A prismatic-etale comparison theorem in the semistable case
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913938056151040 |
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| author | Tian, Yichao |
| author_facet | Tian, Yichao |
| contents | Let $K|\mathbb{Q}_p$ be a complete discrete valuation field with perfect residue field, $O_K$ be its ring of integers. Consider a semistable $p$-adic formal scheme $X$ over $\mathrm{Spf}(O_K)$ with smooth generic fiber $X_η$. Du--Liu--Moon--Shimizu showed recently that the category of analytic prismatic $F$-crystals on the absolute log prismatic site of $X$ is equivalent to the category of semistable étale $\mathbb{Z}_p$-local systems on the adic generic fiber $X_η$. In this article, we prove a comparison between the Breuil--Kisin cohomology of an analytic log prismatic $F$-crystal on $X$ and the étale cohomology of its corresponding étale $\mathbb{Z}_p$-local system. This generalizes Guo--Reneicke's prismatic--étale comparison for crystalline $\mathbb{Z}_p$-local systems to the semi-stable case |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08451 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A prismatic-etale comparison theorem in the semistable case Tian, Yichao Algebraic Geometry Number Theory 14F30, 14F20 Let $K|\mathbb{Q}_p$ be a complete discrete valuation field with perfect residue field, $O_K$ be its ring of integers. Consider a semistable $p$-adic formal scheme $X$ over $\mathrm{Spf}(O_K)$ with smooth generic fiber $X_η$. Du--Liu--Moon--Shimizu showed recently that the category of analytic prismatic $F$-crystals on the absolute log prismatic site of $X$ is equivalent to the category of semistable étale $\mathbb{Z}_p$-local systems on the adic generic fiber $X_η$. In this article, we prove a comparison between the Breuil--Kisin cohomology of an analytic log prismatic $F$-crystal on $X$ and the étale cohomology of its corresponding étale $\mathbb{Z}_p$-local system. This generalizes Guo--Reneicke's prismatic--étale comparison for crystalline $\mathbb{Z}_p$-local systems to the semi-stable case |
| title | A prismatic-etale comparison theorem in the semistable case |
| topic | Algebraic Geometry Number Theory 14F30, 14F20 |
| url | https://arxiv.org/abs/2507.08451 |