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
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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.

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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.