On the prismatization of $O_K$ beyond the Hodge-Tate locus
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913490855264256 |
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| author | Liu, Zeyu |
| author_facet | Liu, Zeyu |
| contents | Let $X=\mathrm{Spf}(\mathcal{O}_K)$. We classify perfect complexes of $n$-truncated prismatic crystals on the prismatic site of $X$ when $n\leq 1+\frac{p-1}{e}$ by studying perfect complexes on the $n$-truncated prismatization of $X$, which are certain nilpotent thickenings of the Hodge-Tate stack of $X$ inside the prismatization of $X$. We describe the category of continuous semilinear representations and their cohomology for $G_K$ with coefficients in $B_{\mathrm{dR},n}^+$ via rationalization of vector bundles on the slight shrinking of the $n$-truncated prismatization of $X$. Along the way, we classify certain integral models for de Rham prismatic crystals studied in arXiv:2205.14914. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02051 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the prismatization of $O_K$ beyond the Hodge-Tate locus Liu, Zeyu Algebraic Geometry Number Theory Let $X=\mathrm{Spf}(\mathcal{O}_K)$. We classify perfect complexes of $n$-truncated prismatic crystals on the prismatic site of $X$ when $n\leq 1+\frac{p-1}{e}$ by studying perfect complexes on the $n$-truncated prismatization of $X$, which are certain nilpotent thickenings of the Hodge-Tate stack of $X$ inside the prismatization of $X$. We describe the category of continuous semilinear representations and their cohomology for $G_K$ with coefficients in $B_{\mathrm{dR},n}^+$ via rationalization of vector bundles on the slight shrinking of the $n$-truncated prismatization of $X$. Along the way, we classify certain integral models for de Rham prismatic crystals studied in arXiv:2205.14914. |
| title | On the prismatization of $O_K$ beyond the Hodge-Tate locus |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2409.02051 |