An algebraic characterisation of non-Archimedean Stein spaces

TL;DR

This paper provides a comprehensive algebraic framework for non-Archimedean Stein spaces, characterizing them via Liu algebras and resolving key conjectures in non-Archimedean geometry.

Contribution

It introduces Liu algebras and establishes a functorial characterization of non-Archimedean Stein spaces, extending Serre's criterion and generalizing the Gerritzen-Grauert Theorem.

Findings

Characterization of non-Archimedean Stein spaces as Berkovich spectra

Resolution of a conjecture of Michael Temkin

Generalization of the Gerritzen-Grauert Theorem

Abstract

We introduce Liu algebras as Banach algebras which are 'locally affinoid', and define non-Archimedean Stein algebras as suitable inverse limits of these. We show that this gives rise to a complete functorial characterisation of non-Archimedean Liu and Stein spaces as Berkovich spectra of their respective algebras, thereby resolving a conjecture of Michael Temkin. This can be interpreted as a non-Archimedean analytic version of Serre's criterion for affineness. Furthermore, we prove a criterion that distinguishes affinoid algebras within the category of Liu algebras, answering another conjecture of Temkin. We also prove a generalisation of the Gerritzen-Grauert Theorem for non-Archimedean Stein spaces.