A note on secondary K-theory

Gonçalo Tabuada
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引用次数: 11

Abstract

We prove that Toen's secondary Grothendieck ring is isomorphic to the Grothendieck ring of smooth proper pretriangulated dg categories previously introduced by Bondal, Larsen and Lunts. Along the way, we show that those short exact sequences of dg categories in which the first term is smooth proper and the second term is proper are necessarily split. As an application, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injective properties: in the case of a commutative ring of characteristic zero, it distinguishes between dg Azumaya algebras associated to non-torsion cohomology classes and dg Azumaya algebras associated to torsion cohomology classes (=ordinary Azumaya algebras); in the case of a field of characteristic zero, it is injective; in the case of a field of positive characteristic p>0, it restricts to an injective map on the p-primary component of the Brauer group.
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关于次k理论的说明
证明了Toen的二次Grothendieck环与Bondal, Larsen和Lunts先前引入的光滑的适当的预三角化dg类的Grothendieck环是同构的。在此过程中,我们证明了那些第一项为光滑固有项和第二项为固有项的dg类的精确短序列是必然分裂的。作为应用,我们证明了从派生的Brauer群到次级Grothendieck环的正则映射具有以下的内射性质:在特征为零的交换环的情况下,它区分了与非扭转上同调类相关的dg Azumaya代数和与扭转上同调类相关的dg Azumaya代数(=普通Azumaya代数);对于特征为0的域,它是内射;对于正特征p>0的域,则限定为Brauer群的p-原分量上的内射映射。
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