The twenty-third post is the September 5, 2012 version in the usual place. The significant new additions deal with local complete intersections, regular sequences, depth, and Cohen-Macaulayness. In particular, the third-last chapter to be public is now out (Chapter 28). (Still to come: formal functions and related notions; and etale/smooth/unramified morphisms.)
As always, 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.)
The new chapter (Chapter 28) is only 10 pages long, but there are notable changes earlier on. Moreover, you may be surprised that the file is shorter — the last page is now p. 690, not 692. This is all a sign that my understanding of these topics has greatly improved while placing them into the architecture of these notes. (Expressions of gratitude: Long ago, Karen Smith and Mike Roth separately explained how to think about these ideas in a good way. More recently, Burt Totaro helped me clean up my thinking, and pointed me to some good references.) Many of the things I wanted to add were better discussed as part of earlier discussions, rather than waiting so late. (I am conscious that this makes the earlier chapters even longer. I am aware of how foie gras is made, and I do not want to do the same thing to these notes, or to the reader.)
Here is a list of changes, culminating with the new chapter 28 on “Depth and Cohen-Macaulayness”. Some specific suggestions for learners are included, as well as some questions for other readers, especially about whether some exercises are gettable given what I say.
9.4 Regular sequences and locally complete intersections
is a new section in the chapter on closed subschemes. Regular sequences are introduced here. Effective Cartier divisors are now introduced here (rather than being some stray comment wherever they were before), as a first example. The reason for the “local” nature of regular sequences is discussed, and the key hard result is that regular sequences remain regular upon any reordering if the ring is local (Theorem 9.4.4). This section is not hard, although I fear it may be distracting. But I feel happier that local complete intersections have a key place in the exposition. (They did not come up naturally for me when I first learned algebraic geometry, and I paid a price for this.)
13.4 Regular local rings are integral domains
is a new section, whose main purpose is the theorem stated in the title. This is a surprisingly hard fact, and I now use it later several times. My goal (as always) is to get what I need as quickly as possible, rather developing all the surrounding theory. Length (used later) and associated graded rings (not used later) are introduced.
Some interesting points:
- In the appendix to Fulton’s Intersection Theory, Lemma A.6.2, gives a delightful short proof using blowing-up; it implicitly works only for varieties. Geometers may like it.
- From the inequality ( is the degree of the Hilbert polynomial of a local ring ), we can get a proof of Krull’s Principal Ideal Theorem. But I need the “multi-equation” version of Krull’s theorem, and I couldn’t see how to prove it in as easy a way, so I’ve kept the proof of Krull’s theorem currently in the notes, which is less motivated, but short and double-starred. But if anyone knows of a short proof of the multi-equation version of Krull’s theorem in this vein, please let me know!
- It is true that , but we don’t need it, so I don’t go through the significantly extra work to show it.
- A question about something earlier in Chapter 13: I am disturbed that I know why a smooth -scheme is nonsingular at closed points only if is perfect. (The statement for general is, for example, in tag 00TT of the stacks project, but I find this a depressingly hard fact.) Does anyone know of an easy explanation?
For learners: please let me know if you can understand this section, and can do the exercises. In particular, 13.4.C, 13.4.G, and 13.4.H are important — can you get them? Exercises 13.4.D and 13.4.E are lots of fun.
Excised sections:
- The old section 13.3 on “two pleasant [unproved] facts” is now cannibalized, as the theorem that localizations of regular local rings are regular is now stated back in 13.2.14 (and we now prove the case of localizations of finite type algebras over perfect fields, see Thm. 13.2.15 and the Chapter 22 discussion below), and the theorem that regular local rings are UFDs is stated here in 13.4.
- Section 13.7 on completions is now removed, as it is no longer used. (It used to be used to show that regular local rings were integral domains in the case of particularly nice varieties.) The only downside is that I’ve lost the definition of “node”, which at some point will have to be put back in.
- On a related note: I removed the section on flatness and completion (in the flatness chapter), because it never was used, and it allowed me to remove 13.7.
