In this note, making use of the recent theory of noncommutative motives, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injectivity properties: in the case of a regular integral quasi-compact quasi-separated scheme, it is injective; in the case of an integral normal Noetherian scheme with a single isolated singularity, it distinguishes any two derived Brauer classes whose difference is of infinite order. As an application, we show that the aforementioned canonical map is injective in the case of affine cones over smooth projective plane curves of degree $\geq 4$ as well as in the case of Mumford’s (famous) singular surface.

中文翻译：

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将派生的Brauer组嵌入到次级$ K $-理论环中

在本说明中，利用最新的非交换动机理论，我们证明了从派生的Brauer群到次级Grothendieck环的典范图具有以下可注射性：在规则积分拟紧凑拟分开方案下，是单射的；在具有单个孤立奇异性的积分正态Noether格式的情况下，它可以区分任意两个派生的Brauer类，它们的差是无限次的。作为一个应用，我们证明了上述经典图在仿射锥在光滑的投影平面度为\\ geq 4 $的仿射锥的情况下，以及在Mumford（著名的）奇异表面的情况下都是可射的。