# Relative Brauer Groups and \'etale cohomology

## TL;DR

This paper introduces a natural homomorphism linking the relative Brauer group to étale cohomology for affine maps of schemes, and explores its properties and related sequences.

## Contribution

It constructs a natural homomorphism from the relative Brauer group to étale cohomology and proves a relative Kummer sequence, advancing understanding of Brauer groups in algebraic geometry.

## Key findings

- Constructed a natural homomorphism from Br(f) to étale cohomology.
- Analyzed the behavior of Brauer groups under subintegral maps.
- Proved a relative version of Kummer's exact sequence.

## Abstract

In this article, we construct a natural group homomorphism $$ \psi: \text{Br}(f)\to H^{1}_{et}(S, f_{*}\mathcal{O}_{X}^{\times}/\mathcal{O}_{S}^{\times})$$ 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.