Chem classes and Lie-Rinehart algebras

Let A be a F-algebra where F is a field, and let W be an A-module of finite presentation. We use the linear Lie-Rinehart algebra VW of W to define the first Chern-class c1(W) in H2(VW|U,OU), where U in Spec(A) is the open subset where W is locally free. We compute explicitly algebraic VW-connections on maximal Cohen-Macaulay modules W on the hypersurface-singularities Bmn2 = xm + yn + z2, and show that these connections are integrable, hence the first Chern-class c1(W) vanishes. We also look at indecomposable maximal Cohen-Macaulay modules on quotient-singularities in dimension 2, and prove that their first Chern-class vanish.