#### Vol. 14, No. 10, 2020

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

### Tong Liu and Yong Suk Moon

Vol. 14 (2020), No. 10, 2773–2789
##### 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 $W\left(k\right)⟨{t}_{1}^{±1},\dots ,{t}_{m}^{±1}⟩$ satisfying some mild conditions, and let $R={R}_{0}{\otimes }_{W\left(k\right)}{\mathsc{𝒪}}_{K}$. We show that if $e, then every crystalline representation of ${\pi }_{1}^{\stackrel{́}{e}t}\left(Spec\phantom{\rule{0.3em}{0ex}}R\left[\frac{1}{p}\right]\right)$ with Hodge–Tate weights in $\left[0,1\right]$ arises from a $p$-divisible group over $R$.

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crystalline representation, $p$-divisible group, relative $p$-adic Hodge theory