Images directes I: Espaces rigides analytiques et images directes

TL;DR

This paper proves the overconvergence of direct images of isocrystals without Frobenius structure for proper smooth morphisms in rigid analytic geometry, addressing a conjecture of Berthelot.

Contribution

It establishes overconvergence of direct images in the absence of Frobenius structure, advancing the understanding of isocrystals in rigid analytic spaces.

Findings

Overconvergence of direct images proven for proper smooth morphisms.

Partial resolution of Berthelot's conjecture on overconvergence.

Framework for future inclusion of Frobenius structures in subsequent articles.

Abstract

This article is the first one of a series of three articles devoted to direct images of isocrystals: here we consider isocrystals without Frobenius structure; in the second one (resp. the third one), we will introduce a Frobenius structure in the convergent (resp. overconvergent) context. For a liftable proper smooth morphism we establish the overconvergence of direct images, owing to a base change theorem for a proper morphism between rigid analytic spaces. This result partially answers a conjecture of Berthelot on the overconvergence of direct images under a proper smooth morphism.