#### Volume 22, issue 5 (2018)

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Additive invariants of orbifolds

### Gonçalo Tabuada and Michel Van den Bergh

Geometry & Topology 22 (2018) 3003–3048
##### Abstract

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second author and Polishchuk concerning Hochschild homology, and of Baranovsky and Petrov, and Cǎldǎraru and Arinkin (unpublished), concerning twisted Hochschild homology; in the case of topological Hochschild homology and periodic topological cyclic homology, the aforementioned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 $D\phantom{\rule{0.3em}{0ex}}$, 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.

##### Keywords
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
##### Publication
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
##### Authors
 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 Lisboa Portugal http://math.mit.edu/~tabuada Michel Van den Bergh Department of Mathematics Universiteit Hasselt Diepenbeek Belgium http://hardy.uhasselt.be/personal/vdbergh/Members/~michelid.html