In Chapter 22 (Differentials):
- The conormal sheaf to a local complete intersection is easily and quickly shown to be locally free, in Proposition 22.2.16.
- The conormal exact sequence is shown to be left-exact for closed embeddings of smooth varieties in Theorem 22.2.26. I was surprised at how easy this was. (Is this similarly easy to learners? Admittedly, it uses a lot of things from earlier on.) I did not do left-exactness in more general situations, but gave references (Remark 22.2.27). It is possible I will do some of this in the forthcoming chapter on smoothness, but I very possibly will not — we don’t need it, and right now it seems like hard work.
- The fact that localizations of regular local rings are regular for localizations of finitely generated algebras over perfect fields (Theorem 13.2.15) is proved in 22.4.10 and 22.4.J. This was surprisingly easy to me. (Do you agree? Disagree?) So we finally know that affine space over a perfect field is nonsingular!
- Learners: can you get Exercise 22.3.D on differentials of discrete valuation rings? I’ve reworked it, following an excellent suggestion of John Pardon.
Chapter 23: Blowing up
This starred chapter used to be Chapter 19, but is now moved after differentials, because the conormal sheaf/cone/bundle plays an important role in later subsections. Now added: for a local complete intersection, the exceptional divisor is the projectivized normal bundle (and related facts, see Exercise 23.3.D), and the blow-up of a smooth subvariety of a smooth variety is smooth (Theorem 23.3.10). This relies on the fact that for a local complete intersection, is an isomorphism. This requires a little work; I followed Fulton’s slick argument in A.6.1 in Intersection Theory.
We finally come to:
Chapter 28: Depth and Cohen-Macaulayness.
Although Koszul complexes are a central tool for understanding depth and Cohen-Macaulayness, I avoid using them, following my usual philosophy of moving as briskly as possible to what we need, and not developing surrounding theory.
18.1 is introduces depth of Noetherian local rings. Because I find it an algebraic rather than geometric concept, we concentrate on developing some geometric sense of what it means. The main technical result in this section is the argument the cohomological interpretation of maximal sequences.
18.2 introduces Cohen-Macaulayness. A number of results are shown (e.g. equidimensionality, no embedded points, slicing criterion). Important but unneeded results are only stated in 28.2.13. A highlight of this section is the short proof of the “miracle flatness theorem”, which we make important use of in the two last chapters. (It is highly possible that this name is due to Brian Conrad. I like it.)
18.3 gives Serre’s criterion for normality. It seemed worth including, because we would like to know that regular local rings are integrally closed (with proof, not just invoking a fact we haven’t proved), so we can use all of our foundational work on normal schemes. But I’ve starred this section, in the hopes that people will read it only if they really want to.
Question for experts: In 18.3, to show that regular local rings are normal, I need the fact that they are R1. Is there an easy explanation for this fact? Currently, I quote the hard fact that localizaton of regular local rings are local, which I actually prove in the case most interesting to most people (finite type over a perfect field, see Theorem 13.2.15 above).
For learners:
Please let me know which exercises you find difficult! (And, if you have the time, even which exercises you solved!)
Exercises you should certainly try: 28.1.A, 28.1.F, 28.2.A, 28.2.B, 28.2.D, 28.2.E (the most fun exercise in this chapter), 28.2.F, 28.3.B.
Please tell me how hard you find these, and if you get them: 28.2.D and 28.2.F.
September 10, 2012 at 8:18 pm
Interesting point in 13.4, on nonsingularity of k-smooth schemes:
Goertz and Wedhorn (Lemma 6.26, p159) have a beautiful short
proof using theory very much like that presented in the Notes,
plus an easy fact involving very dense sets.
September 17, 2012 at 8:32 am
In response to the first comment: thanks Peter! I’ve been thinking about this point recently, and this gives me an excuse to collect my thoughts.
This issue was the last one to think through in the exposition of Chapter 26. (I would like to have a public draft of this by Friday, as I have now figured out what I want to say, but that may be optimistic because of a couple of talks to give out of town. And if I don’t manage it by this Friday, then it may be some time, as our quarter is starting, so there are many duties coming up right now.) This is one of the few points that I would like to cover despite the fact that most people won’t care, because it makes statements much cleaner.
