The twenty-fourth post is the September 25, 2012 version in the usual place. The discussion of smoothness is now incorporated. In particular, the second-last chapter to be public is now out (Chapter 26).
As always, these changes caused ripples of changes throughout the earlier chapters. I continue to be fascinated by how intricately interconnected algebraic geometry is. There are exercises worded in a particular way early on because they come into play 20 chapters later.
As usual, I am very interested in comments, corrections, and suggestions, no matter how small. (Except: as usual, I am not yet interested in issues of formatting, figures, indexing, and other things that are best dealt with en masse later.)
Larger changes, and philosophy
Smooth morphisms are surprisingly complex. There are a number of possible definitions, and it is nontrivial to connect them all. Examples include EGA (first define formal smoothness, then add local finite presentation), Mumford’s red book (flat, geometric fibers are nonsingular, and implicitly local finite presentation), and the Stacks Project (in terms of the naive cotangent complex). I decided to take as simple a definition as possible, that I could motivate and get as much out of as cheaply as possible. (My definition: is said to be smooth of relative dimension if it is locally finitely presented, flat of relative dimension , and is locally free of rank .) However for a number of reasons (both philosophical and because of desired consequences), I also wanted to have a clean local model of smooth morphisms (of relative dimension ), which is: , where the Jacobian matrix of the with respect to the first of the is nonzero. The main difficult (important) result in the chapter is essentially this (or essentially equivalently, the connection to Mumford’s definition), Theorem 26.2.4. I found no way of making this easier. [Update Sept. 26, 2012: I forgot to mention that in the non-Noetherian case, the proof that such a map of affine schemes is flat uses a version of the local criterion for flatness that I didn’t prove. I refer to tag 046Z in the stacks project for a proof. This is yet another sign that this connection is difficult.]
(To preempt some objections: I fully agree that the definition in EGA is important, and I don’t know how to get at the left-exactness of the relative cotangent or conormal sequences otherwise. But if you want to know about it, it should be relatively short but hard work after reading this chapter. And if you don’t feel the need to know about it, there is no advantage to forcing it on you. I also think the approach of the stacks project is very nice and clean, and very possibly my preferred approach after one first meets smoothness — do not be intimidated by the phrase “cotangent complex” in this context!)
In 26.1 I motivate the definitions. In 26.2 I prove their main properties. In 26.3 I prove generic smoothness (in the source and target), and the Kleiman-Bertini theorem.
There are also a number of changes in chapter 13, on nonsingularity. If you want to revisit this chapter to see what changed, look at 13.2-4. The change with the fewest repercussions is the addition of Bertini’s theorem in 13.3. (I had originally put it in the smoothness chapter; I now realize that it is so elementary that it can be done as soon as nonsingularity is introduced.) More seriously, I realized that we can prove, and need to prove (for later exposition), some things I’d stated as facts. Most notably, we now prove what I call the Smooth-Nonsingularity Comparison Theorem, which basically says that over perfect fields, smoothness equals nonsingularity (not just over closed points), and over arbitrary fields, smoothness implies nonsingularity. So 13.2 and 13.4 are rearranged (they used to be one section). 13.4 is now called “More Sophisticated Facts”.
In particular, in previous versions I made a big deal about the fact that smoothness was important, and we really didn’t care about nonsingularity. I now realize that we do care about nonsingularity when you actually prove foundational things — for example, when we use the fact that local rings are integral domains, or “slicing” inductive arguments.
- In 22.4, I define Fano, Calabi-Yau, and general type varieties, and K3 surfaces. They are easy to define, and it is good to see some examples.
- The discussion of Cohomology and Base Change is moved out of the flatness chapter into a new chapter (30) near the end, so readers don’t make the mistake of reading it before reading about smoothness.
- The chapter on the 27 lines is moved back to Chapter 27, before some of the facts needed are established. This is basically because it is morally necessary that the 27 lines chapter be chapter 27. But I will (in a later edit) encourage the reader to read this chapter before getting all the background. It is a reasonable end to a long period of learning this material, and I’d hope people jump to it as soon as they are able.
Still to come
There is only one substantive chapter left, on formal functions and related ideas (Stein factorization and Stein morphisms, Zariski’s Main Theorem, Castelnuovo’s criterion, …). The rest of the notes rely on this chapter remarkably little, so I can spend some time smoothing what is already there. I doubt I will get to this last chapter before January (after I am done teaching, and when I begin my first sabbatical). I’ll get smaller units of time to think about this, which are more suited to working through the (shrinking) to-do list.
There are a number of exercises I would very much like people to try. They contain a lot of insight, and I hope they are gettable, and if they are not gettable, I want to improve them. If anyone tries these, please let me know. I am happy to give hints, and help in all ways (as I am with all exercises). The ones I am most curious about, in order of appearance, are:
- 13.2.F (“smoothness is insensitive to extension of base field”). Update October 10, 2012: as discussed in the 25th post, one of the two directions is not gettable at this point; this half has been moved.
- 22.4.J(b) (a sister to 26.2.J below)
- 26.2.G(b) (in particular, I want to be sure I’m not mistaken in thinking that local finite presentation is not needed in the argument)
- 26.2.J (especially (b), which is used in Kleiman-Bertini)
- 26.2.K (I am hoping this is straightforward)