A note on secondary $K$-theory

Abstract

We prove that Toën’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 injectivity properties: in the case of a commutative ring of characteristic zero, it distinguishes between dg Azumaya algebras associated to nontorsion 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 $p$, it restricts to an injective map on the $p$-primary component of the Brauer group.