  We study the periodic cyclic homology groups of the cross-product of a finite
type algebra $A$ by a discrete group $\Gamma$. In case $A$ is commutative and
$\Gamma$ is finite, our results are complete and given in terms of the singular
cohomology of the strata of fixed points. These groups identify our cyclic
homology groups with the \dlp orbifold cohomology\drp\ of the underlying
(algebraic) orbifold. The proof is based on a careful study of localization at
fixed points and of the resulting Koszul complexes. We provide examples of
Azumaya algebras for which this identification is, however, no longer valid. As
an example, we discuss some affine Weyl groups.
