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
–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and Arinkin (unpublished), concerning twisted Hochschild homology; in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
and
, 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.
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