Equivariant crystalline cohomology and base change

TL;DR

This paper establishes a connection between the reduction modulo p of crystalline cohomology and de Rham cohomology for smooth proper schemes over a perfect field, incorporating group actions and base change properties.

Contribution

It introduces a base change theorem for virtual G-representations in the derived category and relates crystalline and de Rham cohomologies under group actions.

Findings

Reduction modulo p of crystalline cohomology equals de Rham cohomology for G-equivariant objects.

Proves a base change theorem for virtual G-representations.

Crystalline cohomology can be understood via de Rham cohomology in this setting.

Abstract

Given a perfect field $k$ of characteristic $p>0$, a smooth proper $k$-scheme $Y$, a crystal $E$ on $Y$ relative to $W(k)$ and a finite group $G$ acting on $Y$ and $E$, we show that, viewed as virtual $k[G]$-module, the reduction modulo $p$ of the crystalline cohomology of $E$ is the de Rham cohomology of $E$ modulo $p$. On the way we prove a base change theorem for the virtual $G$-representions associated with $G$-equivariant objects in the derived category of $W(k)$-modules.