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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



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

Authors: Valentina Di Proietto, Fabio Tonini and Lei Zhang
Journal: J. Algebraic Geom.
Published electronically: January 12, 2022
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Abstract | References | Additional Information

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 the Künneth formula for the crystalline fundamental group scheme.

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Additional Information

Valentina Di Proietto
Affiliation: College of Engineering, Mathematics and Physical Sciences, University of Exeter, Streatham Campus, Exeter, EX4 4RN, United Kingdom
MR Author ID: 1028772
ORCID: 0000-0003-1237-1506

Fabio Tonini
Affiliation: Dipartimento di Matematica e Informatica Ulisse Dini, Università degli Studi di Firenze, Viale Morgagni 67/a, Firenze, 50134 Italy
MR Author ID: 931746
ORCID: 0000-0001-7784-7750

Lei Zhang
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
ORCID: 0000-0001-5451-8102

Received by editor(s): June 28, 2019
Received by editor(s) in revised form: May 19, 2021
Published electronically: January 12, 2022
Additional Notes: This work was supported by the European Research Council (ERC) Advanced Grant 0419744101 and the Einstein Foundation. Part of the revision of this work has been done while the first author was a guest of the IMPAN: her stay was supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015–2019 Polish MNiSW fund. The second author was supported by GNSAGA of INdAM
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