I will not do much more than mention the definition of formally smooth (and formally etale, and the EGA definition of formally unramified), because although this approach to smoothness (etc.) is much more powerful, it makes simpler things harder, and violates my goal of trying to make everything fit within a year, and be as elementary and complete and self-contained as possible.
Because this comment is long, I will highlight two points. (i) I want to show that
(*) smooth varieties over a field are nonsingular.
(My definition of smoothness of relative dimension is: locally finitely presented, flat of relative dimension , and is locally free of rank . So over a field, this just means finite type, pure dimension , and is locally free of rank .) (ii) I am currently unable to do this if is not perfect in a way that is not “hard”; I can think of a number of different ways of doing this, but all require something difficult. There seems to be a “hard wall” to get through, and one can get through it in a number of ways, but I cannot see how to get around it.
Let me describe what happens in some of my favorite references in the literature.
(a) A number just go ahead and prove it. Grothendieck=EGA takes formal smoothness as part of the definition, and works hard. Matsumura’s Commutative Algebra follows Grothendieck in using the “formal smoothness” approach. The stacks project has an argument that is “easy” except for one key point: the naive cotangent complex (the “Nederlander complex”) is used to show that if you have a finite presentation of a smooth -algebra cut out by ideal , then is locally free. (This seems to be necessarily subtle — flatness needs to be used, for example. But this remains my preferred approach.)
(b) A number just duck the issue. Hartshorne does of course (as it does with all commutative algebra). Section 2.2 in Neron Models on smoothness ducks an equivalent key issue, by just quoting facts in algebra.
(c) Some seem to approach the result in an “easy” manner, but I get confused trying to follow them. For example, in Mumford’s red book, in III.10 (p 306 of the 1st edition), he works over , and then “lifts” to , with the blithe statement that “In fact, we can even….”. (You’ll have to take a look to see what I mean.) I don’t yet understand why we can do that. In Goertz-Wedhorn’s excellent book, I’m aware of the passage you mention, but the argument appeals to Example 6.5, which seems to have the additional hypothesis that the point in question is -rational. (The issue I get confused about is if the point is not separable over .)
So right now, I just mention the fact that smooth varieties are nonsingular (and prove it if is perfect, and give a reference in the general case) and go from there.
To put this in context, here is a result I like to mention. There is a “local model” of smoothness: is smooth of relative dimension if locally on the source it can be written as , where the Jacobian matrix of the with respect to the first variables is invertible. The stacks project calls this “standard smooth”. This is enlightening in a number of ways. For example, (i) there is no mention of flatness. (ii) It isn’t hard to show that such a model is smooth, given an appropriate local criterion for flatness (which is admittedly hard in the non-Noetherian case). (iii) It is immediate that something standard smooth is locally a standard-etale cover of affine space over . (iv) It readily implies (*). (Method: show that is nonsingular — which is not hard; I was deliberately disingenuous in an earlier version of the notes. Then show that etale covers of nonsingular schemes are nonsingular, using etale = unramified + flat.)
But it seems hard to show that smooth implies locally “standard smooth”. The stacks project does this using the Nederlander complex. Mumford attempts to this by looking at a geometric point (and then I get confused). I currently do this by looking at a point, but using (*), so I can’t use it to prove (*).
I’ll likely explain this in my forthcoming post when Chapter 26 is suitable for public consumption. But I’m writing this comment in the hope that someone will suggest a way around the wall, or at least give me their thoughts on the matter.
September 18, 2012 at 11:27 am
Since there may be some urgency, and this blog is so quiet, I will plunge in again with my non-expert opinions, limited to a few aspects that I understand.
As Ravi pointed out, and my superficial reading failed to detect, there is a gap in [GW] that may well need significant extra work to fill. However, it looks as if Mumford provides one (fairly long) way to bridge that gap.
