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NORMAL AND IRREDUCIBLE ADIC SPACES, THE OPENNESS OF FINITE MORPHISMS, AND A STEIN FACTORIZATION

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  16 December 2022

LUCAS MANN Open the ORCID record for LUCAS MANN [Opens in a new window]
LUCAS MANN*
Affiliation:
Mathematisches Institut Westfälische Wilhelms-Universität Münster Einsteinstraße 62 48149 Münster Germany
*
mann.lucas@uni-muenster.de
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Abstract

We transfer several elementary geometric properties of rigid-analytic spaces to the world of adic spaces, more precisely to the category of adic spaces which are locally of (weakly) finite type over a non-archimedean field. This includes normality, irreducibility (in particular, irreducible components), and a Stein factorization theorem. Most notably, we show that a finite morphism in our category of adic spaces is automatically open if the target is normal and both source and target are of the same pure dimension. Moreover, our version of the Stein factorization theorem includes a statement about the geometric connectedness of fibers which we have not found in the literature of rigid-analytic or Berkovich spaces.

MSC classification

Primary: 14G22: Rigid analytic geometry 11G25: Varieties over finite and local fields
Type
Article
Information
Nagoya Mathematical Journal , Volume 250 , June 2023 , pp. 498 - 510
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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