The Brauer group of a Stein algebra

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 $\mathcal{O}(S)$.