THE DECOMPOSITION OF HOCHSCHILD COHOMOLOGY AND GERSTENHABER OPERATIONS

Let A be a commutative algebra over a held of characteristic zero, and M be a symmetric A-bimodule. Gerstenhaber and Schack have shown that there are Hedge-type decompositions H-n(A, M) = +H-k(k,n - k)(A, M), H-n(A, M) = +H-k(k,n - k)(A, M) Of the Hochschild (co)homology. The first summands H-1,H-n - 1(A, M), H-1,H-n - 1(A, M) are known to be the Harrison (co)homology defined in terms of shuffles. We discuss interpretations of the decompositions in terms of k-shuffles and how these relate to versions of the Poincare-Birkoff-Witt theorem. We then turn to a detailed study of how the decomposition behaves with respect to the Gerstenhaber operations (cup and Lie products) in cohomology. We show by example that neither product is generally graded, but that F-q = +(r greater than or equal to q) H*,(r) (A,A) are ideals for both products with F-p boolean OR F-q subset of or equal to F-p + q and [F-p, F-q] subset of or equal to F-p + q. The results for the cup product were conjectured by Gerstenhaber and Schack.