Absolute calculus and prismatic crystals on cyclotomic rings
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913917829120000 |
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| author | Gros, Michel Stum, Bernard Le Quirós, Adolfo |
| author_facet | Gros, Michel Stum, Bernard Le Quirós, Adolfo |
| contents | Let $p$ be a prime, $W$ the ring of Witt vectors of a perfect field $k$ of characteristic $p$ and $ζ$ a primitive $p$th root of unity. We introduce a new notion of calculus over $W$ that we call absolute calculus. It may be seen as a singular version of the $q$-calculus used in previous work, in the sense that the role of the coordinate is now played by $q$ itself. We show that what we call a weakly nilpotent $\mathbbΔ$-connection on a finite free module is equivalent to a prismatic vector bundle on $W[ζ]$. As a corollary of a theorem of Bhatt and Scholze, we finally obtain that a $\mathbbΔ$-connection with a frobenius structure on a finite free module is equivalent to a lattice in a crystalline representation. We also consider the case of de Rham prismatic crystals as well as Hodge-Tate prismatic crystals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_13790 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Absolute calculus and prismatic crystals on cyclotomic rings Gros, Michel Stum, Bernard Le Quirós, Adolfo Algebraic Geometry 14F30, 14F40 Let $p$ be a prime, $W$ the ring of Witt vectors of a perfect field $k$ of characteristic $p$ and $ζ$ a primitive $p$th root of unity. We introduce a new notion of calculus over $W$ that we call absolute calculus. It may be seen as a singular version of the $q$-calculus used in previous work, in the sense that the role of the coordinate is now played by $q$ itself. We show that what we call a weakly nilpotent $\mathbbΔ$-connection on a finite free module is equivalent to a prismatic vector bundle on $W[ζ]$. As a corollary of a theorem of Bhatt and Scholze, we finally obtain that a $\mathbbΔ$-connection with a frobenius structure on a finite free module is equivalent to a lattice in a crystalline representation. We also consider the case of de Rham prismatic crystals as well as Hodge-Tate prismatic crystals. |
| title | Absolute calculus and prismatic crystals on cyclotomic rings |
| topic | Algebraic Geometry 14F30, 14F40 |
| url | https://arxiv.org/abs/2310.13790 |