On higher direct images of convergent isocrystals

TL;DR

This paper provides a new proof of Frobenius descent for convergent isocrystals, enabling a Frobenius-free de Rham comparison and supporting Berthelot's conjecture on higher direct images.

Contribution

It introduces a novel proof technique for Frobenius descent that does not require Frobenius lifting, advancing the understanding of isocrystals and their behavior under proper morphisms.

Findings

Established a Frobenius descent without Frobenius lifting.

Derived an analogue of the de Rham complexes comparison theorem.

Proved a version of Berthelot's conjecture on higher direct images.

Abstract

Let k be a perfect field of characteristic p>0 and W the ring of Witt vectors of k. In this article, we give a new proof of the Frobenius descent for convergent isocrystals on a variety over k relative to W. This proof allows us to deduce an analogue of the de Rham complexes comparaison theorem of Berthelot without assuming a lifting of the Frobenius morphism. As an application, we prove a version of Berthelot's conjecture on the preservation of convergent isocrystals under the higher direct image by a smooth proper morphism of k-varieties.