Additive invariants of orbifolds

G. Tabuada, M. Van Den Bergh

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)

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second author and Polishchuk concerning Hochschild homology, and of Baranovsky and Petrov, and Cǎldǎraru and Arinkin (unpublished), concerning twisted Hochschild homology; in the case of topological Hochschild homology and periodic topological cyclic homology, the aforementioned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.

Original languageEnglish
Pages (from-to)3003–3048
Number of pages46
JournalGeometry and Topology
Volume22
Issue number5
DOIs
Publication statusPublished - 2018

Keywords

  • Algebraic K-theory
  • Azumaya algebra
  • Cyclic homology
  • Noncommutative algebraic geometry
  • Orbifold
  • Standard conjectures
  • Topological Hochschild homology

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