Geometric syntomic cohomology and vector bundles on the Fargues-Fontaine curve

Nizioł, Wiesława

doi:10.1090/jag/742

Author: Wiesława Nizioł
Journal: J. Algebraic Geom. 28 (2019), 605-648
DOI: https://doi.org/10.1090/jag/742
Published electronically: June 28, 2019
MathSciNet review: 3994308
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Abstract | References | Additional Information

Abstract: We show that geometric syntomic cohomology lifts canonically to the category of Banach-Colmez spaces and study its relation to extensions of modifications of vector bundles on the Fargues-Fontaine curve. We include some computations of geometric syntomic cohomology spaces: they are finite rank $\mathbf {Q}_p$-vector spaces for ordinary varieties, but in the nonordinary case, these cohomology spaces carry much more information; in particular they can have a nontrivial $C$-rank. This dichotomy is reminiscent of the Hodge-Tate period map for $p$-divisible groups.

References

Yves André, Slope filtrations, Confluentes Math. 1 (2009), no. 1, 1–85. MR 2571693, DOI https://doi.org/10.1142/S179374420900002X
A. A. Beĭlinson, Notes on absolute Hodge cohomology, Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 35–68. MR 862628, DOI https://doi.org/10.1090/conm/055.1/862628
A. Beilinson, $p$-adic periods and derived de Rham cohomology, J. Amer. Math. Soc. 25 (2012), no. 3, 715–738. MR 2904571, DOI https://doi.org/10.1090/S0894-0347-2012-00729-2
A. Beilinson, On the crystalline period map, Camb. J. Math. 1 (2013), no. 1, 1–51. MR 3272051, DOI https://doi.org/10.4310/CJM.2013.v1.n1.a1
Laurent Berger, Construction de $(\phi ,\Gamma )$-modules: représentations $p$-adiques et $B$-paires, Algebra Number Theory 2 (2008), no. 1, 91–120 (French, with English and French summaries). MR 2377364, DOI https://doi.org/10.2140/ant.2008.2.91
Pierre Colmez, Espaces de Banach de dimension finie, J. Inst. Math. Jussieu 1 (2002), no. 3, 331–439 (French, with English and French summaries). MR 1956055, DOI https://doi.org/10.1017/S1474748002000099
Pierre Colmez, Espaces vectoriels de dimension finie et représentations de de Rham, Astérisque 319 (2008), 117–186 (French, with English and French summaries). Représentations $p$-adiques de groupes $p$-adiques. I. Représentations galoisiennes et $(\phi ,\Gamma )$-modules. MR 2493217
Pierre Colmez and Wiesława Nizioł, Syntomic complexes and $p$-adic nearby cycles, Invent. Math. 208 (2017), no. 1, 1–108. MR 3621832, DOI https://doi.org/10.1007/s00222-016-0683-3
Frédéric Déglise and Wiesława Nizioł, On $p$-adic absolute Hodge cohomology and syntomic coefficients. I, Comment. Math. Helv. 93 (2018), no. 1, 71–131. MR 3777126, DOI https://doi.org/10.4171/CMH/430
Laurent Fargues and Jean-Marc Fontaine, Courbes et fibrés vectoriels en théorie de Hodge $p$-adique, Astérisque 406 (2018), xiii+382 (French, with English and French summaries). With a preface by Pierre Colmez. MR 3917141
Laurent Fargues and Jean-Marc Fontaine, Vector bundles on curves and $p$-adic Hodge theory, Automorphic forms and Galois representations. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 415, Cambridge Univ. Press, Cambridge, 2014, pp. 17–104. MR 3444231
Laurent Fargues, Quelques résultats et conjectures concernant la courbe, Astérisque 369 (2015), 325–374 (French, with English and French summaries). MR 3379639
Jean-Marc Fontaine, Presque $C_p$-représentations, Doc. Math. Extra Vol. (2003), 285–385 (French, with English summary). Kazuya Kato’s fiftieth birthday. MR 2046603
Annette Huber, Mixed motives and their realization in derived categories, Lecture Notes in Mathematics, vol. 1604, Springer-Verlag, Berlin, 1995. MR 1439046
Kiran S. Kedlaya, Slope filtrations revisited, Doc. Math. 10 (2005), 447–525. MR 2184462
Arthur-César Le Bras, Espaces de Banach-Colmez et faisceaux cohérents sur la courbe de Fargues-Fontaine, Duke Math. J. 167 (2018), no. 18, 3455–3532 (French, with English and French summaries). MR 3881201, DOI https://doi.org/10.1215/00127094-2018-0034
Jan Nekovář and Wiesława Nizioł, Syntomic cohomology and $p$-adic regulators for varieties over $p$-adic fields, Algebra Number Theory 10 (2016), no. 8, 1695–1790. With appendices by Laurent Berger and Frédéric Déglise. MR 3556797, DOI https://doi.org/10.2140/ant.2016.10.1695
Bernadette Perrin-Riou, Représentations $p$-adiques ordinaires, Astérisque 223 (1994), 185–220 (French). With an appendix by Luc Illusie; Périodes $p$-adiques (Bures-sur-Yvette, 1988). MR 1293973
Jean-Pierre Schneiders, Quasi-abelian categories and sheaves, Mém. Soc. Math. Fr. (N.S.) 76 (1999), vi+134 (English, with English and French summaries). MR 1779315
Peter Scholze, $p$-adic Hodge theory for rigid-analytic varieties, Forum Math. Pi 1 (2013), e1, 77. MR 3090230, DOI https://doi.org/10.1017/fmp.2013.1

Additional Information

Wiesława Nizioł
Affiliation: UMPA, École Normale Supérieure de Lyon, 46, allée d’Italie, 69007 Lyon, France
Email: wieslawa.niziol@ens-lyon.fr

Received by editor(s): June 9, 2016
Received by editor(s) in revised form: February 12, 2019, and April 9, 2019
Published electronically: June 28, 2019
Additional Notes: The author’s research was supported in part by the grant ANR-14-CE25.
Article copyright: © Copyright 2019 University Press, Inc.