Relative Brauer Groups and étale cohomology

Abstract

In this article, we construct a natural group homomorphism $$\psi :\text{Br}(f)\to H_{et}^1(S,f_{\ast }{\mathcal{O}}_X^{\times }/{\mathcal{O}}_S^{\times })$$ for a faithful affine map $f:X\to S$ of noetherian schemes. Here $\text{Br}(f)$ denotes the relative Brauer group of $f.$ We also discuss the behavior of Brauer groups for a subintegral map. Furthermore, we prove a relative version of Kummer's exact sequence.

Some changes made due to referee report. Final version and accepted for publication