The seventh post is the November 24 version here. The next reading is 9.3-10.6, which is mostly about fibered products. Section 11.1 is included in case you want to read ahead. As always: skip the starred bits (unless you are moved to read them), and skip the double-starred bits (unless you are really really moved to read them).

Example of Fiber Product

Example of Fiber Product

As promised, I will be slow a little longer: I expect the eighth posting to be in three weeks, on December 18. The ninth will likely be three weeks later (roughly January 8), and then I will begin catching up with many loose ends. Some brief thanks to people who have emailed me corrections and suggestions: I’ve incorporated significant chunks of ideas from the Duke group communicated by Ezra Miller, and Lindsay Erickson at Washington.

For learners.

9.3 consists of constructions related to “smallest closed subschemes”. The hardest thing about this is the argument that the scheme-theoretic image can be computed affine-locally (on the target) in reasonable circumstances.
10.1 shows that fibered products of schemes always exist. The proof is intended to set up how to think about similar arguments in the future. This argument is key, and quite subtle the first time you learn about it, so it is crucial that this section be readable. A double-starred section interprets this in a way that is actually quite pretty.
10.2 is about how to compute fibered products in practice, and is useful and not too hard.
10.3 is about pulling back families, and fibers of morphisms. It is mainly language.
10.4 is about properties preserved by base change, and the non-starred portion is short.
10.5 is on the Segre map.
10.6, on normalization, is here because the argument that it exists parallels the existence of the fibered product. It is short, and there are concrete problems to work on.

“Theory” exercises most worth doing: 10.1.B, 10.2.A, 10.2.B, 10.2.E, 10.4.B, 10.4.E, 10.6.A, 10.6.B, 10.6.I.

(It is also worth getting hands-on practice. Exercises of this sort include: 10.2.K, 10.3.C, 10.3.D, 10.6.F, 10.6.H, 10.6.J, 10.6.K.)

As usual, I have some questions for you:
(a) If you read the (double-starred) section 10.1.5 on describing the existence of fiber products in the language of representable functors, can you please tell me how it went, and what was most confusing (and what perhaps was enlightening)?
(b) If you try Exercise 10.2.L (that various definitions of the graph of a rational map are the same), could you please let me know how hard it is (e.g. in an email or a comment below)?
(c) Can anyone solve Exercise 10.4.C (the fact that quasifinite morphisms are preserved by base change) from the hints given, without referring to some other source?
(d) If anyone is reading the starred sections on “geometric fibers” etc., could they please try 10.4.J, and let me know how hard it is for them? I’m hoping it is easy.
(e) If anyone tries 10.6.K(b) (normalization of P^1 in a function field extension), please let me know how hard it is. I have reworded it so it is hopefully more gettable.

For experts.

I’ll begin with some language questions.
(L1) Dan Abramovich emailed me the following: I noticed that you are following Hartshorne’s translation of the French “immersion ouverte” as “open immersion”. I believe the correct translation is “open embedding”. A question to people whose french is better than mine: is he right? If so, a possible radical proposal is to be originalist, and use “open embedding” (and “closed embedding” and “locally closed embedding”). This would be making a case for a cultural change, so I presume I shouldn’t do this.
(L2) Is it reasonable to say “X is finite type” when one means to say “X is of finite type”? How about “X is a finite type A-scheme”?
(L3) Is a geometic point of X a map from Spec of an algebraically closed field into X (which is what I’d thought, and Mumford backs me up in lecture 3 of Curves of an Algebraic Surface), or the map from a separably closed field?

Next, some remarks.
(R1) Brian Osserman once told me an interesting cultural note (cf. 9.3): “EGA seems to have missed the fact that scheme-theoretic image always exists, although the proof isn’t too hard.”
(R2) The difficult point in section 9.3.1 on scheme-theoretic image is that in good circumstances, it can be computed affine-locally. Behind some of the discussion is the key fact that what matters is that (for f: X –> Y) f_* O_X is quasicoherent. But I don’t mention this — the reader doesn’t yet know about quasicoherent sheaves. I never come back to this point, and currently don’t feel guilty about it.
(R3) No one complained (in response to my previous post) when I proposed the phrase “the reduced subscheme structure on a closed subset” rather than the “reduced induced subscheme structure”. I have also realized that this concept is never used (aside from the definition of the “reduction” of a scheme, where we can work around it)! (It will get used only in the proof of the valuative criteria, but you will see that that in turn never gets used, and is mentioned only for cultural reasons, see below.) This makes me want to remove this notion — we shouldn’t keep things only for the sake of nostalgia. But it is short enough that so far it has stayed. But feel free to vote it off the boat (or to keep it on…).

Next, some questions for you.
(Q1) Is the description of the existence of fibered products using functors (10.1.5) reasonable? (Of course, it could be done very quickly and concisely; but it would be great if mortals could learn this in a comfortable and natural way soon after first seeing fibered products.)
(Q2) Notice my question for learners: Can anyone solve Exercise 10.4.C (the fact that quasifinite morphisms are preserved by base change) from the hints given, without referring to some other source? I’d like to ask experts this too: Is there a slick proof I’m missing?
(Q3) I do normalization of reduced schemes, using a universal property. (a) What is the right universal property of normalization?
(b) (less important) I could imagine that one could define normalization more generally; is there any value in this (even in the sense of giving cleaner statements)? My current opinion is “no”.

Finally, I want to give an open challenge about valuative criteria. I’ll save this for a separate post.

For everyone.

I’m thinking of including arguments (in a starred or double-starred section) showing that to check geometric reducedness, it suffices to check after extending only to the perfect closure, and the analogous statements for geometric irreducibility and geometric integrality, based on this note (thanks to Greg Brumfiel and Brian Conrad — revised link Nov. 27 thanks to Brian’s suggestions). Votes or opinions for or against? I’m partially motivated by the fact that I realized after learning some algebraic geometry that I didn’t know why the product of two irreducible complex varieties was also irreducible.

If anyone reads the (sketched, double-starred) proof of finiteness of integral closure (10.6.4), please let me know if it is readable. (I hope someone reads it…)

Chevalley’s Theorem

The earlier Chevalley exposition of 8.4 was indeed botched, and I am grateful to many of you (notably Charles and D.H.) for helping me sort this out. It is rewritten in this version, but I realize I’ve tried everyone’s patience too much to ask you to look at it again. My rearrangement of the proof of closedness of finite morphisms (thanks to Matt) has led to a rearrangement here: I no longer use elimination theory to prove Chevalley, and just use closedness of finite morphisms. I want the proof of Chevalley to have all the steps handed to the reader on a silver platter. I hope this is now the case, and I want to tweak it until I get it right.

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