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

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).