Relative Fontaine-Messing theory over power series rings

TL;DR

This paper extends Fontaine-Messing theory to power series rings over Witt vectors, establishing comparison theorems between torsion crystalline and étale cohomology for smooth proper schemes over such rings.

Contribution

It introduces a relative Fontaine-Messing theory over power series rings, providing new comparison theorems in this setting.

Findings

Established comparison theorems between torsion crystalline and étale cohomology.

Extended classical Fontaine-Messing theory to a new base ring setting.

Provided foundational results for p-adic Hodge 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 \'etale cohomology in this setting.