Hochschild (Co)Homology of Differential Operator Rings,

We show that the Hochschild homology of a differential operator k-algebra E = A#fU(g) is the homology of a deformation of the Chevalley–Eilenberg complex of with coefficients in (M [formula], 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 reduces to the one obtained by C. Kassel (1988, Invent. Math. 91, 221–251) for the homology of filtrated algebras whose associated graded algebras are symmetric algebras. In the last section we give similar results for the cohomology.