Chern classes and Lie-Rinehart algebras

Abstract

Classically the Chern-classes of a locally free coherent A-module W are defined using the curvature of a connection. If we more generally consider the problem of defining Chern-classes where W is a coherent A-module, a connection might not exist. In this paper we use the linear Lie-algebroid of W, where W is any coherent A-module of finite presentation, to define the first Chern-class of W. We also do explicit calculations of Chern-classes for maximal Cohen-Macaulay modules on isolated hypersurface-singularities and 2-dimensional quotient-singularities.