A revised version is now posted at the usual place (the March 25, 2012 version). We have reached the end of the second quarter of our academic year, so I want to pause and look back on where we are, and fill in those who are just watching the notes evolve. (The course webpage is here.)

If we continue at the current breakneck pace, we will finish all the central material I have claimed can be covered in a single-year course. We may not succeed, but it will not be because the goal is impossible. (Instead: I have some material still to think through and prepare, and I may not manage it to my satisfaction.) I am well aware that I have 30 weeks to work with (longer than the academic year at most universities), and the people in the class are not typical, in many ways.

More precisely: in the notes, we’ve reached elliptic curves (we will begin the next quarter showing that they are group schemes). I consider everything up to 21.8 to be in very good shape. There are things that still need fixing, but I have an explicit finite list, which is large, but shrinking. I have no sections that (in my mind) need serious revision before 21.9.

Here are some rambling thoughts, both large and small, in the order in which they appear in the text. Before I begin, I should say that there are many many improvements, due to people in my class, but also a large number of sending emails from elsewhere on the globe, and also posting here. I want to repeatedly thank you for the huge number of comments you have sent in.

The section on valuative criteria (13.5) is now in potential “final form”. In other words, it is now self-contained, and open for criticism. I state the criteria (6 in total: valuative criteria for separatedness, universal closure, and properness, each in “DVR” and “general” versions), but do not prove them. I sketch the proof of the valuative criterion for separatedness in the DVR case (I basically give the proof). This is based on the discussion in the post on valuative criteria here. Please feel free to complain! (Any attempt to give a complete proof of the valuative criterion of properness ended up being longer than I wanted to include at this point.)

Fun fact (14.5.B): suppose you have a short exact sequence of quasicoherent sheaves. If the first and third are locally free, then so is the second. If the second and third are locally free and of finite rank, then so is the first. I had wondered about a counterexample if the “finite rank” hypotheses were removed. Daniel Litt has given me one, and posted it here. (Perhaps this or something like it is in the literature? Perhaps this should be added to the stacks project?)

I am mildly curious about the following (cf. 16.4). (Not curious enough that I’ve given it any thought, but curious enough that I’m hopeful someone has a very fast answer.) If S_* is a graded ring, and M_* is a graded S_*-module, if M_* is finite type, is the corresponding quasicoherent sheaf finite type? And similarly for coherence? Presumably yes.  (Update June 29, 2012:  Fred Rohrer has explained this now, see below.)

The way in which I first discuss pullbacks has evolved (17.3); three different approaches all come into it (the affine-local picture; the universal property; and the “inverse image then tensor with structure sheaf” definition). (Feedback I’d earlier gotten: one expert prefers a more general approach, doing things for ringed spaces; two learners found the exercises surprisingly straightforward. So far I’m sticking with straightforward over general.)

The notion “generated by global sections” is slightly awkward, especially when relativized. I’m using the terms “globally generated” (16.3), “finitely globally generated” (16.3), and “relatively globally generated” (18.3.7). If this potentially bothers you, please complain. Ideally make a counteroffer, or at least an argument.

Relative Proj is now done differently (see 18.2). I am now quite happy with the approach, because I have (sadly) given up on dealing with any universal property, as without it, the construction is very easy (when done in the right way). If anyone reads it, please let me know what you think, and tell me what is still confusing. (Summary of feedback to date: people find this an uninspiring topic, but the exercises are gettable.)

In Exercise 18.3.B, we show that the composition of projective morphisms is projective if the final target is quasicompact. (That wacky hypothesis is part of the sign that the notion of projective notion is not great.) I am curious: does anyone know a counterexample without the quasicompactness hypothesis? This isn’t important (it will undoubtedly never come up for me in real life). [Update August 21, 2012: I’ve now asked it on mathoverflow.]

(Update March 27, 2012: there were many typos in the Chow’s Lemma section, so a revised version is now here.) In 20.8, I prove the following form of Chow’s Lemma: if \pi: X \rightarrow \text{Spec} A is proper, and A is Noetherian, then there exists \rho: X' \rightarrow X surjective and projective, with \pi \circ \rho also projective, and with \rho an isomorphism on a dense open subset of X. I want to include all other versions that reasonable people (or even reasonably unreasonable people) might reasonably use — with references, but most likely without proofs. The versions I can think of are: (i) weaken “proper” to “finite type and separated”, and weaken the conclusion to “\pi \circ \rho is quasiprojective” (rather than projective), and (ii) a generalization where \text{Spec} A is replaced by a Noetherian scheme, and (iii) = (ii)+(i) (EGA II.5.6.1). If X is reduced, or irreducible, or integral, then we can obviously take X' to be as well. EGA II.5.6 has a variant where the target is quasicompact and separated, with a finite number of irreducible components. Are there any other variants I should care about?

Advertisements