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. 32 (2023), 93-141
DOI:
https://doi.org/10.1090/jag/789
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.
References
- Giulia Battiston, Gieseker conjecture for homogeneous spaces, arXiv:1612.02154, 2016.
- Pierre Berthelot, Cohomologie cristalline des schemas de caractéristique $p>0$, French, Lect. Notes Math. 407, 1974.
- Pierre Berthelot, Géométrie rigide et cohomologie des variétés algébriques de caractéristique $p$. Mém. Soc. Math. France (N.S.) 23 (1986), no. 3, 7–32.
- Marcel Bökstedt and Amnon Neeman, Homotopy limits in triangulated categories. Compositio Math. 86 (1993), no. 2, 209–234.
- Pierre Berthelot and Arthur Ogus, Notes on crystalline cohomology, English. Princeton University Press, Princeton, N.J., 1978.
- Daniel Caro, Sur la préservation de la surconvergence par l’image directe d’un morphisme propre et lisse, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 1, 131–169.
- Marco D’Addezio, Slopes of $F$-isocrystals over abelian varieties, arXiv:2101.06257, 2021.
- P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977.
- P. Deligne, Catégories tannakiennes. (Tannaka categories), French, The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. II, Prog. Math. 87, 111–195, 1990.
- Vladimir Drinfeld, A stacky approach to crystals, arXiv:1810.11853, 2018.
- Valentina Di Proietto and Atsushi Shiho, On the homotopy exact sequence for log algebraic fundamental groups, Doc. Math. 23 (2018), 543–597.
- B. Eckmann and P. J. Hilton, Group-like structures in general categories. I. Multiplications and comultiplications, Math. Ann. 145 (1961/62), 227–255.
- Hélène Esnault, Phùng Hô Hai, and Xiaotao Sun, On Nori’s fundamental group scheme, English, Geometry and Dynamics of Groups and Spaces. Progress in Mathematics 265 (2008), 377–398.
- Hélène Esnault and Atsushi Shiho, Existence of locally free lattices of crystals, url: http://page.mi.fu-berlin.de/esnault/preprints/helene/119b_esn_shi.pdf, 2015.
- Hélène Esnault and Atsushi Shiho, Convergent isocrystals on simply connected varieties, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 5, 2109–2148.
- Hélène Esnault and Atsushi Shiho, Chern classes of crystals, Trans. Amer. Math. Soc., 371 (2019), no. 2, 1333–1358.
- Jean-Yves Etesse, Images directes I: Espaces rigides analytiques et images directes, J. Théor. Nombres Bordeaux 24 (2012), no. 1, 101–151.
- Robin Hartshorne, On the de Rham cohomology of algebraic varieties, English, Publications Mathématiques de l’IHÉS 45 (1975), 5–99.
- Luc Illusie, Grothendieck’s existence theorem in formal geometry, Fundamental algebraic geometry. volume 123. Math. Surveys Monogr. With a letter (in French) of Jean-Pierre Serre. Amer. Math. Soc., Providence, RI, 179–233, 2005.
- Efstathia Katsigianni, A note on rank 1 log extendable isocrystals on simply connected open varieties, arXiv:1703.04503, 2018.
- Nicholas M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, English, Publ. Math., Inst. Hautes Étud. Sci. 39 (1970), 175–232.
- Kiran S. Kedlaya, Errata to “Good formal structures for flat meromorphic connections. I: Surfaces,” [Duke Math. J. 154 (2010), 343-418], English, Duke Math. J. 161 (2012), no. 4, 733–734.
- Christopher Lazda, Incarnations of Berthelot’s conjecture, J. Number Theory 166 (2016), 137–157.
- Christopher Lazda and Ambrus Pál, A homotopy exact sequence for overconvergent isocrystals, arXiv:1704.07574, 2017.
- Matthew Morrow, A variational Tate conjecture in crystalline cohomology, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 11, 3467–3511.
- Arthur Ogus, F-isocrystals and de Rham cohomology. II: Convergent isocrystals, English, Duke Math. J. 51 (1984), 765–850.
