Prismatic cohomology and $A_{\inf}$-cohomology with coefficients
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915481413222400 |
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| author | Tsuji, Takeshi |
| author_facet | Tsuji, Takeshi |
| contents | For a smooth $p$-adic formal scheme over the ring of integers of a perfectoid field of mixed characteristic $(0,p)$ containing all $p$-power roots of unity, we prove that the prismatic cohomology of a locally finite free prismatic crystal is isomorphic to the $A_{\inf}$-cohomology of the corresponding relative Breuil-Kisin-Fargues module, which is a certain type of locally finite free $\mathbb{A}_{\inf}$-module, on the proétale site of the generic fiber. We use a global description of the former in terms of $q$-Higgs modules via cohomological descent. We also discuss its compatibility with inverse image functors, scalar extensions under Frobenius, and tensor products. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04954 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Prismatic cohomology and $A_{\inf}$-cohomology with coefficients Tsuji, Takeshi Algebraic Geometry Number Theory For a smooth $p$-adic formal scheme over the ring of integers of a perfectoid field of mixed characteristic $(0,p)$ containing all $p$-power roots of unity, we prove that the prismatic cohomology of a locally finite free prismatic crystal is isomorphic to the $A_{\inf}$-cohomology of the corresponding relative Breuil-Kisin-Fargues module, which is a certain type of locally finite free $\mathbb{A}_{\inf}$-module, on the proétale site of the generic fiber. We use a global description of the former in terms of $q$-Higgs modules via cohomological descent. We also discuss its compatibility with inverse image functors, scalar extensions under Frobenius, and tensor products. |
| title | Prismatic cohomology and $A_{\inf}$-cohomology with coefficients |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2509.04954 |