The Brauer group of tori

TL;DR

This paper proves the surjectivity of the Brauer group map for tori over certain fields, advancing understanding of their algebraic and cohomological properties in number theory.

Contribution

It establishes the surjectivity of the Brauer group map for tori over local, global, or quasi-trivial cases, extending previous results in algebraic geometry and number theory.

Findings

Surjectivity of the Brauer group map for tori over local and global fields.

Extension of known results to quasi-trivial tori.

Implications for the arithmetic of algebraic tori.

Abstract

We show that the map $\operatorname{Br} T \to (\operatorname{Br} T_{\bar k})^{\Gamma_k}$ is surjective for a torus $T$ defined over a field $k$ of characteristic $0$ when $k$ is a local or global field or $T$ is quasi-trivial.