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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.13205 |
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| _version_ | 1866929734815842304 |
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| author | Pham, Dat |
| author_facet | Pham, Dat |
| contents | Let $K$ be a complete discretely valued field of mixed characteristic $(0,p)$ with perfect residue field, and let $E$ be a finite extension of $\mathbf{Q}_p$ contained in $K$. We show that the category of prismatic $F$-crystals on $\mathcal{O}_K$ (relative to $E$ in a suitable sense) is equivalent to the category of $\mathcal{O}_E$-lattices in $E$-crystalline representations defined by Kisin--Ren, extending the main result of \cite{arxiv:2106.14735} in the case $E=\mathbf{Q}_p$. As a key ingredient in the proof, by adapting a lemma of Du--Liu, we prove a general full faithfulness result for certain vector bundles on the prismatic site, which simplifies and refines the key descent step in the approach of Bhatt--Scholze without invoking the Beilinson fibre sequence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_13205 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Prismatic $F$-crystals and $E$-crystalline Galois representations Pham, Dat Number Theory Let $K$ be a complete discretely valued field of mixed characteristic $(0,p)$ with perfect residue field, and let $E$ be a finite extension of $\mathbf{Q}_p$ contained in $K$. We show that the category of prismatic $F$-crystals on $\mathcal{O}_K$ (relative to $E$ in a suitable sense) is equivalent to the category of $\mathcal{O}_E$-lattices in $E$-crystalline representations defined by Kisin--Ren, extending the main result of \cite{arxiv:2106.14735} in the case $E=\mathbf{Q}_p$. As a key ingredient in the proof, by adapting a lemma of Du--Liu, we prove a general full faithfulness result for certain vector bundles on the prismatic site, which simplifies and refines the key descent step in the approach of Bhatt--Scholze without invoking the Beilinson fibre sequence. |
| title | Prismatic $F$-crystals and $E$-crystalline Galois representations |
| topic | Number Theory |
| url | https://arxiv.org/abs/2306.13205 |