The fourth post is the October 9 version here. This fortnight’s reading is 5.3-6.4. At the end of it, you will begin to have some experience with schemes. If you are comfortable with it, you may wish to read ahead, so I’ve included 6.5 (on associated points — as suggested in the first paragraph of 6.5, just consider the case where A is reduced), and 7.1 (an introduction to morphisms).

As I feared, I am falling behind in responding to comments. I’ve chosen to post notes and move forward rather than wait to respond to earlier comments, but I promise to respond to comments before too long. (The fall of 2010 will be the worst time time of the academic year for me. Last week was one of the worst weeks of the fall. When the season of recommendation-writing ends, I expect to be better.)

For learners.

I’m not happy with my description of projective schemes (5.5); it seems much harder than it needs to be. If you have any suggestions (or even references where the development seems easier or more natural — in the same generality of course, where the ring is not necessarily generated in degree 1), please let me know! The key idea is that one wants to understand this in terms of covers of affines. I hope this is clear.

If I had to pick the eleven most important exercises, they would be 5.3.A, 5.3.B, 5.3.F, 5.4.A, 5.4.B, 5.4.H, 5.5.B, 6.1.E, 6.2.E, 6.4.F, and 6.4.H. But these are necessarily notably harder than many of the other problems. The best exercises to get your hands dirty with explicit examples are 5.4.B, 5.4.E, 5.4.F, 6.2.A, and 6.4. H.

I’d also like you to ask you for some advice about four exercises.

Exercise 5.5.B is important and tricky. Is it too hard? Is the hint sufficient?

Exercise 5.4.A is essential. Are the hints sufficient for you to figure it out (after a lot of thought)? This is often done explicitly (I think for example in Eisenbud and Harris), but I find that it is something that you have to do yourself to understand. If this exercise is far too hard, I should merge it into the text. If it is only a little too hard, I shouldn’t.

Is Exercise 6.4.K do-able (notably the non-UFD-ness)? I haven’t thought this part through.

For experts.

See my plea about the discussion of projective schemes above.

I call a ring graded by the nonnegative integers, with S_0 = A and the irrelevant ideal finitely generated over A, a finitely generated graded ring over A. That is slightly unwieldy, but I can’t think of anything better.

The important and tricky exercise 5.5.B is to identify a bijection between the primes of ((S_.)_f)_0 and the homogeneous prime ideals of (S_.)_f. This seems quite tricky if S_. is not generated in degree 1. (Hartshorne says that “the properties of localization” show that this is a bijection, but I don’t understand this.) If there is a better way to proceed than my hint, I’d be interested in hearing it.

Very unimportant point: Exercise 6.2.I says that a locally Noetherian scheme is integral if and only if it is connected, and all stalks (of the structure sheaf) are integral domains. Is there a counterexample if the Noetherian hypothesis is removed? (Update July 25 2011: Here is a link to an answer.) Also, I was very surprised to hear that “locally Noetherian” is not a stalk-local condition — Joe Rabinoff gave me a short neat example of a scheme that is not locally Noetherian, yet has Noetherian stalks.

I’d like to give a name (other than “Nike’s Lemma”) to the proposition that the intersection of 2 affine opens can be covered by opens that are simultaneously distinguished in both of the big affine opens (Prop. 6.3.1).

Unimportant question (cf. Ex. 6.4.G): is there a reference that if R is integrally closed, so is R[x]?

What’s the easiest example to show that factoriality is not affine-local? (Of course, I don’t just want an example stated. That’s not hard, and one is given in 6.4.K. I want an example that is easy to prove with bare hands.)

For everyone.

I’ve gradually come around to the idea that when learning about some category for the first time, the notion of isomorphism is pedagogically prior to the notion of morphism. I won’t argue about whether it is logically prior; that’s not my point. When learning groups, students first propose the notion of isomorphism (as they figure out what they mean by the intuition of two groups being the same) before the notion of how you map from one group to another. With schemes too isomorphisms come first.

In mathematics notation, we have a symbol for “is isomorphic to” (\cong). But this is the wrong notion in general: we need a symbol for “a map that is an isomorphism”. We already have a reasonable answer: a right arrow with a \sim on top. But might it be nicer to make it look slightly different, and have a symbol that looks like a \cong, where the bottom is a rightarrow? I like this idea because although it is (slightly) new notation, it is patently clear what it means, and also useful.

When dealing with projective space P^n, with “projective coordinates” x_0, …, x_n, we often use the n+1 “standard” affine patches U_0, …, U_n, and it is convenient giving names to the coordinates on the affine patches that helps avoid confusion. I’ve used “x_{i/j}” on patch U_j (with convention/definition x_{j/j}=1).

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