Images directes III: F-isocristaux surconvergents

TL;DR

This paper proves the overconvergence of direct images of overconvergent F-isocrystals under proper smooth morphisms, advancing understanding of p-adic cohomology and confirming a conjecture by Berthelot.

Contribution

It establishes the overconvergence of direct images of overconvergent F-isocrystals for liftable proper smooth morphisms, partially confirming Berthelot's conjecture.

Findings

Overconvergence of direct images proven for certain morphisms

Supports Berthelot's conjecture on overconvergent F-isocrystals

Extends previous work on direct images in p-adic cohomology

Abstract

This article is the third one of a series of three articles devoted to direct images of isocrystals: here we consider overconvergent isocrystals with Frobenius structure. For a liftable proper smooth morphism we establish the overconvergence of direct images, owing to the first article and the existence of lifts of Frobenius. This result partially answers a conjecture of Berthelot on the overconvergence of direct images of overconvergent F-isocrystals under a proper smooth morphism.