F-isocrystals of Higher Direct Images of $p$-Divisible Groups

TL;DR

This paper proves that for a p-divisible group over a smooth projective variety, the higher direct images form isogenous p-divisible groups and relates their Dieudonné crystals to slope parts of crystalline cohomology, answering a question of Artin--Mazur.

Contribution

It establishes a canonical isomorphism between the Dieudonné crystal of the divisible part and the slope-[0,1] component of crystalline cohomology, linking formal groups and F-isocrystals.

Findings

Higher direct images of p-divisible groups are isogenous to p-divisible groups.

Dieudonné crystals correspond to slope-[0,1] parts of crystalline cohomology.

Provides an answer to the rational form of Artin--Mazur's question.

Abstract

For a $p$-divisible group $G$ over a smooth projective variety $X$ over $k$, where $k$ is a field finitely generated over a perfect field of characteristic $p$, we show that the formal group $R^i f_{\fppf*} G$ is isogenous to a $p$-divisible group. The Dieudonn\'e crystal of its divisible part is canonically isomorphic to the slope-$[0,1]$ part of $R^i f_{\crys*} \cM^{cr}(G)$ in the category of $F$-isocrystals over $k$. This provides an answer to the rational form of a question of Artin--Mazur regarding the enlarged formal Brauer groups.