The twelfth post is the March 10 version in the usual place. This post covers up to 20.3 (although more is included). The next post should appear on April 2. Corrections from numerous people, including Darij Greenberg (in the spectral sequences section) and Jason Ferguson (in chapters up to 3, with more changes still to be made). I continue to be way way behind on responding to comments, including a number from the fall (!).

This installment includes chapter 18, on relative Spec and Proj, and projective morphisms; the double-starred chapter 19, on blowing-up; and chapter 20, on Cech cohomology.

I’ve not looked over Chapter 19 in a very long time, so look at it at your own risk. (But comments are still welcome of course!) Update: it has now been edited, see this March 16 post.

I am somewhat surprised to find that there isn’t much to go: we’re through most of the notes. If you look at the table of contents, you’ll see that there are about 100 pages of the current rough draft still to come. There are one of two more chapters still to be written (one on the “formal functions” theorem and consequences, and very possibly one one Cohen-Macaulayness). So we’re almost there! (Of course, there are lots of things that need revisiting, and lots of comments still to respond to.)

For learners.

18.1 gives the relative version of Spec, and re-interprets affine morphisms. It is pleasant.

On the other hand, 18.2 gives the relative version of Proj, and like Proj, it is less pleasant. I am not sure if I’m giving the “right” definition, and will ask the experts below. Any suggestions on how to make this better would be appreciated.

As a result of the weirdness of Proj, the notion of projective morphisms (18.3) is surprisingly tricky. But it isn’t so terrible. Notice in particular that finite morphisms are projective.

In 18.4, we see some of these ideas in action, applied to curves. (We’ll soon be talking about curves at great length, likely in the 13th notes.)

Skip Chapter 19 (unless you really don’t want to). It deals with blow-ups, which we won’t use a huge amount.

Chapter 20 is about Cech cohomology. Cohomology is lots of fun, and for me the highlight of the course. It is surprisingly simple (certainly compared to what you have done so far). It is amazing what you can do with it right away; we’ll start exploring that in 20.4 (and 20.5 and 20.6), and when we look at curves in Chapter 21. In 20.1, we first discuss what we want cohomology to do for us.

In 20.2 we actually define it, and verify that these key properties hold.

In 20.3, we do a key computation, of the cohomology of line bundles in projective space.

And then you are set to do some fun applications, which we’ll get into next time. But if you want to read ahead, the rest of Chapter 20 is included.

Here is a selection of interesting and/or important problems: 18.1.A, 18.1.B, 18.1.F, 18.2.A, 18.2.C, 18.2.F, 18.2.G, 18.3.C, 18.3.E (surprisingly but pleasantly intricate), 18.3.G, 18.3.H, 18.4.E, 20.1.B, 20.1.D, 20.2.C, 20.2.D, 20.2.G. As always, I hope you try more.

Here are some problems dealing with explicit examples: 18.1.H, 18.4.D (especially for number theorists), 20.2.A, 20.3.A, 20.3.B. And many more examples are coming up in 20.4 onwards!

If you solve 20.2.E, please let me know; it seems surprisingly tricky. (But people have solved it, for example Nick Haber, Mike Lipnowski, and I think Jeremy Miller.)

For experts.

18.2: I feel like I go through contortions in defining SheafProj. The right universal property would make this better. Do you have expositions you prefer?

18.3 on projective morphisms: there are multiple definitions of projective morphism in the literature, which are not all the same. The Stacks Project (as usual) does a good job of sorting this out. I stick with EGA’s definition (rather than Hartshorne’s): I really want all finite morphisms to be projective! (I wish O(1) were part of the definition. I don’t like saying “there exists a very ample line bundle” because I don’t like using the weasel phrase “there exists” in a definition, as it is less well behaved than if one makes a choice.) The fact that composition of projective morphisms are projective I only manage to prove if the target is quasicompact. (Is there an easy counterexample otherwise?)

(Aside: I remember when learning the subject being confused for some time as to why a map from one projective curve to another is necessarily projective.)

About cohomology: I’m continually surprised and delighted by how beautiful this subject is. I had been led to think otherwise when first learning it.

An inflammatory comment (that I’ve made enough times that some people are sick of hearing it): cohomology in algebraic geometry seems to be traditionally introduced through derived functors. I remain unconvinced that this is a good way to introduce it to someone learning the subject for the first time (from a typical background — I’d make exceptions from people who had worked with homological algebra a fair bit before). In particular, it makes the subject misleadingly hard, in that you need more work to get to the basic results (i.e. building up the machinery of derived functors). There is little in Hartshorne, for example, that isn’t more easily done with Cech cohomology. (An explicit list: the Leray spectral sequence seems easier to prove in this way. And Serre duality can’t really be discussed without it, because you need derived functors to even define Ext.) Don’t get me wrong: I realize derived functors are important. But I find that they naturally come second, after the learner has had fun with explicit examples.

For everyone

The description of the cohomology of line bundles on projective space was inspired by Daniel Erman’s explanation of the proof given in Miller and Sturmfels’ excellent book “Combinatorial Commutative Algebra”.