Over a normal base scheme, we prove the generalized Theorem of the Cube for 1-motives and that a torsion class of the group ${\mathrm{H}}_{\acute{e}t}^2(M,{\mathbb{G}}_{m,M})$ of a 1-motive M, whose pull-back via the unit section $ϵ:S\to M$ is zero, comes from an Azumaya algebra. In particular, we deduce that over an algebraically closed field of characteristic zero, all classes of ${\mathrm{H}}_{\acute{e}t}^2(M,{\mathbb{G}}_{m,M})$ come from Azumaya algebras.

中文翻译：

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一动机的Brauer组

在一个普通的基本方案上，我们证明了1动机的立方体定理和该组的一个扭转类的广义定理。 ${\mathrm{H}}_{\acute{Ë}Ť}^{\mathrm{2个}}（\mathrm{中号}，{\mathbb{G}}_{米，\mathrm{中号}}）$1动机M，其通过单位部分回撤$ϵ：\mathrm{小号}\to \mathrm{中号}$是零，来自Azumaya代数。特别是，我们推论在特征为零的代数闭合域上，所有类别的${\mathrm{H}}_{\acute{Ë}Ť}^{\mathrm{2个}}（\mathrm{中号}，{\mathbb{G}}_{米，\mathrm{中号}}）$ 来自Azumaya代数。