Relative crystalline representations and $p$-divisible groups in the small ramification case

Abstract

Let $k$ be a perfect field of characteristic $p>2$, and let $K$ be a finite totally ramified extension over $W\left(k\right)\left[\frac{1}{p}\right]$ of ramification degree $e$. Let $R_0$ be a relative base ring over $W\left(k\right)⟨t_1^{\pm 1},\dots ,t_m^{\pm 1}⟩$ satisfying some mild conditions, and let $R=R_0{\otimes }_{W\left(k\right)}{\mathcal{𝒪}}_K$. We show that if $e<p-1$, then every crystalline representation of ${\pi }_1^{\acute{e}t}\left(SpecR\left[\frac{1}{p}\right]\right)$ with Hodge–Tate weights in $\left[0,1\right]$ arises from a $p$-divisible group over $R$.