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The Brauer group of tori

Julian Demeio

Abstract

We show that the map $\operatorname{Br} T \to (\operatorname{Br} T_{\bar k})^{Γ_k}$ 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.

The Brauer group of tori

Abstract

We show that the map is surjective for a torus defined over a field of characteristic when is a local or global field or is quasi-trivial.

Paper Structure

This paper contains 7 sections, 12 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Transcendental Brauer groups of quasi-trivial tori
  4. Proof of Theorem \ref{['Thm2']}
  5. Transcendental Brauer groups of real tori
  6. Functoriality of Grothendieck's spectral sequence
  7. Compatibility of Leray and Hocschild-Serre spectral sequences

Key Result

Theorem 1.1

Let $k$ be a field of characteristic $0$, and $T$ be a $k$-torus. The identity $\mathop{\mathrm{Br}}\nolimits_{tr}X = (\mathop{\mathrm{Br}}\nolimits X_{\overline{k}})^{\Gamma_k}$ holds if either:

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • proof : Proof of Theorem \ref{['Thm2']}
  • Theorem 4.1
  • ...and 17 more