Relative Fontaine-Messing theory over power series rings

Abstract

Let $k$ be a perfect field of characteristic $p>2$, $R:=W(k)[[t_1,\dots ,t_d]]$ be the power series ring over the Witt vectors, and $X$ be a smooth proper scheme over $R$. The main goal of this article is to extend classical Fontaine-Messing theory to the setting where the base ring is $R$. In particular, we obtain comparison theorems between torsion crystalline cohomology of $X/R$ and torsion étale cohomology in this setting.