Stein Domains in Banach Algebraic Geometry

TL;DR

This paper provides a homological framework to characterize the topology of Stein spaces across various valued fields, bridging complex and non-Archimedean analytic geometries within derived geometric foundations.

Contribution

It introduces a homological characterization of Stein space topologies applicable to both complex and non-Archimedean settings, advancing the foundations of derived analytic geometry.

Findings

Homological characterization of complex Stein spaces matches Euclidean topology.

Non-Archimedean Stein algebra topologies align with Berkovich spaces.

Framework unifies complex and non-Archimedean analytic geometries.

Abstract

In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a contribution towards the foundations of derived analytic geometry.