# Brauer groups of 1-motives

## TL;DR

This paper extends the Theorem of the Cube to 1-motives and shows that certain torsion classes in their étale cohomology originate from Azumaya algebras, especially over algebraically closed fields of characteristic zero.

## Contribution

It generalizes the Theorem of the Cube for 1-motives and characterizes torsion classes in their étale cohomology as coming from Azumaya algebras.

## Key findings

- Generalized Theorem of the Cube for 1-motives
- Torsion classes in H^2_ét(M,G_m) originate from Azumaya algebras
- All classes in H^2_ét(M,G_m) over algebraically closed fields of characteristic zero come from Azumaya algebras

## Abstract

Over a normal base scheme, we prove the generalized Theorem of the Cube for 1-motives and that a torsion class of the group H^2_\'et(M,G_m)$ of a 1-motive M, whose pull-back via the unit section is zero, comes from an Azumaya algebra. In particular, we deduce that over an algebraically closed field of characteristic zero, all classes of H^2_\'et(M,G_m) come from Azumaya algebras.