A note on secondary K-theory II

TL;DR

This paper investigates the injectivity of the canonical map from the derived Brauer group to the secondary Grothendieck ring, using noncommutative motives to establish results for specific classes of schemes and singular surfaces.

Contribution

It proves injectivity properties of the canonical map in new cases, including regular schemes and certain singular surfaces, advancing understanding of secondary K-theory.

Findings

Injectivity for regular integral quasi-compact schemes.

Distinguishing classes on schemes with isolated singularities.

Injectivity for affine cones over certain complex curves and singular surfaces.

Abstract

This note is the sequel to [A note on secondary K-theory. Algebra and Number Theory 10 (2016), no. 4, 887-906]. 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 canonical map is injective in the case of affine cones over smooth projective plane complex curves of degree greater than or equal to four as well as in the case of Mumford's (celebrated) singular surface.