Berthelot’s conjecture predicts that under a proper and smooth morphism of schemes in characteristic p, the higher direct images of an overconvergent F-isocrystal are overconvergent 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 ...
Berthelot’s conjecture predicts that under a proper and smooth morphism of schemes in characteristic p, the higher direct images of an overconvergent F-isocrystal are overconvergent 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.