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.
Bibliographical noteFunding 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.
© 2018 Elsevier Inc.
- Berkovich space
- Bornological space
- Nuclear space
- Stein space
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