A note on secondary K-theory

TL;DR

This paper establishes an isomorphism between Toen's secondary Grothendieck ring and a known Grothendieck ring of dg categories, and explores implications for the derived Brauer group and dg Azumaya algebras.

Contribution

It proves the isomorphism between Toen's secondary Grothendieck ring and the Grothendieck ring of smooth proper dg categories, and analyzes the injectivity properties of the derived Brauer group map.

Findings

Secondary Grothendieck ring is isomorphic to the Grothendieck ring of smooth proper dg categories.

Short exact sequences with smooth proper first term are necessarily split.

The canonical map from the derived Brauer group to the secondary Grothendieck ring has injective properties in various cases.

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.