Hochschild (Co)Homology of Differential Operator Rings

Abstract

We show that the Hochschild homology of a differential operator kalgebra E = A#f U(g), is the homology of a deformation of the Chevalley-Eilenberg complex of g with coefficients in (M ⊗A^-∗, b∗). Moreover, when A is smooth and k is a characteristic zero field, we obtain a type of Hochschild-Kostant-Rosenberg theorem for these algebras. When A = k our complex reduce to the one obtained in [K] for the homology of filtrated algebras whose associated graded algebras are symmetric algebras. In the last section we give similar results for the cohomology.