Strongly divisible lattices and crystalline cohomology in the imperfect residue field case

Abstract

Let k be a perfect field of characteristic $p \ge 3$, and let K be a finite totally ramified extension of $K_0 = W(k)[p^{-1}]$. Let $L_0$ be a complete discrete valuation field over $K_0$ whose residue field has a finite p-basis, and let $L = L_0\otimes _{K_0} K$. For $0 \le r \le p-2$, we classify $\textbf{Z}_p$-lattices of semistable representations of $\textrm{Gal}(\overline{L}/L)$ with Hodge–Tate weights in [0, r] by strongly divisible lattices. This generalizes the result of Liu (Compos Math 144:61–88, 2008). Moreover, if $\mathcal {X}$ is a proper smooth formal scheme over $\mathcal {O}_L$, we give a cohomological description of the strongly divisible lattice associated to $H^i_{\acute{\text {e}}\text {t}}(\mathcal {X}_{\overline{L}}, \textbf{Z}_p)$ for $i \le p-2$, under the assumption that the crystalline cohomology of the special fiber of $\mathcal {X}$ is torsion-free in degrees i and $i+1$. This generalizes a result in Cais and Liu (Trans Am Math Soc 371:1199–1230, 2019).