A stacky approach to $p$-adic Hodge theory
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909317710479360 |
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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 known comparison theorems in $p$-adic Hodge theory so as to accommodate coefficients. More precisely, we establish a comparison 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 $X$ which allows for coefficients in any crystalline local system on the generic fibre of $X$; moreover, we also prove a comparison between the Nygaard filtration and the Hodge filtration for coefficients in an arbitrary gauge in the sense of Bhatt--Lurie. In the process, we develop a stacky approach to diffracted Hodge cohomology as introduced by Bhatt--Lurie, 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. This work is the author's master's thesis at the University of Bonn. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10557 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A stacky approach to $p$-adic Hodge theory 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 known comparison theorems in $p$-adic Hodge theory so as to accommodate coefficients. More precisely, we establish a comparison 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 $X$ which allows for coefficients in any crystalline local system on the generic fibre of $X$; moreover, we also prove a comparison between the Nygaard filtration and the Hodge filtration for coefficients in an arbitrary gauge in the sense of Bhatt--Lurie. In the process, we develop a stacky approach to diffracted Hodge cohomology as introduced by Bhatt--Lurie, 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. This work is the author's master's thesis at the University of Bonn. |
| title | A stacky approach to $p$-adic Hodge theory |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2409.10557 |