Computing connections on modules

We consider the notion of a connection on a module over a commutative ring, and deduce an obstruction calculus for determining if there exist such connections. The obstruction calculus is defined using Hochschild cohomology. However, in order to compute with Gröbner bases, we need the non-trivial conversion to a description using free resolutions. We describe our implementation in Singular 3.0, available as the library conn.lib. Finally, we use the library to verify some known results and to obtain a new theorem for maximal Cohen-Macaulay (MCM) modules on isolated singularities. For a simple hypersurface singularity in dimension one and two, all MCM modules admit connections. Based on experiments, we conjecture that only the free MCM modules admit connections in dimension three.