The year’s course has come to an end. (I would like to give huge thanks to the people in the class — their detailed comments and suggestions have led to a vast number of improvements.) A revised version is now posted at **the usual place **(the August 16, 2012 version).

I may later write a post with some thoughts on how the course went. (In short: I think it showed that it is possible to cover the amount of material I wanted to, in a single year. There were two topics I wanted to cover, but did not because I have not yet written up the exposition; but they were replaced with other useful facts.) But right now, I want to post the latest version of the notes, with a substantially new exposition of the proof of Serre duality (the final chapter — the missing chapters are earlier) for projective varieties .

I have reverted to an earlier proof I gave in versions years ago, in terms of finite (often flat) morphisms. (I was partially prompted by an email discussion with Yuhao Huang at Berkeley — thank you Yuhao!)

There are a number of possible statements of Serre duality one might want, with the expected trade-offs: better statements require more work. I found productive (both personally and pedagogically) to discuss these trade-offs, and to make some decisions, and to see (in class) where approaches broke down, and where they worked.

I concentrated on several desiderata. We want some some of duality involving a dualizing sheaf. We will want the dualizing sheaf to be the determinant of the cotangent bundle. This is surprisingly hard, and left to the end.

There will be potentially three versions of each desired type of duality. There is the dimensional (lame) version, saying that the dimension of two cohomology groups are the same. Better, there should be some duality between the cohomology groups, that should be functorial in the sheaf/bundle (“functorial Serre duality”). And best of all, it should arise from a cup product of some sort (“trace version” of Serre duality), which requires defining the cup product.

The first kind of duality one might ask for (because we used it in discussing curves and surfaces) is Serre duality for vector bundles. Better yet, one can have it for coherent sheaves. (And we can get better still: Serre duality for families; dualizing complexes; etc. But the brutal demands of doing it all within a course, with proofs, means that we do not go there.)

In order to move toward proving these things (or even to move toward making some of the statements precise), we have to discuss Ext groups, which wasn’t hard given what we had done earlier. As an aside (double-starred), I discussed the cup product for Ext, which of course is needed for the best statements of Serre duality. (But it is not needed for what I prove, and what I need!) The method of proof is to do Serre duality for projective space, and then to go to finite flat covers. The key trick is to use a version of the “upper shriek” construction, an occasional adjoint to (even if isn’t quite “upper shriek”, so I denoted it rather than ). A complication is added by the fact that we are working with O-modules, but the construction I give only works for quasicoherent sheaves; and it works best for closed immersions. I won’t say much more here, because it will only confuse you — it is best to read the notes.

With this approach, Serre duality (in the versions we use) ends up being surprisingly easy — it seems to come out of nowhere. (This was independently stated to me by three people in the class, and I agree.) But there is a surprising amount of subtlety hidden in the exposition here. The subtlety isn’t in the arguments, or what is said; it is in the choice of what to say, and what paths to take. There were many points where I tried to take a different route, and then something bad happened, forcing me to retract. (The version from earlier this year proved Serre duality for projective schemes using closed immersions into projective space, but I got myself into difficulty, as observed by a number of people, including Yuhao Huang, Yuncheng Lin, Preston Wake, Charles Staat, and Yifei Zhu. This approach ended up being much cleaner.)

Finally, this approach does not easily show that the dualizing sheaf of a smooth projective variety is the “sheaf of algebraic volume forms” (the determinant of the cotangent bundle). To do this requires more work, and a different approach, which also allows us to prove the adjunction formula. (However, it *is* possible to prove this directly using the “finite flat cover” approach. Matt Baker and Janos Csirik worked it out in **this note**. (I thank both Matt and Janos for permission to post this here.)

Aside: I realized that alternative expositions can work as well, with different costs and benefits. For example, the category of quasicoherent sheaves on a variety actually has enough injectives, so one can work directly in that category.

**What comes next. **My to-do list has been finite and shrinking, but there are many things left to do. There are a number of loose ends in the chapters already done. I had hoped to finish the three final chapters this summer, and now I am just going hoping to finish a draft of one of them (on regular sequences).