In my first post, we showed how important information about a PDE on could be turned into a certain module over the ring of regular differential operators on . Our focus today will be about extending the idea of differential operators to other kinds of objects.

The first step will be to define the ring of regular differential operators on an arbitrary complex affine variety. The construction is a bit mysterious, but for smooth varieties it is the ring generated by multiplying by regular functions and differentiating along regular vector fields. Then, we will show how there is a natural restriction map from the differential operators on an affine variety to the differential operators on a Zariski open set in that variety. With this important tool, we can compare differential operators in two different affine subsets of a non-affine variety, and see if we can patch them together into a differential operator over the union. Effectively, this will define the sheaf of differential operators on any variety. We will close by computing the global differential operators on , and showing that they are in agreement with the universal enveloping algebra of the Lie algebra of regular vector fields on .