Stein domains in Banach algebraic geometry

Federico Bambozzi; Oren Ben-Bassat; Kobi Kremnizer

doi:10.1016/j.jfa.2018.01.003

Stein domains in Banach algebraic geometry

Federico Bambozzi, Oren Ben-Bassat, Kobi Kremnizer

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a derived perspective on analytic geometry.

Original language	English
Pages (from-to)	1865-1927
Number of pages	63
Journal	Journal of Functional Analysis
Volume	274
Issue number	7
DOIs	https://doi.org/10.1016/j.jfa.2018.01.003
State	Published - 1 Apr 2018

Bibliographical note

Funding Information:
The first author acknowledges the support of the University of Padova by MIUR PRIN2010-11 “Arithmetic Algebraic Geometry and Number Theory”, and the University of Regensburg with the support of the DFG funded CRC 1085 “Higher Invariants. Interactions between Arithmetic Geometry and Global Analysis” that permitted him to work on this project. The second author acknowledges the University of Oxford and the support of the European Commission under the Marie Curie Programme for the IEF grant STACKSCATS which enabled this research to take place. The contents of this article reflect the views of the authors and not the views of the European Commission. We would like to thank Konstantin Ardakov, Francesco Baldassarri, Elmar Grosse-Klönne, Frédéric Paugam, and Jérôme Poineau for interesting conversations.

Publisher Copyright:
© 2018 Elsevier Inc.

Keywords

Berkovich space
Bornological space
Nuclear space
Stein space

ASJC Scopus subject areas

Analysis

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10.1016/j.jfa.2018.01.003

Cite this

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abstract = "In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a derived perspective on analytic geometry.",

keywords = "Berkovich space, Bornological space, Nuclear space, Stein space",

author = "Federico Bambozzi and Oren Ben-Bassat and Kobi Kremnizer",

note = "Funding Information: The first author acknowledges the support of the University of Padova by MIUR PRIN2010-11 “Arithmetic Algebraic Geometry and Number Theory”, and the University of Regensburg with the support of the DFG funded CRC 1085 “Higher Invariants. Interactions between Arithmetic Geometry and Global Analysis” that permitted him to work on this project. The second author acknowledges the University of Oxford and the support of the European Commission under the Marie Curie Programme for the IEF grant STACKSCATS which enabled this research to take place. The contents of this article reflect the views of the authors and not the views of the European Commission. We would like to thank Konstantin Ardakov, Francesco Baldassarri, Elmar Grosse-Kl{\"o}nne, Fr{\'e}d{\'e}ric Paugam, and J{\'e}r{\^o}me Poineau for interesting conversations. Publisher Copyright: {\textcopyright} 2018 Elsevier Inc.",

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AU - Ben-Bassat, Oren

AU - Kremnizer, Kobi

N1 - Funding Information: The first author acknowledges the support of the University of Padova by MIUR PRIN2010-11 “Arithmetic Algebraic Geometry and Number Theory”, and the University of Regensburg with the support of the DFG funded CRC 1085 “Higher Invariants. Interactions between Arithmetic Geometry and Global Analysis” that permitted him to work on this project. The second author acknowledges the University of Oxford and the support of the European Commission under the Marie Curie Programme for the IEF grant STACKSCATS which enabled this research to take place. The contents of this article reflect the views of the authors and not the views of the European Commission. We would like to thank Konstantin Ardakov, Francesco Baldassarri, Elmar Grosse-Klönne, Frédéric Paugam, and Jérôme Poineau for interesting conversations. Publisher Copyright: © 2018 Elsevier Inc.

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N2 - In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a derived perspective on analytic geometry.

AB - In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a derived perspective on analytic geometry.

KW - Berkovich space

KW - Bornological space

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KW - Stein space

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Stein domains in Banach algebraic geometry

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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