Rinehart complexes and Batalin-Vilkovisky algebras

For a Lie-Rinehart algebra (A,L) such that, as an A-module, L is finitely generated and projective of finite constant rank, the relationship between generators of the Gerstenhaber bracket and connections on the highest A-exterior power of L given in an earlier paper arises from the canonical pairing between the exterior A-powers of L. Thus, given an exact generator for the corresponding Gerstenhaber algebra, the chain complex underlying the resulting Batalin-Vilkovisky algebra coincides with the Rinehart complex computing the corresponding Lie-Rinehart homology.