Harrison Homology, Hochschild Homology and Triples*l

We consider the following situation: a field k, a commutative k-algebra R and a left R-module M. Since R is commutative, M may also be considered as an R-R bimodule with the same operation on each side (such modules are often termed symmetric). With these assumptions we have the Harrison (co-) homology groups Harr,(R, M) (Harr*(R, M)), the Hochschild (co-) homology groups Hoch, (R, M) (Hoch*(R, M)) and the symmetric algebra triple (co-) homology groups Sym,(R, M) (Sym*(R, M)). (Harrison cohomology is introduced in [8], homology may be defined similarly; Hochschild cohomology is introduced in [9], a good account of the homology and cohomology is found in [IO]; general triple homology and cohomology are described in [4] and [5], and the symmetric algebra cotriple is described at the beginning of Section 3 below.) There are obvious natural transformations