A connectedness result for \(p\)-adic analytic varieties: Privilege and noetherianity. (Un résultat de connexité pour les variétés analytiques \(p\)-adiques: Privilège et noethérianité.) (French) Zbl 1133.14026

Poineau, Jérôme

Compos. Math. 144, No. 1, 107-133 (2008).

Let \(k\) be a non-Archimedean field, let \(X\) be a \(k\)-affinoid space and let \(f_1,\ldots, f_n\) be analytic functions over \(X\). For \((\epsilon_1,\ldots, \epsilon_n)\in{\mathbb R}_{>0}^n\) let \[ V_{\epsilon}=\cup_{1\leq j\leq n}\{x\in X\quad| \quad | f_{j}(x)| \geq\epsilon_j\}. \] The first main result of this paper is that if \(X\) is irreducible and \((\epsilon_1,\ldots, \epsilon_n)\) is small enough, then also \(V_{\epsilon}\) is irreducible. If \(X\) is not necessarily irreducible, the set of irreducible (resp. connected) components of \(V_{\epsilon}\) is shown to vary in a certain tame manner with \(\epsilon\). This second result generalizes a principle used for the ramificaton theory for non-Archimedean fields as developed by A. Abbes and T. Saito [Am. J. Math. 124, 879–920 (2002; Zbl 1084.11064)]. These theorems may be understood and hold true in the context of rigid analytic geometry, but also in the context of Berkovich’s non-Archimedean analytic geometry. In fact, it is in the framework of the latter where the proofs are carried out. Besides the various topological properties of Berkovich’s analytic spaces (shared with classical complex analytic spaces) also some arguments from (formal) algebraic geometry are invoked, e.g. the following fact: for a morphism of finite type, the function which assigns to a point in the base scheme the number of geometric connected components of the corresponding fibre is locally constructible. In order to efficiently apply this fact, the reduced fibre theorem of S. Bosch, W. Lütkebohmert and M. Raynaud [Invent. Math. 119, No. 2, 361–398 (1995; Zbl 0839.14014)] is needed. In an appendix, a noetherianity result for germs of analytic functions from complex analytic geometry is adapted to the non-Archimedean setting. Its proof given here is logically quite independent of the main part of the paper, but an old proof [by J. Frisch, Invent. Math. 4, 118–138 (1967; Zbl 0167.06803)] of this result in the complex analytic setting motivated some of the concepts used in the present paper.

Reviewer: Elmar Große-Klönne (Berlin)

MSC:

14G22

Rigid analytic geometry

Keywords:

Berkovich spaces; rigid geometry; formal schemes; privilege; noetherianity

Citations:

Zbl 1084.11064; Zbl 0839.14014; Zbl 0167.06803