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Additive invariants of orbifolds

Gonçalo Tabuada and Michel Van den Bergh

Geometry & Topology 22 (2018) 3003–3048

Using the recent theory of noncommutative motives, we compute the additive invariants of orbifolds (equipped with a sheaf of Azumaya algebras) using solely “fixed-point data”. As a consequence, we recover, in a unified and conceptual way, the original results of Vistoli concerning algebraic K–theory, of Baranovsky concerning cyclic homology, of the second author and Polishchuk concerning Hochschild homology, and of Baranovsky and Petrov, and Cǎldǎraru and Arinkin (unpublished), concerning twisted Hochschild homology; in the case of topological Hochschild homology and periodic topological cyclic homology, the aforementioned computation is new in the literature. As an application, we verify Grothendieck’s standard conjectures of type C+ and D, as well as Voevodsky’s smash-nilpotence conjecture, in the case of “low-dimensional” orbifolds. Finally, we establish a result of independent interest concerning nilpotency in the Grothendieck ring of an orbifold.

orbifold, algebraic $K$–theory, cyclic homology, topological Hochschild homology, Azumaya algebra, standard conjectures, noncommutative algebraic geometry
Mathematical Subject Classification 2010
Primary: 14A15, 14A20, 14A22, 19D55
Received: 24 April 2017
Revised: 21 December 2017
Accepted: 5 March 2018
Published: 1 June 2018
Proposed: Dan Abramovich
Seconded: Richard Thomas, Haynes R Miller
Gonçalo Tabuada
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Departamento de Matemática e Centro de Matemática e Aplicações
Faculdade de Ciências e Tecnologia
Universidade Nova de Lisboa
Michel Van den Bergh
Department of Mathematics
Universiteit Hasselt