Vol. 10, No. 4, 2016

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A note on secondary $K$-theory

Gonçalo Tabuada

Vol. 10 (2016), No. 4, 887–906
DOI: 10.2140/ant.2016.10.887
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.

Keywords
dg category, semiorthogonal decomposition, Azumaya algebra, Brauer group, Grothendieck ring, noncommutative motives, noncommutative algebraic geometry
Mathematical Subject Classification 2010
Primary: 14A22
Secondary: 14F22, 16E20, 16H05, 16K50, 18D20
Milestones
Received: 21 September 2015
Revised: 4 February 2016
Accepted: 16 March 2016
Published: 20 June 2016
Authors
Gonçalo Tabuada
Departamento de Matemática e Centro de Matemática e Aplicações
Faculdade de Ciências e Tecnologia
Universidade Nova de Lisboa
Quinta da Torre
2829-516 Caparica
Portugal
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139
United States