Espaces de Berkovich sur $\mathbb{Z}$: morphismes \'etales

TL;DR

This paper extends the theory of Berkovich spaces over the integers, establishing properties of unramified, étale, and smooth morphisms, and providing criteria for analytification across various valuation contexts.

Contribution

It develops a unified framework for properties of morphisms between Berkovich spaces over $\

Findings

Properties of morphisms analogous to scheme theory are established.

Analytification criteria are provided for various valuation settings.

Results unify cases over number rings and discrete valuation rings.

Abstract

We develop properties of unramified, \'etale and smooth morphisms between Berkovich spaces over $\mathbb{Z}$. We prove that they satisfy properties analogous to those of morphisms of schemes and we provide analytification criteria. Our results hold for any valued field, rings of integers of a number field and discrete valuation rings. Those cases are treated by a unified way.