Relative Fontaine–Messing Theory Over Power Series Rings

Abstract

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 [ 13] 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.