Geometry & Topology

Additive invariants of orbifolds

Gonçalo Tabuada and Michel Van den Bergh

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second author and Polishchuk concerning Hochschild homology, and of Baranovsky and Petrov, and Cǎldǎraru and Arinkin (unpublished), concerning twisted Hochschild homology; in the case of topological Hochschild homology and periodic topological cyclic homology, the aforementioned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

Article information

Geom. Topol., Volume 22, Number 5 (2018), 3003-3048.

Received: 24 April 2017
Revised: 21 December 2017
Accepted: 5 March 2018
First available in Project Euclid: 26 March 2019

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Zentralblatt MATH identifier

Primary: 14A15: Schemes and morphisms 14A20: Generalizations (algebraic spaces, stacks) 14A22: Noncommutative algebraic geometry [See also 16S38] 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60]

orbifold algebraic $K$–theory cyclic homology topological Hochschild homology Azumaya algebra standard conjectures noncommutative algebraic geometry


Tabuada, Gonçalo; Van den Bergh, Michel. Additive invariants of orbifolds. Geom. Topol. 22 (2018), no. 5, 3003--3048. doi:10.2140/gt.2018.22.3003.

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