Descente fid\`element plate et alg\'ebrisation en g\'eom\'etrie de Berkovich

TL;DR

This paper explores descent theory within Berkovich spaces, establishing conditions for fibered categories to form stacks and demonstrating the local nature of algebraicity in morphisms under faithfully-flat topology.

Contribution

It provides new criteria for stacks in Berkovich geometry and shows the effectiveness of descent data, advancing understanding of algebraic properties in non-Archimedean analytic spaces.

Findings

Conditions for fibered categories to be stacks in Berkovich spaces

Effectiveness of certain descent data

Algebraicity of morphisms is local for faithfully-flat topology

Abstract

This article studies descent theory in the setting of Berkovich spaces. We give sufficient conditions for a given fibered category over the category of k-affinoid algebras to be a stack for the Berkovich analogue of the faithfully-flat topology. We give some applications to the faithfully flat descent of morphisms and show that some descent data are always effective. We also show that the property of being algebraic for a morphism between the analytification of two schemes is a local property for the faithfully-flat topology.