MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This is a follow up to my question about D-modules supported on the nilpotent cone. I got some good answers there but now I have a more basic question.

Consider an affine algebraic variety X, a closed subvariety i:Y-->X, and the intermediate extension of the structure sheaf on Y to all of X (do I denote this i!*OY ? For that matter, explaining either the * or ! extension instead would be a helpful start if its easier).

My question is this: Since X is affine, D(X) is just an associative algebra, generated by O(X) and Vect(X) by the usual construction. My question is how can I understand i!*OY as a module the associative algebra D(X), supposing I understand D(X)?

In other words, what is the vector space underlying i!*OY, how do functions in O(X) and vector fields act?

I probably need to invest some serious time with a textbook to answer this question myself, but any help getting started would be most appreciated!

share|cite|improve this question
Just for completeness: D(X) is only generated by O(X) and derivations, if X is defined over a field of characteristic 0. – Lars Nov 1 '09 at 18:57
And if X is smooth. – Greg Muller Nov 1 '09 at 22:04
Yes, thanks on both counts. I'm really interested in the situation on the link, so both of these are true. – David Jordan Nov 1 '09 at 23:24
up vote 4 down vote accepted

Have you looked at Bernstein's lectures on D-modules? He proves a result relevant to your question in Lec. 3, Sec. 14: for an affine embedding Y --> X with Y irreducible, if E is an OY-coherent DY module (i.e. a vector bundle with a flat connection) then the !* direct image from E can be characterized as the unique irreducible subquotient of either the * or ! direct image which has nonzero restriction to Y. Since it can be easy in such situations to compute the * direct image, this may be a good way to get a handle on the other functors too. For example, I think one can see from this that if we take the embedding of the origin into the affine line, then both the * and !* direct images of OY=k (the ground field) are A(1)/A(1)t, where t is the coordinate on the affine line and A(1)=DX is the 1-dimensional Weyl algebra k[t,d/dt]. This quotient is rightly considered the "delta-function" A(1)-module, since its generator δ satisfies tδ=0. I'm not sure how similar to this case your general situation will be. But certainly by Kashiwara's theorem one knows that the * direct image (and hence the !* direct image) will be supported on the subvariety Y.

share|cite|improve this answer
Thanks! I had looked over these a few years ago in an intro course, but not using it hadn't known exactly where to look. – David Jordan Nov 1 '09 at 19:52
So there is no easy explicit answer in general? Just in terms of a ring O(X) and its quotient ring O(Y)? – მამუკა ჯიბლაძე Feb 24 '15 at 5:17

Since Y is closed, the !, *, and !* extensions coincide, and there should be a straightforward (but confusing for me) way of doing what you want. If Y is smooth, then I wonder if you can identify the vector space you're talking about with something like sections of the conormal bundle to Y in X, but this is not quite right (e.g. when Y = X). When Y is singular, there is a more complicated story, having to do with the fact that the obvious definition of D-module on a singular variety is no good.

When Y is only locally closed, describing the !* extension is a difficult problem. Vilonen's thesis, the only reference I know about this, is available here:

There is even something nontrivial to say about the ! and * extensions--the characteristic cycles of such D-modules were computed by Schmid and Vilonen. Maybe there is a more elementary answer to your question about these extensions, though.

share|cite|improve this answer
Thanks, this is helpful! As you saw in the other post, the ones I'm interested in eventually aren't closed. – David Jordan Nov 1 '09 at 23:26
OK not "sections of the conormal bundle" but "functions on the total space of the conormal bundle." A vector field on X determines a function on the conormal bundle to Y by contraction, and operates on the vector space of such functions by multiplication. – David Treumann Nov 2 '09 at 4:28

I think I can answer my subquestion about * extension after thinking about it, but it's really the ! and !* extensions I'd like to understand. I think I remember the * extension is just the pushforward.

So i*O(Y) = O(X) ⊗i O(Y), and the D-module action is always happening on the O(X) factor.

Sorry to answer (a tiny part of) my own question, but I couldn't do it in a comment because of the formatting, and maybe this will save some kind respondent some time.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.