A prismatic-etale comparison theorem in the semistable case

Let $K\mathrm{\mid }{\mathbb{Q}}_p$ 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 $\mathrm{S}\mathrm{p}\mathrm{f}(O_K)$ 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 étale ${\mathbb{Z}}_p$-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 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