A prismatic-etale comparison theorem in the semistable case

TL;DR

This paper establishes a comparison theorem linking Breuil--Kisin cohomology of log prismatic F-crystals with étale cohomology of associated local systems in the semistable setting, extending previous crystalline case results.

Contribution

It proves a new comparison between Breuil--Kisin cohomology and étale cohomology for semistable schemes, generalizing prior crystalline case results.

Findings

Proves a comparison theorem for semistable schemes.

Extends prismatic-étale comparison to the semistable case.

Links cohomology theories for log prismatic F-crystals and étale local systems.

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 $X_{\eta}$. 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 \'etale $\mathbb{Z}_p$-local systems on the adic generic fiber $X_{\eta}$. In this article, we prove a comparison between the Breuil--Kisin cohomology of an analytic log prismatic $F$-crystal on $X$ and the \'etale cohomology of its corresponding \'etale $\mathbb{Z}_p$-local system. This generalizes Guo--Reneicke's prismatic--\'etale comparison for crystalline $\mathbb{Z}_p$-local systems to the semi-stable case