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
, it restricts to an injective
map on the
-primary
component of the Brauer group.
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