An integral comparison of crystalline and de Rham cohomology
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913956687249408 |
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| author | Abhinandan Youcis, Alex |
| author_facet | Abhinandan Youcis, Alex |
| contents | Let $\mathcal{O}_K$ be a mixed characteristic complete DVR with perfect residue field $k$ and fraction field $K$. It is a celebrated result of Berthelot and Ogus that for a smooth proper formal scheme $X/\mathcal{O}_K$ there exists a comparison between the de Rham cohomology groups $\mathrm{H}^i_\mathrm{dR}(X/\mathcal{O}_K)$ and the crystalline cohomology groups $\mathrm{H}^i_\mathrm{crys}(X_k/W(k))$ of the special fibre, after tensoring with $K$. In this article, we use the stacky perspective on prismatic cohomology, due to Drinfeld and Bhatt--Lurie, to give a version of this comparison result with coefficients in a perfect complex of prismatic $F$-crystals on $X$. Our method is of an integral nature and suggests new tools to understand the relationship between torsion in de Rham and crystalline cohomology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_17631 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An integral comparison of crystalline and de Rham cohomology Abhinandan Youcis, Alex Number Theory Algebraic Geometry 14F30, 14F40 Let $\mathcal{O}_K$ be a mixed characteristic complete DVR with perfect residue field $k$ and fraction field $K$. It is a celebrated result of Berthelot and Ogus that for a smooth proper formal scheme $X/\mathcal{O}_K$ there exists a comparison between the de Rham cohomology groups $\mathrm{H}^i_\mathrm{dR}(X/\mathcal{O}_K)$ and the crystalline cohomology groups $\mathrm{H}^i_\mathrm{crys}(X_k/W(k))$ of the special fibre, after tensoring with $K$. In this article, we use the stacky perspective on prismatic cohomology, due to Drinfeld and Bhatt--Lurie, to give a version of this comparison result with coefficients in a perfect complex of prismatic $F$-crystals on $X$. Our method is of an integral nature and suggests new tools to understand the relationship between torsion in de Rham and crystalline cohomology. |
| title | An integral comparison of crystalline and de Rham cohomology |
| topic | Number Theory Algebraic Geometry 14F30, 14F40 |
| url | https://arxiv.org/abs/2507.17631 |