An algebraic characterisation of non-Archimedean Stein spaces
Tom Biesbrouck
TL;DR
The paper develops an algebraic framework for non-Archimedean analytic geometry by introducing Liu algebras (locally affinoid) and Stein algebras (Fréchet inverse limits), establishing an exact anti-equivalence with Liu spaces and Stein spaces via Berkovich spectra. It provides a complete non-Archimedean analogue of Serre's criterion for affineness, along with a criterion to distinguish affinoid algebras within Liu algebras and a generalisation of the Gerritzen-Grauert theorem for Stein spaces. Key technical advances include Tate-like acyclicity for Liu algebras, a base-change compatible rational localisation theory, and the extension of Berkovich spectra to Fréchet algebras to handle Stein spaces. The results collectively yield a robust, functorial correspondence between algebraic and analytic categories in the non-Archimedean setting, with explicit local-to-global tools such as Runge immersions and Runge-type decompositions.
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.