An arithmetic étale-crystalline comparison with coefficients in crystalline local systems
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913822943477760 |
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| author | Hauck, Maximilian |
| author_facet | Hauck, Maximilian |
| contents | We use the stacky approach to $p$-adic cohomology theories recently developed by Drinfeld and Bhatt--Lurie to generalise a comparison theorem between the rational crystalline cohomology of the special fibre and the rational $p$-adic étale cohomology of the arithmetic generic fibre of any proper $p$-adic formal scheme $X$ due to Colmez--Niziol to the case of coefficients in an arbitrary crystalline local system on the generic fibre of $X$. In the process, we establish a version of the Beilinson fibre square of Antieau--Mathew--Morrow--Nikolaus with coefficients in the proper case and prove a comparison between syntomic cohomology and $p$-adic étale cohomology with coefficients in an arbitrary $F$-gauge. Our methods also yield a description of the isogeny category of perfect $F$-gauges on $\mathbb{Z}_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03294 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An arithmetic étale-crystalline comparison with coefficients in crystalline local systems Hauck, Maximilian Algebraic Geometry Number Theory We use the stacky approach to $p$-adic cohomology theories recently developed by Drinfeld and Bhatt--Lurie to generalise a comparison theorem between the rational crystalline cohomology of the special fibre and the rational $p$-adic étale cohomology of the arithmetic generic fibre of any proper $p$-adic formal scheme $X$ due to Colmez--Niziol to the case of coefficients in an arbitrary crystalline local system on the generic fibre of $X$. In the process, we establish a version of the Beilinson fibre square of Antieau--Mathew--Morrow--Nikolaus with coefficients in the proper case and prove a comparison between syntomic cohomology and $p$-adic étale cohomology with coefficients in an arbitrary $F$-gauge. Our methods also yield a description of the isogeny category of perfect $F$-gauges on $\mathbb{Z}_p$. |
| title | An arithmetic étale-crystalline comparison with coefficients in crystalline local systems |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2505.03294 |