A crystalline incarnation of Berthelot’s conjecture and Künneth formula for isocrystals

Abstract

Berthelot’s conjecture predicts that under a proper and smooth morphism of schemes in characteristic p p , the higher direct images of an overconvergent F F -isocrystal are overconvergent F F -isocrystals. In this paper we prove that this is true for crystals up to isogeny. As an application we prove the Künneth formula for the crystalline fundamental group scheme.