593539072 |
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| author | Ophir, Amit Sorensen, Claus |
| author_facet | Ophir, Amit Sorensen, Claus |
| contents | We investigate under which circumstances there exists nonzero {\it{projective}} smooth $\field[G]$-modules, where $\field$ is a field of characteristic $p$ and $G$ is a locally pro-$p$ group. We prove the non-existence of (non-trivial) projective objects for so-called {\it{fair}} groups -- a family including $\bf{G}(\frak{F})$ for a connected reductive group $\bf{G}$ defined over a non-archimedean local field $\frak{F}$. This was proved in \cite{SS24} for finite extensions $\frak{F}/\Bbb{Q}_p$. The argument we present in this note has the benefit of being completely elementary and, perhaps more importantly, adaptable to $\frak{F}=\Bbb{F}_q(\!(t)\!)$. Finally, we elucidate the fairness condition via a criterion in the Chabauty space of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12867 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projective smooth representations in natural characteristic Ophir, Amit Sorensen, Claus Number Theory Representation Theory 11F70, 22E50, 20C20 We investigate under which circumstances there exists nonzero {\it{projective}} smooth $\field[G]$-modules, where $\field$ is a field of characteristic $p$ and $G$ is a locally pro-$p$ group. We prove the non-existence of (non-trivial) projective objects for so-called {\it{fair}} groups -- a family including $\bf{G}(\frak{F})$ for a connected reductive group $\bf{G}$ defined over a non-archimedean local field $\frak{F}$. This was proved in \cite{SS24} for finite extensions $\frak{F}/\Bbb{Q}_p$. The argument we present in this note has the benefit of being completely elementary and, perhaps more importantly, adaptable to $\frak{F}=\Bbb{F}_q(\!(t)\!)$. Finally, we elucidate the fairness condition via a criterion in the Chabauty space of $G$. |
| title | Projective smooth representations in natural characteristic |
| topic | Number Theory Representation Theory 11F70, 22E50, 20C20 |
| url | https://arxiv.org/abs/2411.12867 |