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 $p$, 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 a Künneth formula for the crystalline fundamental group.

Revised version following a referee report

References in corpus (4)

• Relative log convergent cohomology and relative rigid cohomology I
• The monodromy groups of lisse sheaves and overconvergent $F$-isocrystals
• A note on convergent isocrystals on simply connected varieties
• Gieseker conjecture for homogeneous spaces

Cited by in corpus (1)

• A theorem on meromorphic descent and the specialization of the pro-étale fundamental group