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 ${\mathbf{Z}}_p$-lattices of semistable representations of $\mathrm{Gal}(\overline{L}/L)$ with Hodge-Tate weights in $[0,r]$ by strongly divisible lattices. This generalizes the result of Liu. 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 $H_{\text{eˊt}}^i({\mathcal{X}}_{\overline{L}},{\mathbf{Z}}_p)$ 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-Liu.

arXiv commentAdded the case of semistable representations. To appear in Selecta Math. (N.S.)