The seventeenth version of the notes is the July 21 version in the usual place. This version has a complete exposition (i.e. everything I currently intend to say) of flatness (chapter 25), and a proof of Serre duality (chapter 28). Some content is added earlier (e.g. the Artin-Rees Lemma). The next post may appear in August, depending on baby constraints.

Status report.

There are only three more content chapters still to come, one on smooth/etale/unramified morphisms; one on formal functions and related issues (Zariski’s main theorem, Stein factorization, etc.); and one on regular sequences and related issues (local complete intersections, Cohen-Macaulayness, etc.). I’m 100% sure they will appear, but I’m not sure when (again, due to oncoming family constraints). Of course, a lot of work remains to be done to fill in holes and patch problems in the rest of the notes (and responding to old comments), so I may spend some time doing that.

I also want to take this opportunity to thank Sándor Kovács for advice throughout this project (and before), both technical and otherwise.

For learners.

Flatness is confusing the first time you see it. Also the second and the third. But with each iteration, you will digest and master more aspects of flatness. With most parts of algebraic geometry, when you learn a concept, you get used to one strange thing and then you’re good to go. That’s not true with flatness — when thinking over this chapter, I realized that there are many different types of results and arguments that come up. I’ve done my best to organize them, and to discuss no more than I find brutally necessary. (Many important flatness facts are left unproved or unstated, but hopefully by the end you will know enough to be able to read what you need elsewhere.)

The structure of the chapter is described in 25.1.1, so I won’t repeat it here.

I hope some of you read the proof of cohomology and base change — if you do, please let me know, and please let me know what is most confusing!

Here as always are some suggested problems. Of course, try every exercise marked “easy”. If it isn’t easy, let me know!

Here are twelve problems on flatness: 25.2.E (transitivity of flatnes); 25.2.G (relating flatness in algebra with flatness in scheme theory); 25.2.L (explicit examples), 25.2.M (cohomology commutes with flat base change — this looks hard but isn’t), 25.3.A (explicit and important Tor calculation), 25.3.F (practice with Tor), 25.4.D (flat = torsion-free for a PID), 25.4.F (“finite flat morphisms have locally constant degree”), 25.4.I (an explicit example that will come up later, involving two planes meet at a point), 25.5.D (going-down for flat morphisms), 25.5.F (fibers of flat morphisms have the “expected dimension”), 25.7.B (important! invariance of many important numbers in flat families), some 25.8.A-F (using cohomology and base change),

If you want to work through some pleasant explicit examples, I’d recommend 25.4.8 onwards, on flat limits. Another fun discussion that will help you see if you understand flatness well enough to do something is Hironaka’s example of a proper nonprojective nonsingular threefold, 25.8.6. If you get stuck, please let me know.

If you read 25.10 on flatness and completions, please let me know how it went.

Chapter 28 is a starred proof of Serre duality. I hope some of you try to read it — it is not double-starred, which means that I intend for this to be readable, and not just an indication that a proof exists. As with flatness, I try to prove no more than I really need to given this stage of the notes/course. If you read this, some good problems to try are 28.3.C (relations among Ext, sheaf Ext, and H^i), 28.3.H (Ext and vector bundles), and 28.3.J (the local-to-global spectral sequence for Ext).

For experts (and general discussion)

The Artin-Rees Lemma.

Greg Brumfiel explained the Artin-Rees Lemma to me in a way that made it very natural — enough so that I can no longer forget the proof. I’d never understood it well before. I hope I’ve gotten it across with some semblance of Greg’s clarity (13.6 and parts of 25.10).

Question: suppose $A$ is a Noetherian ring, and $I$ an ideal. Suppose $0 \rightarrow M \rightarrow N \rightarrow P \rightarrow 0$ is an exact sequence of $A$-modules. I find it entertaining that it remains exact when tensoring by $\hat{A}$, but that $0 \rightarrow \hat{M} \rightarrow \hat{N} \rightarrow \hat{P} \rightarrow 0$ need not be exact (without coherence hypotheses on the modules — perhaps just on $P$?). Does anyone have a( reference to a)n example where exactness does not hold? (And as a consequence, we’ll see an example where completion is not the same as tensoring with $\hat{A}$.) Update August 17, 2011: An answer is given in the section “Completion is not exact” in the “Examples” chapter of the Stacks Project. I’m not sure how to find out the tag. But I’ll add this example, in the version to be released around the end of August.

Flatness.

I found the flatness notes of Brian Lehmann (available on his webpage) very nice. Andrew Critch’s enlightening a postiori explanation of how to think about flatness is incorporated into
Remark 25.4.2, just before the equational criterion. (Side remark: we don’t use the equational criterion for anything.)

I hope someone looks closely at my exposition (and proof) of Cohomology and Base Change. That’s a topic where I think I learned the right perspective only by talking to people, and part of my goal is to translate some of the folklore into writing. If you have never bothered fully understanding the proof, and want to, please take a look and let me know where the exposition confuses you. (It is now divided up into some general facts about cohomology of complexes, and a very short argument for the theorem itself.)

