Un résultat de connexité pour les variétés analytiques p-adiques: privilège et noethérianité

Jérôme Poineau

doi:10.1112/S0010437X07003016

Un résultat de connexité pour les variétés analytiques p-adiques: privilège et noethérianité

Part of:Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press: 01 January 2008

Jérôme Poineau

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Jérôme Poineau*: Affiliation:
IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France (email: jerome.poineau@univ-rennes1.fr)

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Abstract

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Abstract

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Let k be a non-Archimedean field, let X be a k-affinoid space and let f1,…,fn, with $n\in \mathbb {N}^*$ , be analytic functions over X. If X is irreducible, we prove that the analytic domain $\bigcup _{1\le j\le n} \{x\in X\mid |f_{j}(x)|\ge \varepsilon _{j}\}$ is still irreducible, provided that $(\varepsilon _{1},\ldots ,\varepsilon _{n}) \in \mathbb {R}_{+}^n$ is small enough. Then, for a general X, we precisely describe how the geometric connected components of the spaces $\{x\in X\mid |f(x)|\ge \varepsilon \}$ behave with regards to ε. Finally, we obtain a result concerning privileged neighbourhoods and adapt a theorem from complex analytic geometry about Noetherianity for germs of analytic functions.