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

TL;DR

This paper classifies certain p-adic Galois representations using strongly divisible lattices in the context of imperfect residue fields and extends cohomological descriptions of these lattices for smooth formal schemes.

Contribution

It generalizes Liu's classification of lattices of semistable representations and provides a cohomological description of strongly divisible lattices in the imperfect residue field case.

Findings

Classifies Z_p-lattices of semistable representations with bounded Hodge-Tate weights.

Provides a cohomological description of strongly divisible lattices for proper smooth formal schemes.

Extends previous results to the imperfect residue field setting.

Abstract

Let $k$ be a perfect field of characteristic $p \geq 3$, 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 $0 \leq r \leq p-2$, 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 $H^i_{\text{\'et}}(\mathcal{X}_{\overline{L}}, \mathbf{Z}_p)$ for $i \leq p-2$, 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.