  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 \'etale cohomology in this setting.
