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We study the periodic cyclic homology groups of the cross-product of a finitetype 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 singularcohomology of the strata of fixed points. These groups identify our cyclichomology groups with the \dlp orbifold cohomology\drp\ of the underlying(algebraic) orbifold. The proof is based on a careful study of localization atfixed points and of the resulting Koszul complexes. We provide examples ofAzumaya algebras for which this identification is, however, no longer valid. Asan example, we discuss some affine Weyl groups.

The periodic cyclic homology of crossed products of finite type algebras