I have at hand only the non-red typo-ridden version of [Mu]. Looking around p222, where R/P is embedded in an alg. closed field Omega, with k the dimension of the fibre F, the A bar there must be A tensor Omega (say first tensor with R/P), where A is the given ideal of R[X1,…,X_n+k]. The k elements f_i bar in A bar chosen to have differentials that span the fibre F are, roughly, Omega-linear combinations of elements g_ij coming from A, so we can use k of these to obtain a new basis of differentials for F and calculate a Jacobian det. in R[X1,…,X_n+k]/A.
I did not try to digest other parts that make up the proof of Theorem 3′. The Lemma at the end, with its elaborate use of flatness and alg. closure, seems to be where this approach has its hard wall. Conceivably its study could cast some light on the approach to smoothness you want to present.
September 21, 2012 at 3:31 pm
Dear Peter,
Thanks very much (as always) for your thoughtful comments. I am now happy with Mumford’s argument, and it can be restated so as to work in complete generality (not just for varieties, and indeed without Noetherian hypotheses). I have partially digested where I think the cleverness comes in. I currently think there are two points, both of which I had earlier tried to use, but which I had not combined in the right way. I will mention them rather briefly (which will probably be useless to everyone but myself). First, there is the useful fact about flat morphisms that when you pull back ideal sheaves as quasicoherent sheaves, you get the same thing as when you pull them back “as closed subschemes”. Second, there is “faithfully flat descent” in the trivial case of field extensions.
But the upshot is now this: I think the proof (and the latest round of additions, including the rest of this chapter) is now complete, and I’ll smooth it out a little bit before posting. In particular, there is now a proof that smooth varieties are always nonsingular.
September 21, 2012 at 6:47 am
A suggestion for an example to add: Let be a nonperfect field, and not a power. The curve , at the closed point , is regular but not smooth.
Your current example of a ring which is regular but not smooth is zero dimensional and depends on specifying what ground field you consider it over. I think I would have found it “too small” to be clarifying. The above example is the one I banged my head against for a few days until I got it,.
Still thinking about your search for a short proof that smooth implies regular at points defined over inseparable extensions!
September 21, 2012 at 3:34 pm
Hi David,
Thanks! I have now added this example. I don’t fully understand one thing about what you write, though — even in this example, you have to specify the ground field, and indeed “smoothness” is always with respect to some target, which in this case has to be some chosen field. Could you say a bit more about that.
And about the proof that smooth implies regular: as I mentioned in the just-posted response to Peter Johnson, I now have *a* proof (which is better than just a reference, which in turn is better than no proof at all). But I still wish there were some way of seeing it in a way that wasn’t so hard. I now have the feeling that this isn’t possible.
September 23, 2012 at 5:59 pm
Thanks for making me think through what I mean about specifying the ground field. Let
The ring contains (in fact is) the field . So it makes sense to consider as a scheme over either or . It is smooth in the second sense but not the first.
The ring does not contain . If you are going to consider as a scheme over a ground field, you have to use ; there is no other option.
Further thinking about why I had phrased this badly made me realize that the following is probably worth spelling out explicitly if it isn’t already. Regularity is preserved by restriction of scalars: If is a algebra, and , then is regular whether we think of it as a algebra or a algebra.
Smoothness is preserved by change of base field. If is a algebra, then is smooth over if and only if is smooth over . If one conceptualizes an algebraic variety as a set of equations; the same equations will be smooth or singular over any field . (Caveats: must contain the coefficients of the equations, and we must think of the coefficients as living in some field with a definite characteristic, so the fact that is smooth in odd characteristic and not in characteristic is not a counterexample).
September 25, 2012 at 9:33 pm
Thanks, David, I now see what you are getting at, and it is a good point. Section 13.2.8 in the version I just posted (September 25, 2012) now has your example.
I found your point that regularity is preserved by restriction of scalars, while smoothness is preserved by extension of base field, to be interesting, and novel to me. I decided not to include it because I felt it might confuse the issue; I’m trying to present them as two completely different animals with very little in common.
If you are unhappy with the new 13.2.8, please let me know! (I guess you can let me know if you are happy too, but I’ll optimistically interpret silence as happiness. That’s how I know that 7 billion people are very happy with the current version…)