September 19, 2011
March 25, 2013
A new version is now posted at the usual place (the Mar. 23, 2013 version). There’s not much to report. I’ve responded to the advice you’ve given in the previous post, done the bibliography (so in particular, you are free to criticize it), given a little more geometric motivation for completions following the advice of Andrei, and responded to more suggestions sent by email, and by in this year’s Stanford reading group.
Still to do: As usual, the figures, index, and formatting have not yet been thought about. (Again: the bibliography is off this list! I’ve gotten advice from a number of people on the index, notably Rob Lazarsfeld, and I have at least some idea of how I want to proceed.) I have a to-do list of precisely 50 items. (Perhaps in the next posted version, I may have “TODO” appearing in the text indicating where there is still work in progress.)
Questions for you: it may soon come time to make figures. Do you have recommendations on how to make pictures? You might guess my criteria: I would like to make them look reasonably nice, but they needn’t be super-fancy; and the program should be easy for me to use (I am a moron about these things), and ideally cheap or free. I’ve used xfig in the past in articles (and the figures currently made), and like it a lot, but I’ve found it imperfect for more complicated figures (with curvy things), and a little primitive for somewhat complicated figures. I’m remotely considering finding someone who is good at this, and seeing how much they might cost.
Added April 26, 2013: Charles Staats sent me this beautiful picture of a blow-up. (I currently haven’t imported it into the file, because for compiling reasons the file goes through dvi, not pdflatex; but don’t bother telling me how to fix this, as I can always ask later if it becomes urgent.) Caution: it didn’t view well on my browser; you may want to download it to view it properly.
February 26, 2013
(Update May 15, 2013: at this point, to my amazement, all of my questions have been answered, although more good answers to the rest would be most welcome. I have learned a great deal of neat stuff I should have known long before by asking these questions!)
A new version is now posted at the usual place (the Feb. 25, 2013 version). There are many small improvements and patches, but no important changes. Because I want to give you something new to look at, here is a newly added diagram of chapter dependencies. I was a little surprised by what it showed.
The “to-do” list of things to be worked on is now at about 100 items. With the exception of formatting, figures, the bibliography, and the index, I continue to be very interested in hearing of any suggestions or corrections you might have, no matter how small. We are clearly nearing the endgame.
I now want to ask advice on a number of issues all at once. These are mostly small things, and are along the lines of “what is the best reference to point learners to on this topic”. Some of the questions are “unimportant”, in the sense that I doubt they will affect my exposition (although they may be important in some larger sense). For those questions relating to some particular part of the notes, I will give the section number. Please feel free to respond by email or in the comments. Here we go!
Peter Johnson strongly preferred using “fibre” instead of “fiber”. Does anyone else feel strongly?
1.4.1 Latex question: how do you get the \varprojlim subscript in the right place? (answer here)
1.6.12 (unimportant) Do right-exact functors always commute with colimits? (For example, commutes with direct sums, which is what we use, but that is easy to check directly.) (answer here)
5.4.M Can anyone get this exercise (that, basically, says that normality descends under finite field extensions)? I think it should be gettable, but not easy, but I’ve had clues that this is harder than I thought. I want to be sure I have the level correctly gauged. (Feel free to respond by email if you get stuck.) (two positive response received so far, including this one)
5.4.N is not a unique factorization domain, but its Spec can be covered with 2 (distinguished) affine subsets, each of which are Specs of UFD’s. Is there some good reference for this? (Presumably it becomes a UFD upon inverting either 2 or 3, but I can’t see why this is the case. And of course I don’t just want to know what is true; I’d like a reference for why it is true.) Added later: I should also have added, is there a well-loved reference that shows that the class group of is ? (answer here)
6.3.K A compact complex variety can have only one algebraic structure. What is a reference? (A number of sources mention this fact, but I want an actual proof.) On a related point, in 10.3: A variety over is proper if and only if it is compact in the “usual” topology. What is a reference? (answer here and here)
6.7 In this section, I mention the Schubert cell decomposition of the Grassmannian. The key idea is that any -dimensional subspace of (where is a field; and say , …, is the standard basis of ) has a canonical basis, where the first to appear in each basis element appears with coefficient 1, and that appears in no other basis element, and that special for that basis element is “to the right” of the of the previous one. Is there a standard name for this? (Normal form? Row-reduced echelon form?) Is there a good (fairly standard) reference for it? (Perhaps this gets too far into how linear algebra is taught in different countries, and I should just not give a reference, and instead give it as an exercise.) (answer here, although I’m also happy to get more references)
8.4.H Interesting fact: I almost wanted to say that effective Cartier divisors are the same as codimension 1 regular embeddings. But I could only show this in the locally Noetherian situation (or more generally, when the structure sheaf is coherent). The reason for the problem is that the definition of effective Cartier divisor is in terms of open subsets (for good reason), while the definition of regular embedding is in terms of stalks (for good reason), and getting from the latter to the former requires Nakayama. If you think I’m not giving the right definition of one of these two notions, please let me know! (see here for an interesting follow-up, thanks to Laurent Moret-Bailly)
9.1.7 Peter Johnson did not like my use of the phrase “open subfunctor” in 9.1.7. Is anyone else bothered? How seriously? (current plan after discussing with Peter: leave as is)
