|
|||||
 |
|
|
|
|
|
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 , it restricts to an injective map on the -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 |
|
