The sixth post is the November 5 version here. The next reading is 8.1-9.2. Sections 9.3 and 10.1 are bonuses. I had pondered including 9.3 in this batch of required reading, but I wanted to have only one “hard” section per batch if possible.

About my slowness. As promised, this fall is much busier than the rest of the academic year will be, so I continue to be very very slow on responses. I promise to start catching up by the end of the academic year, and intend to respond to essentially everything! (In the last while I’ve been working through emailed comments.) In order to accommodate my slowness, the next posting will be in three weeks (target: Nov. 27), and the posting after that will be three weeks later (very tentative target: Dec. 18). I intend to then continue posting fortnightly.

For learners.
8.1 (open immersions) is short and straightforward.
8.2 (algebraic interlude) has one pretty algebraic trick which can be used in lots of circumstances. You should make friends with Nakayama’s lemma, and in particular try to get some sense of when it can be used.
8.3 (finiteness conditions on morphisms) has a number of definitions (many taking advantage of 8.2), but isn’t too bad.
8.4 (Chevalley’s theorem and elimination theory) is supposed to be pretty, but people have found earlier versions of this section harder than I would like. How do you find it? (Feel free to email me privately.) Can anything help make it better (e.g. a particular extra sentence in some proof, or a particular exercise)?
Chapter 9: closed immersions are surprisingly tricky, and much of the trickiness is encoded in a particular exercise (9.1.F).

Exercises: Here are some good ones, although as usual you should try a random sampling of others that catch your eye. 8.1.A, 8.2.D, 8.2.H, 8.3.G and 8.3.O (practice with the Affine Communication Lemma), 8.3.H, 8.3.K, 8.3.Q, 9.1.F (hard but key), 9.1.G, 9.2.C, 9.2.N.

For worthwhile problems involving getting your hands dirty with examples (very much worth doing!), you could try 8.4.C, 9.2.B, 9.2.F, 9.2.J, 9.2.M, and 9.2.N (yes, that last one is a repeat from the above list).

I have some questions for you too, about other exercises you may try.

Can you do the Exercise 8.3.H on classifying finite morphisms to Spec k by bare hands, without having seen it before? Or would more of a hint help?

If someone (likely arithmetic or non-Noetherian) tries Exercise 8.3.R (affine local nature of locally finitely presented morphisms), please let me know. I haven’t had a chance to think it through, and I don’t have a sense of how hard it is (as an exercise). If it is too hard, I’ll just give a reference. Or if it could be easy with an outline/hint, I’d prefer to do that instead. (If you solve it, please send me your solution!)

Do you have opinions on the do-ability of Exercise 8.4.C on elimination of quantifiers over an algebraically closed field? In an earlier incarnation, people didn’t like it, but I’d botched the exposition, and I think it is now repaired.

For experts.

It seems to me that the fact that the composition of two quasiseparated morphisms is quasiseparated is from the (“wrong”) definition (that the preimage of any affine is a quasiseparated scheme) is surprisingly difficult. If true, this is another good sales pitch for thinking of things in terms of the diagonal map (coming soon, after fiber product).

My definition of quasifinite (after 8.3.Q) includes a finite type hypothesis, which I believe agrees with everyone but Hartshorne.

I use the phrase “the reduced subscheme structure on a closed subset” rather than the “reduced induced subscheme structure”. If this is terrible, please let me know.

I prove that finite morphisms are closed (8.4.4) using elimination theory. One can also prove this with the Going-Up Theorem, but I find that it takes a little care, and that the elimination theory argument is kind of pretty (using just that finite morphisms are projective, which is useful later too). [Update Nov. 15 2010: useful comments below changed my opinion on this.]

I wonder if I should mention the fact that you can check surjectivity of maps of varieties by just checking closed points, which is a nice application of Chevalley for people thinking mainly of varieties (most people). But perhaps they are already sold on how it is a geometrically important fact. (I should say that I think that Chevalley is currently undervalued in the way we often learn the subject. I only learned to appreciate it from Matthew Emerton, long after I should have.)

For everyone.

A visiting topologist asked the reasonable question: where are all the examples? That’s a fair question. In some sense we have lots of examples (we have all varieties! all schemes!), but we don’t have the ability to do too much with that. Hopefully by this posting, examples have started to appear (both geometric and arithmetic). But if any “obvious” or particularly enlightening examples are missing, please let me know!

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