Musings on D-modules, part 2

So we have moved along in our D-modules seminar. It’s been surprisingly successful. I would have never guessed we would get 8 people to come to a seminar in the middle of summer. I want to share with you some of what we have learned.

When we left off last time (see my last post), I explained that a D-module on a smooth complex variety X was a sheaf of modules M over the sheaf of differential operators D_X. Then I gave some motivation. Now, I want to get into what you actually do with these D-modules.

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Musings on D-modules

Here in Berkeley, we are having a small learning seminar on D-modules, trying to read Bernstein’s notes. Thursday we had an organizational meeting and on Monday Anton is giving the first talk.

If I had been more organized, I would have given a motivational talk on Thursday trying to explain why D-modules are interesting. Here is what I would have said.

First, just so that we are on the right page, let me explain the basic concepts. The ring D_X is the (sheaf of) ring(s) of algebraic differential operators on a smooth algebraic variety and a D-module is a (sheaf of) module(s) over this ring. For example, we could take X to be the affine line over \mathbb{C}, in which case D_X = \mathbb{C}\langle x, \partial \rangle / \partial x - x \partial = 1. Here \partial is the differential operator d/dx and x is multiplication by x. In general the ring D_X is a kind of universal enveloping algebra of vector fields over functions.

So here are three reasons why D-modules are interesting.

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