Notions of Stein spaces in non-Archimedean geometry
Authors:
Marco Maculan and Jérôme Poineau
Journal:
J. Algebraic Geom. 30 (2021), 287-330
DOI:
https://doi.org/10.1090/jag/764
Published electronically:
July 27, 2020
MathSciNet review:
4233184
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Abstract |
References |
Additional Information
Abstract:
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
- 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 https://doi.org/10.1007/s00229-003-0382-4
- O. Goldman and N. Iwahori, The space of ${\mathfrak p}$-adic norms, Acta Math. 109 (1963), 137–177. MR 144889, DOI https://doi.org/10.1007/BF02391811
- 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 https://doi.org/10.1007/BF01425404
- 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 https://doi.org/10.2748/tmj/1178227611
- 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 https://doi.org/10.24033/bsmf.2648
- 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 https://doi.org/10.1007/BF02786625
References
- 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. (1993), no. 78, 5–161 (1994). MR 1259429
- S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis: A systematic approach to rigid analytic geometry, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984. 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 https://doi.org/10.1007/s00229-003-0382-4
- O. Goldman and N. Iwahori, The space of ${\mathfrak {p}}$-adic norms, Acta Math. 109 (1963), 137–177. MR 144889, DOI https://doi.org/10.1007/BF02391811
- Hans Grauert and Reinhold Remmert, Theory of Stein spaces, translated from the German by Alan Huckleberry, reprint of the 1979 translation, Classics in Mathematics, Springer-Verlag, Berlin, 2004. 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 https://doi.org/10.1007/BF01425404
- Ludger Kaup and Burchard Kaup, Holomorphic functions of several variables, An introduction to the fundamental theory, with the assistance of Gottfried Barthel, translated from the German by Michael Bridgland, De Gruyter Studies in Mathematics, vol. 3, Walter de Gruyter & Co., Berlin, 1983. 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 https://doi.org/10.2748/tmj/1178227611
- 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 https://doi.org/10.24033/bsmf.2648
- 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 https://doi.org/10.1007/BF02786625
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
Email:
marco.maculan@imj-prg.fr
Jérôme Poineau
Affiliation:
Laboratoire de mathématiques Nicolas Oresme, Université de Caen Normandie, BP 5186, F-14032 Caen Cedex, France
Email:
jerome.poineau@unicaen.fr
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.