- Arthur Ogus, The convergent topos in characteristic $p$, The Grothendieck Festschrift, Vol. III. volume 88. Progr. Math. Birkhäuser Boston, Boston, MA, 133–162, 1990.
- João Pedro dos Santos, The homotopy exact sequence for the fundamental group scheme and infinitesimal equivalence relations, Algebr. Geom. 2 (2015), no. 5, 535–590.
- Atsushi Shiho, Crystalline fundamental groups. I: Isocrystals on log crystalline site and log convergent site, English, J. Math. Sci., Tokyo 7 (2000), no. 4, 509–656.
- Atsushi Shiho, Crystalline fundamental groups. II. Log convergent cohomology and rigid cohomology, J. Math. Sci. Univ. Tokyo 9 (2002), no. 1, 1–163.
- Atsushi Shiho, Relative log convergent cohomology and relative rigid cohomology I, arXiv:0707.1742, 2018.
- Atsushi Shiho, Relative log convergent cohomology and relative rigid cohomology II, arXiv:0707.1743, 2008.
- Atsushi Shiho, Relative log convergent cohomology and relative rigid cohomology III, arXiv:0805.3229, 2008.
- Atsushi Shiho, A note on convergent isocrystals on simply connected varieties, arXiv:1411.0456, 2014.
- The Stacks project authors, The Stacks project, https://stacks.math.columbia.edu, 2019.
- Nobuo Tsuzuki, On base change theorem and coherence in rigid cohomology, Doc. Math. Extra Vol. (2003), 891–918.
- Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 2004.
- Daxin Xu, On higher direct images of convergent isocrystals, Compos. Math. 155 (2019), no. 11, 2180–2213.
- Lei Zhang, The homotopy sequence of the algebraic fundamental group, English, Int. Math. Res. Not. 2014 (2014), no. 22, 6155–6174.
References
- Giulia Battiston, Gieseker conjecture for homogeneous spaces, arXiv:1612.02154, 2016.
- Pierre Berthelot, Cohomologie cristalline des schemas de caractéristique $p>0$, French, Lect. Notes Math. 407, 1974.
- Pierre Berthelot, Géométrie rigide et cohomologie des variétés algébriques de caractéristique $p$. Mém. Soc. Math. France (N.S.) 23 (1986), no. 3, 7–32.
- Marcel Bökstedt and Amnon Neeman, Homotopy limits in triangulated categories. Compositio Math. 86 (1993), no. 2, 209–234.
- Pierre Berthelot and Arthur Ogus, Notes on crystalline cohomology, English. Princeton University Press, Princeton, N.J., 1978.
- Daniel Caro, Sur la préservation de la surconvergence par l’image directe d’un morphisme propre et lisse, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 1, 131–169.
- Marco D’Addezio, Slopes of $F$-isocrystals over abelian varieties, arXiv:2101.06257, 2021.
- P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977.
- P. Deligne, Catégories tannakiennes. (Tannaka categories), French, The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. II, Prog. Math. 87, 111–195, 1990.
- Vladimir Drinfeld, A stacky approach to crystals, arXiv:1810.11853, 2018.
- Valentina Di Proietto and Atsushi Shiho, On the homotopy exact sequence for log algebraic fundamental groups, Doc. Math. 23 (2018), 543–597.
- B. Eckmann and P. J. Hilton, Group-like structures in general categories. I. Multiplications and comultiplications, Math. Ann. 145 (1961/62), 227–255.
- Hélène Esnault, Phùng Hô Hai, and Xiaotao Sun, On Nori’s fundamental group scheme, English, Geometry and Dynamics of Groups and Spaces. Progress in Mathematics 265 (2008), 377–398.
- Hélène Esnault and Atsushi Shiho, Existence of locally free lattices of crystals, url: http://page.mi.fu-berlin.de/esnault/preprints/helene/119b_esn_shi.pdf, 2015.
