The eighth post is the December 17 version here. The next reading is 11.1-12.2. Section 12.3 is included in case you want to read ahead, and the double-starred Section 12.4 (proof of Krull) is included just so the chapter is complete. As always, you should 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).

As promised last time, I expect the next post to be on or about January 8, and I then hope to return to a fortnightly schedule. Many pages of changes from Daniel Murphy; Ezra Miller and the Duke group; and Brian Osserman and the Davis group are incorporated, but there are more to go. Thanks to them, and to many others!

For learners.

11.1 is mainly about separatedness, and is surprisingly straightforward. (This is one of the many ideas in algebraic geometry that inspired a musical tribute.) The Cancellation Theorem 11.1.19 is fun, has a fun proof, and is surprisingly useful.
11.2 is about rational maps. The key (easy) result is the fact that two morphisms from a reduced scheme to a separated scheme are determined by their behavior on a dense open set. I find Proposition 11.2.3 strangely confusing, and the hardest part of the chapter (see below)
11.3 is on proper morphisms. Again, it is surprisingly easy.
12.1 is on the general theory of (co)dimension.
12.2 connects dimension to transcendence theory, via Noether normalization.
(12.3 and 12.4 are about two ways in which codimension 1 behaves miraculously well.)

Eleven problems worth doing (including four involving explicit examples): 11.1.B, 11.1.C, 11.1.J, 11.2.A, 11.3.A, 12.1.A, 12.1.C, 12.1.E, 12.2.B, 12.2.E, 12.2.H.

Comments on three more problems:

  • If you aren’t very comfortable with transcendence theory, you should do 12.2.A. Please tell me where you got stuck — I want to include enough hints so the problem is as do-able as possible.
  • I want to add a hint to 12.2.D, so don’t try it unless you want a challenge.
  • Can you do 12.2.F, the exercise on “a first example of the utility of dimension theory”? This is very enlightening, but will likely be a fair bit of work. If someone manages it (or has interesting comments while failing), I’d be interested in hearing about it.

For experts.

Radiciel morphisms are now added (10.4.1, 10.4.5), but are optional.

Reference question: what’s the right reference for the Noether-Lefschetz theorem implying that the only curves on a very general surface of degree more than 3 are complete intersections?

Notation question: Ezra Miller proposes that I use colons rather than semicolons for points of projective space. He makes a convincing case. Opinions?

Terminology question: “Constructible” or “constructable”? I’m currently using “constructible”, following Bjorn Poonen (and my instincts), but my spellchecker disagrees (and some readers).

Random question: Mumford (in the red book?) mentions the following result: if f: X –> Y is surjective, and h: Y –> Z is separated and finite type, and h \circ f is proper, then h is proper. I’m happy to mention this if it is useful, but I’ve never used it. Is this anything that anyone finds important?

The proof of Prop 11.2.3 (that two birational integral separated schemes must have isomorphic open sets) is not transparent. Is there a better explanation? (Bjorn sent me an improved version just over a month ago, and I just looked at it, and I’ve just asked him about some details. So this may soon be fixed anyway.)

For everyone.

I find Theorem 11.1.19 (the “cancellation theorem for a property of morphisms”) remarkably and unexpectedly useful, in much the same way as the magic diagram. I hope I’ve made this clear. I want to give it a memorable name, and this is the best I could come up with. Does anyone have a better name?

Can someone (e.g. someone who already knows it) read the proof of Algebraic Hartogs’ Lemma (12.3.11) and Krull (12.4), and point out the parts that need renovation? I want these proofs to be short but readable (with work), so people won’t need to read them, but feel reassured by their presence.

Question: can you give a short clean complete proof that quasifinite morphisms to (Spec of) a field are finite? (Not acceptable: “Use the Nullstellensatz.”)