Lie–Rinehart cohomology and integrable connections on modules of rank one

Let k be an algebraically closed field of characteristic 0, let R be a commutative k-algebra, and let M be a torsion free R-module of rank one with a connection ∇. We consider the Lie–Rinehart cohomology with values in EndR(M) with its induced connection, and give an interpretation of this cohomology in terms of the integrable connections on M. When R is an isolated singularity of dimension d⩾2, we relate the Lie–Rinehart cohomology to the topological cohomology of the link of the singularity, and when R is a quasi-homogenous hypersurface of dimension two, we give a complete computation of the cohomology.