- Hélène Esnault and Atsushi Shiho, Convergent isocrystals on simply connected varieties, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 5, 2109–2148.
- Hélène Esnault and Atsushi Shiho, Chern classes of crystals, Trans. Amer. Math. Soc., 371 (2019), no. 2, 1333–1358.
- Jean-Yves Etesse, Images directes I: Espaces rigides analytiques et images directes, J. Théor. Nombres Bordeaux 24 (2012), no. 1, 101–151.
- Robin Hartshorne, On the de Rham cohomology of algebraic varieties, English, Publications Mathématiques de l’IHÉS 45 (1975), 5–99.
- Luc Illusie, Grothendieck’s existence theorem in formal geometry, Fundamental algebraic geometry. volume 123. Math. Surveys Monogr. With a letter (in French) of Jean-Pierre Serre. Amer. Math. Soc., Providence, RI, 179–233, 2005.
- Efstathia Katsigianni, A note on rank 1 log extendable isocrystals on simply connected open varieties, arXiv:1703.04503, 2018.
- Nicholas M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, English, Publ. Math., Inst. Hautes Étud. Sci. 39 (1970), 175–232.
- Kiran S. Kedlaya, Errata to “Good formal structures for flat meromorphic connections. I: Surfaces,” [Duke Math. J. 154 (2010), 343-418], English, Duke Math. J. 161 (2012), no. 4, 733–734.
- Christopher Lazda, Incarnations of Berthelot’s conjecture, J. Number Theory 166 (2016), 137–157.
- Christopher Lazda and Ambrus Pál, A homotopy exact sequence for overconvergent isocrystals, arXiv:1704.07574, 2017.
- Matthew Morrow, A variational Tate conjecture in crystalline cohomology, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 11, 3467–3511.
- Arthur Ogus, F-isocrystals and de Rham cohomology. II: Convergent isocrystals, English, Duke Math. J. 51 (1984), 765–850.
- Arthur Ogus, The convergent topos in characteristic $p$, The Grothendieck Festschrift, Vol. III. volume 88. Progr. Math. Birkhäuser Boston, Boston, MA, 133–162, 1990.
- João Pedro dos Santos, The homotopy exact sequence for the fundamental group scheme and infinitesimal equivalence relations, Algebr. Geom. 2 (2015), no. 5, 535–590.
- Atsushi Shiho, Crystalline fundamental groups. I: Isocrystals on log crystalline site and log convergent site, English, J. Math. Sci., Tokyo 7 (2000), no. 4, 509–656.
- Atsushi Shiho, Crystalline fundamental groups. II. Log convergent cohomology and rigid cohomology, J. Math. Sci. Univ. Tokyo 9 (2002), no. 1, 1–163.
- Atsushi Shiho, Relative log convergent cohomology and relative rigid cohomology I, arXiv:0707.1742, 2018.
- Atsushi Shiho, Relative log convergent cohomology and relative rigid cohomology II, arXiv:0707.1743, 2008.
- Atsushi Shiho, Relative log convergent cohomology and relative rigid cohomology III, arXiv:0805.3229, 2008.
- Atsushi Shiho, A note on convergent isocrystals on simply connected varieties, arXiv:1411.0456, 2014.
- The Stacks project authors, The Stacks project, https://stacks.math.columbia.edu, 2019.
- Nobuo Tsuzuki, On base change theorem and coherence in rigid cohomology, Doc. Math. Extra Vol. (2003), 891–918.
- Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 2004.
- Daxin Xu, On higher direct images of convergent isocrystals, Compos. Math. 155 (2019), no. 11, 2180–2213.
- Lei Zhang, The homotopy sequence of the algebraic fundamental group, English, Int. Math. Res. Not. 2014 (2014), no. 22, 6155–6174.
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
Email:
valentina.diproietto@gmail.com
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
Email:
jacobbb84@gmail.com
Lei Zhang
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
ORCID:
0000-0001-5451-8102
Email:
cumt559@gmail.com
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
Article copyright:
© Copyright 2022
University Press, Inc.
