Simple connectivity of Fargues–Fontaine curves
Annales Henri Lebesgue, Volume 4 (2021) , pp. 1203-1233.

Metadata

Keywordsperfectoid spaces, Fargues–Fontaine curves, Drinfeld’s lemma

Abstract

We show that the Fargues–Fontaine curve associated to an algebraically closed field of characteristic p is geometrically simply connected; that is, its base extension from p to any complete algebraically closed overfield admits no nontrivial connected finite étale covering. We then deduce from this an analogue for perfectoid spaces (and some related objects) of Drinfeld’s lemma on the fundamental group of a product of schemes in characteristic p.


References

[Ber90] Berkovich, Vladimir G. Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, 1990 | MR 1070709 | Zbl 0715.14013

[BGR84] Bosch, Siegfried; Güntzer, Ulrich; Remmert, Reinhold Non-Archimedean Analysis. A systematic approach to rigid analytic geometry, Grundlehren der Mathematischen Wissenschaften, 261, Springer, 1984 | Zbl 0539.14017

[Bou07] Bourbaki, Nicolas Éléments de mathématique. Algèbre. Chapitres 4 à 7, Springer, 2007 | Zbl 1139.12001

[CD94] Christol, Gilles; Dwork, Bernard Modules différentielles sur les couronnes, Ann. Inst. Fourier, Volume 44 (1994) no. 3, pp. 663-701 | Article | Zbl 0859.12004

[CKZ21] Carter, Annie T.; Kedlaya, Kiran S.; Zábrádi, Gergely Drinfeld’s lemma for perfectoid spaces and overconvergence of multivariate (φ,Γ)-modules (2021) (https://arxiv.org/abs/1808.03964v4)

[CTT16] Cohen, Adina; Temkin, Michael; Trushkin, Dmitri Morphisms of Berkovich curves and the different function, Adv. Math., Volume 303 (2016), pp. 800-858 | Article | MR 3552539 | Zbl 1375.14089

[Duc14] Ducros, Antoine La Structure des Courbes Analytiques, 2014 (draft, available at https://webusers.imj-prg.fr/~antoine.ducros/livre.html)

[FF18] Fargues, Laurent; Fontaine, Jean-Marc Courbes et fibrés vectoriels en théorie de Hodge p-adique, Astérisque, 406, Société Mathématique de France, 2018 | Zbl 07005651

[Gro57] Grothendieck, Alexander Sur la classification des fibrés holomorphes sur la sphère de Riemann, Am. J. Math., Volume 79 (1957), pp. 121-138 | Article | Zbl 0079.17001

[HP04] Hartl, Urs; Pink, Richard Vector bundles with a Frobenius structure on the punctured unit disc, Compos. Math., Volume 140 (2004) no. 3, pp. 689-716 | Article | MR 2041777 | Zbl 1074.14028

[Hub96] Huber, Roland Étale Cohomology of Rigid Analytic Varieties and Adic Spaces, Aspects of Mathematics, E30, Vieweg & Sohn, 1996 | MR 1734903 | Zbl 0868.14010

[Kap42] Kaplansky, Irving Maximal fields with valuation, Duke Math. J., Volume 9 (1942), pp. 303-321 | MR 6161 | Zbl 0063.03135

[Kap45] Kaplansky, Irving Maximal fields with valuation. II, Duke Math. J., Volume 12 (1945), pp. 243-248 | MR 12276 | Zbl 0061.05506

[Ked04] Kedlaya, Kiran S. A p-adic local monodromy theorem, Ann. Math., Volume 160 (2004), pp. 93-184 | Article | MR 2119719 | Zbl 1088.14005

[Ked05] Kedlaya, Kiran S. Slope filtrations revisited, Doc. Math., Volume 10 (2005), pp. 447-525 errata, ibid. 12 (2007), 361–362; additional errata at http://kskedlaya.org/papers/ | MR 2184462 | Zbl 1081.14028

[Ked07] Kedlaya, Kiran S. Swan conductors for p-adic differential modules, I: A local construction, Algebra Number Theory, Volume 1 (2007) no. 3, pp. 269-300 | Article | MR 2361935 | Zbl 1184.11051

[Ked10] Kedlaya, Kiran S. p-adic Differential Equations, Cambridge Studies in Advanced Mathematics, 125, Cambridge University Press, 2010 | MR 2663480 | Zbl 1213.12009

[Ked11] Kedlaya, Kiran S. Swan conductors for p-adic differential modules, II: Global variation, J. Inst. Math. Jussieu, Volume 10 (2011) no. 1, pp. 191-224 | Article | MR 2749575 | Zbl 1272.11128

