Abstract:
Let A be a commutative unital ring and E a left A-module. There is a ring end(E) - the ring of projectivity of E with the property that end(E)=0 if and only if E is finitely generated and projective. If end(E)=0 we get a projective basis for the finitely generated and projective module E.
I will use this basis to give explicit formulas for a class of l'th order differential operators on E with l>=1.
I will study l'th order differntial operators,l'th order connections and l'th order curvature for projective modules and show that such l'th order connections are seldom flat. I will relate this study to the following general problem:
The ring of differential operators on A Diff(A) is an associative subring of EndZ(A) containing the ring diff(A) - the associative subring of EndZ(A) generated by the derivations. The ring of differential operators on E Diff(E) is a subring of EndZ(E)). I will use l'th order connections on E to construct a map
If the map f is a ring homomorphism it defines an extension of the left A-structure on E to a left Diff(A)-structure on E.
I will indicate that the map f is seldom a ring-homomorphism and talk about the relationship between this and the curvature of l'th order connections on E. If there is time I will give explicit examples.