A prismatic-etale comparison theorem in the semistable case

Abstract

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 $X_{\hat{I}}\cdot$. 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 $X_{\hat{I}}\cdot$. 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

61 pages