The Brauer group of a Stein algebra

TL;DR

This paper studies the Brauer group of holomorphic function rings on Stein spaces, providing topological computations, comparison theorems with cohomology, and results on purity and index of classes.

Contribution

It offers a topological approach to computing the Brauer group of Stein algebras and establishes new links with étale and singular cohomology.

Findings

Topological computation of the Brauer group of Stein algebras

Comparison theorem between étale and singular cohomology in degree 2

Purity theorem for nonsingular Stein spaces

Abstract

We investigate the Brauer group of the ring $\mathcal{O}(S)$ of holomorphic functions on a finite-dimensional Stein space S. We provide a purely topological computation of this group and deduce a comparison theorem between the \'etale cohomology of $\textrm{Spec}(\mathcal{O}(S))$ and the singular cohomology of S in degree 2. Furthermore, we prove a purity theorem when S is nonsingular and study the index of classes in the Brauer group of $\mathcal{O}(S)$.