The Brauer group of a valuation ring

TL;DR

This paper proves that for a valuation ring, the natural map from its Brauer group to the Brauer group of its field of fractions is injective, clarifying the relationship between these algebraic structures.

Contribution

It establishes the injectivity of the canonical Brauer group map for valuation rings, a result not previously known in this generality.

Findings

The canonical map r(V) r(K) is injective for valuation rings.

Provides new insights into the structure of Brauer groups over valuation rings.

Enhances understanding of algebraic properties relating valuation rings and their fraction fields.

Abstract

Let $V$ be a valuation ring and $K$ be its field of fraction. We show that the canonical map $\Br(V) \to \Br(K)$ is injective.