zbMATH — the first resource for mathematics

Variation de la dimension relative en géométrie analytique $$p$$-adique; Variation of relative dimension in $$p$$-adic analytic geometry. (French) Zbl 1161.14018
Author’s abstract: Let $$k$$ be a complete, non-Archimedean valued field (the trivial absolute value is allowed) and let $$\varphi :X\rightarrow Y$$ be a morphism between two Berkovich $$k$$-analytic spaces; we show that, for any integer $$n$$, the set of points of $$X$$ at which the local dimension of $$\varphi$$ is at least equal to $$n$$ is a Zariski-closed subset of $$X$$. In order to establish it, we first prove an analytic analogue of Zariski’s Main Theorem, and we also introduce, and study, the notion of an analytic system of parameters at a point.

MSC:
 14G22 Rigid analytic geometry 14A99 Foundations of algebraic geometry
Full Text: