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 V_W of W to define the first Chern-class c₁(W) in , where U in Spec(A) is the open subset where W is locally free. We compute explicitly algebraic V_W-connections on maximal Cohen-Macaulay modules W on the hypersurface-singularities B_mn2 = x^m + 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.