The periodic cyclic homology of crossed products of finite type algebras

We study the periodic cyclic homology groups of the cross-product of a finite type algebra A by a discrete group Γ. In case A is commutative and Γ is finite, our results are complete and given in terms of the singular cohomology of the sets of fixed points. These groups identify our cyclic homology groups with the “orbifold cohomology” of the underlying (algebraic) orbifold. The proof is based on a careful study of localization at fixed points and of the resulting Koszul complexes. This is achieved by extending to our class of noncommutative algebras the concept of an “infinitesimal neighborhood” that plays such an important role in commutative algebra. We provide examples of Azumaya algebras for which this identification is, however, no longer valid. As an example, we discuss some affine Weyl groups.