I was looking for a simple example of a covariant derivative on a bundle, where the bundle is not projective. If necessary, the example could be from complex or noncommutative geometry, but I would hope for a simple example with the usual calculus on a manifold.
In noncommutative geometry, there is a theorem by Cuntz and Quillen saying that for the universal calculus, a module on an algebra has a covariant derivative if and only if the module is projective. The universal calculus should be the most restrictive in this context, there should be other calculi with non-projective modules having covariant derivatives. I suspect that algebraic geometry has lots of these, but I would like an easy one to explain!
Added: I guess that an example might be based around a skyscraper sheaf, but I don't see how to give such a thing a connection! (I am likely being stupid here...)