Les espaces de Berkovich sont excellents

TL;DR

This paper proves that local rings of Berkovich analytic spaces are excellent, introduces analytically separable extensions, and explores geometric properties like irreducible components and normalization in non-archimedean geometry.

Contribution

It establishes the excellence of local rings in Berkovich spaces and develops foundational geometric concepts such as irreducible components and normalization.

Findings

Local rings of Berkovich spaces are excellent.

Stability of algebraic properties under analytically separable extensions.

Existence and basic properties of irreducible components and normalization.

Abstract

In this paper, we first study the local rings of a Berkovich analytic space from the point of view of commutative algebra. We show that those rings are excellent ; we introduce the notion of a an analytically separable extension of non-archimedean complete fields (it includes the case of the finite separable extensions, and also the case of any complete extension of a perfect complete non-archimedean field) and show that the usual commutative algebra properties (Rm, Sm, Gorenstein, Cohen-Macaulay, Complete Intersection) are stable under analytically separable ground field extensions; we also establish a GAGA principle with respect to those properties for any finitely generated scheme over an affinoid algebra. A second part of the paper deals with more global geometric notions : we define, show the existence and establish basic properties of the irreducible components of analytic space ; we define, show the existence and establish basic properties of its normalization ; and we study the behaviour of connectedness and irreducibility with respect to base change.