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Additive invariants of orbifolds

Author(s)
Tabuada, Gonçalo; Van den Bergh, Michel
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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second author and Polishchuk concerning Hochschild homology, and of Baranovsky and Petrov, and Cǎldǎraru and Arinkin (unpublished), concerning twisted Hochschild homology; in the case of topological Hochschild homology and periodic topological cyclic homology, the aforementioned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, 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
Date issued
2018-06
URI
https://hdl.handle.net/1721.1/124685
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
Geometry & Topology
Publisher
Mathematical Sciences Publishers
Citation
Tabuada, Gonçalo and Van den Bergh, Michel. "Additive invariants of orbifolds." Geometry & Topology 22, 5 (2018): 3003–3048 © 2018 Mathematical Science Publishers
Version: Final published version
ISSN
1465-3060

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