Syntomification and crystalline local systems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914579803537408 |
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| author | Pentland, Dylan |
| author_facet | Pentland, Dylan |
| contents | Let $p$ be a prime, and let $\mathrm{X}$ be a smooth $p$-adic formal scheme over $\mathrm{Spf} \mathcal{O}_K$ where $K/\mathbf{Q}_p$ is a finite extension. We show that reflexive sheaves on the stack $\mathrm{X}^{\mathrm{Syn}}$ are equivalent to $\mathbf{Z}_p$-lattices in crystalline local systems on the rigid generic fiber $\mathrm{X}_η$, and then use this to study the essential image of the étale realization functor on the isogeny category of perfect complexes on $\mathrm{X}^{\mathrm{Syn}}$. We also show that when $\mathrm{X}/\mathrm{Spf} \mathcal{O}_K$ is smooth and proper that $\mathsf{Perf}(\mathrm{X}^{\mathrm{Syn}})[1/p]$ is equivalent to a category of admissible filtered $F$-isocrystals in perfect complexes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16961 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Syntomification and crystalline local systems Pentland, Dylan Number Theory Let $p$ be a prime, and let $\mathrm{X}$ be a smooth $p$-adic formal scheme over $\mathrm{Spf} \mathcal{O}_K$ where $K/\mathbf{Q}_p$ is a finite extension. We show that reflexive sheaves on the stack $\mathrm{X}^{\mathrm{Syn}}$ are equivalent to $\mathbf{Z}_p$-lattices in crystalline local systems on the rigid generic fiber $\mathrm{X}_η$, and then use this to study the essential image of the étale realization functor on the isogeny category of perfect complexes on $\mathrm{X}^{\mathrm{Syn}}$. We also show that when $\mathrm{X}/\mathrm{Spf} \mathcal{O}_K$ is smooth and proper that $\mathsf{Perf}(\mathrm{X}^{\mathrm{Syn}})[1/p]$ is equivalent to a category of admissible filtered $F$-isocrystals in perfect complexes. |
| title | Syntomification and crystalline local systems |
| topic | Number Theory |
| url | https://arxiv.org/abs/2510.16961 |