Max Lieblich told me that he first figured out Cohomology and Base Change by translating to local rings, and working there, which has the advantange that you can make short exact sequences split. I could imagine that this would yield an even faster exposition. (One worry: the statement I want of cohomology and base change involves an honest Zariski open set, see part (i) of my statement. But I bet Max’s approach would give that too.) Partially because I’d already written this, I haven’t tried to piece together how Max’s argument should go. But if someone does, or someone thinks I should because it would make things more transparent, please let me know.

The flatness chapter is disappointingly long — and I even didn’t prove (for example) that the flat locus is open (under reasonable hypotheses). I didn’t prove Grothendieck’s generic freeness lemma because I didn’t use it (but I stated it). I didn’t prove the fibral flatness theorem, but stated it. Are there things that I really should include? Are there things I’ve included that you think could reasonably be tossed in a first course? (You’ll noticed that lots of the chapter is already starred or double-starred.) One fact that isn’t there but will be (in a later chapter) is what Brian Conrad calls “miracle flatness”, about a morphism $\pi: X \rightarrow Y$ to a nonsingular scheme, and relating the flatness of $\pi$, the equidimensionality of the fibers, and the Cohen-Macaulayness of $X$.

Serre duality

[Update August 2012: more discussion is in the 22nd post.]

I’m going to upset some people here, by not proving the “right” statement. My goal, given that this discussion comes at the end of a long set of notes, and at the end of a long course, is to prove just enough to justify the statements made earlier in the notes.

Here’s what gets used:
(i) we need a perfect pairing (28.1.1.1) $H^i(X, \mathcal{F}) \times H^{n-i}(X, \mathcal{F}^{\vee} \otimes \omega_x) \rightarrow k$ in good circumstances.
(ii) We need the dualizing sheaf to be the determinant of the cotangent bundle if $X$ is smooth.
(iii) Perfect pairing (i) often arises from something better (which I call Strong Serre duality) which is an isomorphism $\rm{Ext}^i_X(\mathcal{F}, \omega_X) \rightarrow H^{n-i}(X, \mathcal{F})^{\vee}$.
I show (iii), but don’t show that these maps are well-behaved in any way at all — for what we do, we don’t need the perfect pairing (28.1.1.1) to be “natural” in any way — we just need dimensions. The reason I can’t show any sort of naturality (in a pedagogically easy way) is that it isn’t worth the trouble of defining the natural maps $\rm{Ext}^i_X( \mathcal{F},\mathcal{G}) \times H^j(X, \mathcal{F}) \rightarrow H^{i+j}(X, \mathcal{G})$. In 28.3.4, I do mention where these maps come from, and outline the Yoneda cup product for Ext’s (following Grothendieck’s Theoreme de dualite pour les faiscaux algebriauqes coherents — apologies for lack of accents).

The advantage of my approach is that we can prove the statement we actually use relatively easily (although not so easily that I’d remove the star from the chapter). Keep in mind that we are the end of the course; I want to prove what we use as easily as possible.

(The disadvantage is that we clearly prove the wrong statement!)

Side remark: in an earlier version of the course, I proved Serre duality via duality for finite flat morphisms. This results in a proof which is much easier and shorter. (To apply it, we need the “miracle flatness theorem” I mentioned above; but that will be included.) The serious downside of this approach was that I was unable to prove (ii). So instead I decided to go with the current exposition, which requires more work.

Random questions for experts

1. I’ve proved uppersemicontinuity of fiber dimension on the target (for a projective morphism). But I haven’t proved uppersemicontinuity of fiber dimension on the source (for locally finite type morphisms to locally Noetherian schemes; or if you really care, for locally finitely presented morphisms in general, but that’s just an easy generalization once you’ve got the hard part). I don’t know an easy proof (i.e. short given what is already done in the notes). Does anyone know one (or have a reference)? It seems to be surprisingly hard work. (I also asked for a trick solution here.)

2. A reference questions about the category of O-modules on a scheme. I have heard that they don’t have enough projectives. (I asked a variant of this question here.) Does anyone have a reference (ideally with a proof)? I’ve heard that locally free sheaves on a scheme are not necessarily projective in the category of O-modules. Reference with proof? (Update August 2, 2011: see David Speyer’s comment below.)

3. (unimportant; maybe better suited to mathoverflow) It was in grad school that I first heard about the Lefschetz principle, allowing you to reduce all statements over an algebraically closed field of char 0 to $\mathbf{C}$. Even now I’m not sure precisely what this principle is supposed to be (except in a rather baby case, where it is basically elimination of quantifiers). Is there a reference somewhere? Here is an interesting article complaining about it. (Warning: you need access to jstor to access it. Bibliographic info: A. Seidenberg, Comments on Lefschetz’s Principle, The American Mathematical Monthly, Vol. 65, No. 9 (Nov., 1958), pp. 685-690.) Here is a possible reference (that I’ve not read): Frey, Gerhard and Rück, Hans-Georg, The strong Lefschetz principle in algebraic geometry, Manuscripta Math. 55 (1986), no. 3-4, 385–401.