Prismatic $G$-displays and descent theory
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916856500060160 |
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| author | Ito, Kazuhiro |
| author_facet | Ito, Kazuhiro |
| contents | For a smooth affine group scheme $G$ over the ring of $p$-adic integers $\mathbb{Z}_p$ and a cocharacter $μ$ of $G$, we study $G$-$μ$-displays over the prismatic site of Bhatt-Scholze. In particular, we obtain several descent results for them. If $G=\mathrm{GL}_n$, then our $G$-$μ$-displays can be thought of as Breuil-Kisin modules with some additional conditions. The relation between our $G$-$μ$-displays and prismatic $F$-gauges introduced by Drinfeld and Bhatt-Lurie is also discussed. In fact, our results are formulated and proved for smooth affine group schemes over the ring of integers $\mathcal{O}_E$ of any finite extension $E$ of $\mathbb{Q}_p$ by using $\mathcal{O}_E$-prisms, which are $\mathcal{O}_E$-analogues of prisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_15814 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Prismatic $G$-displays and descent theory Ito, Kazuhiro Algebraic Geometry Number Theory Primary 14F30, Secondary 14G45, 14L05 For a smooth affine group scheme $G$ over the ring of $p$-adic integers $\mathbb{Z}_p$ and a cocharacter $μ$ of $G$, we study $G$-$μ$-displays over the prismatic site of Bhatt-Scholze. In particular, we obtain several descent results for them. If $G=\mathrm{GL}_n$, then our $G$-$μ$-displays can be thought of as Breuil-Kisin modules with some additional conditions. The relation between our $G$-$μ$-displays and prismatic $F$-gauges introduced by Drinfeld and Bhatt-Lurie is also discussed. In fact, our results are formulated and proved for smooth affine group schemes over the ring of integers $\mathcal{O}_E$ of any finite extension $E$ of $\mathbb{Q}_p$ by using $\mathcal{O}_E$-prisms, which are $\mathcal{O}_E$-analogues of prisms. |
| title | Prismatic $G$-displays and descent theory |
| topic | Algebraic Geometry Number Theory Primary 14F30, Secondary 14G45, 14L05 |
| url | https://arxiv.org/abs/2303.15814 |