Les espaces de Berkovich sont angéliques

Jérôme Poineau (1)

1 IRMA - Institut de Recherche Mathématique Avancée

Although Berkovich spaces may fail to be metrizable when defined over too big a field, we prove that a large part of their topology can be recovered through sequences: for instance, cluster points are limits of sequences and compact subsets are sequentially compact. Our proof uses extension of scalars in an essential way and we need to investigate some of its properties. We show that a point in a disc may be defined over a subfield of countable type and that, over algebraically closed fields, every point is universal: in a extension of scalars, there is a canonical point above it.

Mots clés: Espaces de Berkovich; Géométrie analytique p-adique; Espaces de Fréchet-Urysohn; Espaces séquentiels; Espaces angéliques

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]

Jérôme Poineau. Les espaces de Berkovich sont angéliques. 2011. ⟨hal-00588936v2⟩