Skip to Main Content
Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Notions of Stein spaces in non-Archimedean geometry

Authors: Marco Maculan and Jérôme Poineau
Journal: J. Algebraic Geom. 30 (2021), 287-330
Published electronically: July 27, 2020
MathSciNet review: 4233184
Full-text PDF

Abstract | References | Additional Information


Let $k$ be a non-Archimedean complete valued field and let $X$ be a $k$-analytic space in the sense of Berkovich. In this note, we prove the equivalence between three properties: (1) for every complete valued extension $k’$ of $k$, every coherent sheaf on $X \times _{k} k’$ is acyclic; (2) $X$ is Stein in the sense of complex geometry (holomorphically separated, holomorphically convex), and higher cohomology groups of the structure sheaf vanish (this latter hypothesis is crucial if, for instance, $X$ is compact); (3) $X$ admits a suitable exhaustion by compact analytic domains considered by Liu in his counter-example to the cohomological criterion for affinoidicity.

When $X$ has no boundary the characterization is simpler: in (2) the vanishing of higher cohomology groups of the structure sheaf is no longer needed, so that we recover the usual notion of Stein space in complex geometry; in (3) the domains considered by Liu can be replaced by affinoid domains, which leads us back to Kiehl’s definition of Stein space.

References [Enhancements On Off] (What's this?)

  • Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990. MR 1070709
  • Vladimir G. Berkovich, Étale cohomology for non-Archimedean analytic spaces, Inst. Hautes Études Sci. Publ. Math. 78 (1993), 5–161 (1994). MR 1259429
  • S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry. MR 746961
  • H. Cartan, Faisceaux analytiques sur les variétés de Stein, vol. 4, Séminaire Cartan, no. 18, 1951/1952, pp. 1–10.
  • H. Cartan, Faisceaux analytiques sur les variétés de Stein : démonstration des théorèmes fondamentaux, vol. 4, Séminaire Cartan, no. 19, 1951/1952, pp. 1–15.
  • H. Cartan, Théorie de la convexité (II), vol. 4, Séminaire Cartan, no. 9, 1951/1952, pp. 1–12.
  • Antoine Ducros, Parties semi-algébriques d’une variété algébrique $p$-adique, Manuscripta Math. 111 (2003), no. 4, 513–528 (French, with English summary). MR 2002825, DOI
  • O. Goldman and N. Iwahori, The space of ${\mathfrak p}$-adic norms, Acta Math. 109 (1963), 137–177. MR 144889, DOI
  • Hans Grauert and Reinhold Remmert, Theory of Stein spaces, Classics in Mathematics, Springer-Verlag, Berlin, 2004. Translated from the German by Alan Huckleberry; Reprint of the 1979 translation. MR 2029201
  • Laurent Gruson, Théorie de Fredholm $p$-adique, Bull. Soc. Math. France 94 (1966), 67–95 (French). MR 226381
  • Reinhardt Kiehl, Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie, Invent. Math. 2 (1967), 256–273 (German). MR 210949, DOI
  • Ludger Kaup and Burchard Kaup, Holomorphic functions of several variables, De Gruyter Studies in Mathematics, vol. 3, Walter de Gruyter & Co., Berlin, 1983. An introduction to the fundamental theory; With the assistance of Gottfried Barthel; Translated from the German by Michael Bridgland. MR 716497
  • Qing Liu, Un contre-exemple au “critère cohomologique d’affinoïdicité”, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 2, 83–86 (French, with English summary). MR 954265
  • Qing Liu, Sur les espaces de Stein quasi-compacts en géométrie rigide, Sém. Théor. Nombres Bordeaux (2) 1 (1989), no. 1, 51–58 (French, with English summary). MR 1050264
  • Qing Liu, Sur les espaces de Stein quasi-compacts en géométrie rigide, Tohoku Math. J. (2) 42 (1990), no. 3, 289–306 (French). MR 1066662, DOI
  • Werner Lütkebohmert, Steinsche Räume in der nichtarchimedischen Funktionentheorie, Schr. Math. Inst. Univ. Münster (2) 6 (1973), ii+55 (German). MR 330507
  • Jérôme Poineau, Les espaces de Berkovich sont angéliques, Bull. Soc. Math. France 141 (2013), no. 2, 267–297 (French, with English and French summaries). MR 3081557, DOI
  • J. Poineau and A. Pulita, Banachoid spaces, in preparation.
  • M. Temkin, On local properties of non-Archimedean analytic spaces. II, Israel J. Math. 140 (2004), 1–27. MR 2054837, DOI

Additional Information

Marco Maculan
Affiliation: Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4 place Jussieu, F-75252 Paris, France
MR Author ID: 1200158

Jérôme Poineau
Affiliation: Laboratoire de mathématiques Nicolas Oresme, Université de Caen Normandie, BP 5186, F-14032 Caen Cedex, France

Received by editor(s): December 22, 2018
Published electronically: July 27, 2020
Additional Notes: The first author was partially supported by ANR grant ANR-18-CE40-0017. The second author was partially supported by the ANR project “GLOBES”: ANR-12-JS01-0007-01 and ERC Starting Grant “TOSSIBERG”: 637027.
Article copyright: © Copyright 2020 University Press, Inc.