[Ked15] Kedlaya, Kiran S. Local and global structure of connections on nonarchimedean curves, Compos. Math., Volume 151 (2015) no. 6, pp. 1096-1156 | Article | MR 3357180 | Zbl 1379.12007

[Ked16a] Kedlaya, Kiran S. Convergence polygons for connections on nonarchimedean curves, Nonarchimedean and Tropical Geometry. Based on two Simons Symposia, Island of St. John, March 31–April 6, 2013 and Puerto Rico, February 1–7, 2015 (Simons Symposia), Springer, 2016, pp. 51-97 | Zbl 1366.14024

[Ked16b] Kedlaya, Kiran S. Noetherian properties of Fargues–Fontaine curves, Int. Math. Res. Not., Volume 2016 (2016) no. 8, 2544–2567 | MR 3519123 | Zbl 1404.13026

[Ked19] Kedlaya, Kiran S. Sheaves, stacks, and shtukas, Perfectoid Spaces: Lectures from the 2017 Arizona Winter School (Mathematical Surveys and Monographs), Volume 242, American Mathematical Society, 2019, pp. 58-205 (with an introduction by Peter Scholze) | Zbl 1453.14074

[KL15] Kedlaya, Kiran S.; Liu, Ruochan Relative p-adic Hodge theory: Foundations, Astérisque, 371, Société Mathématique de France, 2015 | Zbl 1370.14025

[KL19] Kedlaya, Kiran S.; Liu, Ruochan Relative p-adic Hodge theory, II: Imperfect period rings (2019) (https://arxiv.org/abs/1602.06899v3)

[Kra66] Krasner, Marc Prolongement analytique uniforme et multiforme dans les corps valués complets, Les Tendances Géométrique et Algébrique et Théorie des Nombres (Colloques Internationaux du Centre National de la Recherche Scientifique), Volume 143, Centre National de la Recherche Scientifique, 1966, pp. 97-141 | Zbl 0139.26202

[Kra74] Krasner, Marc Rapport sur le prolongement analytique dans les corps valués complets par la méthode des éléments analytique quasi-connexes, Bull. Soc. Math. Fr., Suppl., Mém., Volume 39–40 (1974), pp. 131-254 | Numdam | Zbl 0295.12104

[NS65] Narasimhan, M.S.; Seshadri, C.S. Stable and unitary vector bundles on a compact Riemann surface, Ann. Math., Volume 82 (1965), pp. 540-567 | Article | MR 184252 | Zbl 0171.04803

[Sch12] Scholze, Peter Perfectoid spaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 116 (2012), pp. 245-313 | Article | Numdam | MR 3090258 | Zbl 1263.14022

[Sch17] Scholze, Peter Étale cohomology of diamonds (2017) (preprint available at http://www.math.uni-bonn.de/people/scholze/EtCohDiamonds.pdf)

[Sta17] Stacks Project Authors Stacks Project, 2017 http://stacks.math.columbia.edu (retrieved Nov 2017)

[SW20] Scholze, Peter; Weinstein, Jared Berkeley Lectures on p-adic Geometry, Annals of Mathematics Studies, 207, Princeton University Press, 2020 | Zbl 07178476

[Tem17] Temkin, Michael Wild coverings of Berkovich curves, Actes de la conférence “Non-Archimedean analytic geometry: theory and practice” (Publications Mathématiques de Besançon. Algèbre et Théorie des Nombres), Volume 2017, Presses Universitaires de Franche-Comté, 2017 no. 1, pp. 127-135 | Numdam | MR 3752490 | Zbl 1428.14038

[Wei17] Weinstein, Jared Gal( ¯ p / p ) as a geometric fundamental group, Int. Math. Res. Not., Volume 2017 (2017) no. 10, pp. 2964-2997 | MR 3658130 | Zbl 1405.14048

[Wei19] Weinstein, Jared Adic spaces, Perfectoid Spaces. Lectures from the 20th Arizona Winter School, University of Arizona, Tucson, AZ, USA, March 11–17, 2017 (Mathematical Surveys and Monographs), Volume 242, American Mathematical Society, 2019, pp. 14-57 (with an introduction by Peter Scholze) | MR 3970252 | Zbl 1451.14083

[Xia10] Xiao, Liang On ramification filtrations and p-adic differential equations, I: equal characteristic case, Algebra Number Theory, Volume 4 (2010) no. 8, pp. 969-1027 | Article | Zbl 1225.11152

[Xia12] Xiao, Liang On ramification filtrations and p-adic differential equations, II: mixed characteristic case, Compos. Math., Volume 148 (2012) no. 2, pp. 415-463 | Article | MR 2904193 | Zbl 1266.11122