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
Categories: math.AG math.NT