Additive invariants of orbifolds

Tabuada, Gonçalo¹ ; Van den Bergh, Michel²

¹ 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
² Department of Mathematics, Universiteit Hasselt, Diepenbeek, Belgium

Geometry & topology, Tome 22 (2018) no. 5, pp. 3003-3048.

Résumé

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second author and Polishchuk concerning Hochschild homology, and of Baranovsky and Petrov, and Cǎldǎraru and Arinkin (unpublished), concerning twisted Hochschild homology; in the case of topological Hochschild homology and periodic topological cyclic homology, the aforementioned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

Détail

DOI : 10.2140/gt.2018.22.3003

Classification : 14A15, 14A20, 14A22, 19D55
Keywords: orbifold, algebraic $K$–theory, cyclic homology, topological Hochschild homology, Azumaya algebra, standard conjectures, noncommutative algebraic geometry

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