A Tannakian framework for prismatic $F$-crystals
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866917691762147328 |
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| author | Imai, Naoki Kato, Hiroki Youcis, Alex |
| author_facet | Imai, Naoki Kato, Hiroki Youcis, Alex |
| contents | We develop the Tannakian theory of (analytic) prismatic $F$-crystals on a smooth formal scheme $\mathfrak{X}$ over the ring of integers of a discretely valued field with perfect residue field. Our main result gives an equivalence between the $\mathcal{G}$-objects of prismatic $F$-crystals on $\mathfrak{X}$ and $\mathcal{G}$-objects on a newly-defined category of $\mathbb{Z}_p$-local systems on $\mathfrak{X}_η$: those of prismatically good reduction. Additionally, we develop a shtuka realization functor for (analytic) prismatic $F$-crystals on $p$-adic (formal) schemes and show it satisfies several compatibilities with previous work on the Tannakian theory of shtukas over such objects. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08259 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Tannakian framework for prismatic $F$-crystals Imai, Naoki Kato, Hiroki Youcis, Alex Number Theory Algebraic Geometry We develop the Tannakian theory of (analytic) prismatic $F$-crystals on a smooth formal scheme $\mathfrak{X}$ over the ring of integers of a discretely valued field with perfect residue field. Our main result gives an equivalence between the $\mathcal{G}$-objects of prismatic $F$-crystals on $\mathfrak{X}$ and $\mathcal{G}$-objects on a newly-defined category of $\mathbb{Z}_p$-local systems on $\mathfrak{X}_η$: those of prismatically good reduction. Additionally, we develop a shtuka realization functor for (analytic) prismatic $F$-crystals on $p$-adic (formal) schemes and show it satisfies several compatibilities with previous work on the Tannakian theory of shtukas over such objects. |
| title | A Tannakian framework for prismatic $F$-crystals |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2406.08259 |