9.4.E Can anyone get this exercise (that, basically, says the product of integral varieties over an algebraically closed field is also an integral variety)? I think it should be gettable, but no one I know has gotten it (possibly because I haven’t asked it in homework sets). I want to be sure I have the level correctly gauged. (Feel free to respond by email if you get stuck.) (two positive response received so far, including from Gyujin Oh)
10.3.9 Is there an example of a non-smooth group variety over a field , i.e. a finite type reduced group scheme over that is not smooth? Translation: is there a group variety that is not an algebraic group? (answer: yes! example here)
11.3.13 Over an algebraically closed field, every smooth hypersurface of degree at least in is not uniruled. What is a good reference? (I know why it is true! As with many of these questions, I’d like to know where to point people to.) (answer here)
13.8 I mention Tate’s theory of non-archimedean analytic geometry. Is there a “right” source to point the interested reader (who is just starting out) to? (possible answer here)
19.9.B In (22.214.171.124), we have , and the discussion is away from characteristic . I want to say that the normalization factor is because of characteristic , but I couldn’t convince myself that this was true. Presumably it is. Is there a good reference? (Remark for comparison: one can also write in terms of : . Here the prime factors of are and ; but the reason for the is not characteristic .) (answer: yes, see here)
20.2.H Suppose is a complex elliptic curve. Then is always or . It is if there is a nontrivial endomorphism from to itself (i.e. not just multiplication by followed by translation); the additional class comes from the graph of this endomorphism. Is there a reference for this fact that I can/should direct learners to? (answer: yes, see here)
21.5.9 Is there a good reference for the Lefschetz principle? (Examples currently mentioned: Kodaira vanishing in characteristic 0; and non-jumping of hodge numbers in characteristic 0.) (good answer here)
21.7.8 (not needed) It is a nontrivial fact that irreducible smooth projective curves of
genus have finite automorphism groups. I know three arguments: using the Neron-Severi theorem (and the Hodge index theorem) (see Hartshorne V.1.9, for example); the fact that the automorphisms are reduced and form a scheme (too hard); and by action on Weierstrass points. I am surprised that this is so hard. (Note: I know that the idea can be quickly outlined to someone learning. But I want an easy complete rigorous proof. As long as I am asking, I also want someone to give me a Tesla Roadster.)
21.7.9 Smooth curves in positive characteristic can have way more than automorphisms. Is there a “best” reference?
28.1.L Is there a canonical reference for Tsen’s theorem, that any proper flat morphism to a curve, whose geometric fibers are isomorphic to is a Zariski -bundle? Follow-up question (posted March 5), in response to David Speyer’s comment here: Does anyone have a (loved) reference for the fact that the universal plane conic (over the space of smooth plane conics) is not a -bundle? (See David Speyer’s comment for a little more detail.) (possible answer to the first question here; answer to the second question here)
29.3.B I currently define node only in the case of a variety over an algebraically closed field, in which case I say that it is something formally isomorphic to . I gesture toward the definition in other cases. For example, if is not algebraically closed, I define it as , where is a quadratic with no repeated roots. I want to say that if is reducible, then this is said to be a split node, and otherwise it is a non-split node. I’d thought this was standard notation, but google suggests otherwise. Does anyone have strong feelings about this?
29.5 (This is a follow-up to discussion in the 27th post.) I am reluctant to introduce new terminology in a well-established field, but there is a notion that I think deserves a name. Suppose is a proper morphism. (For the technically-minded, it is likely that “finitely presented” should also be added, but I will play it safe, and not include this.) Then I want to say that is [something] if the natural map is an isomorphism. Not EinStein suggested the name -connected, and I quite like this — it suggests that this notion is even stronger than connected, and suggests in what way it is stronger. Another possibility is -isomorphic (which I suggested, but which I currently like less well). Opinions? (Are you offended by giving this a new name? Or do you like one of these suggestions? Or do you have another idea?)
30.3.4 Is there a canonical (“introductory”) reference for (which will require an introduction to derived categories)? (Brian Conrad’s book Grothendieck duality and base change perhaps?) (possible answer here)
February 19, 2013
The twenty-seventh post is the February 19, 2013 version in the usual place.
Drafts of the final two chapters are now complete. At this point, all the mathematical material is essentially done. The list of things to be worked on is now strongly finite (well under 150 items). With the exception of formatting, figures, the bibliography, and the index, I am very interested in hearing of any suggestions or corrections you might have, no matter how small.
Here are the significant changes from the earlier version, in order.
The first new chapter added is the Preface. There is no mathematical content here, but I’d appreciate your comments on it. These notes are perhaps a little unusual, and I want the preface to get across the precise mission they are trying to accomplish, without spending too much time, and without sounding grandiose. I’ve noticed that people who have used the notes understand well what they try to accomplish, but those who haven’t seen them are sometimes mystified.
In 10.3.9 there is a new short section on group varieties, and in particular abelian varieties are defined, and the rigidity lemma is proved. Although it isn’t possible to give an interesting example of an abelian variety in these notes other than an elliptic curve, it seemed sensible to at least give a definition.
In 20.2.6, a short proof of the Hodge Index Theorem is given (on the convincing advice of Christian Liedtke).
And the last new chapter is Chapter 29, on completions.
29.1 is a short introduction.
29.2 gives brief algebraic background. It concludes with one of the two tricky parts of the chapter, a theorem relating completion with exactness (and flatness).
In 29.3 we finally define various sorts of singularities.
In 29.4, the Theorem of Formal Functions is stated; this is the key result of the chapter. Note: the proof is hard (and deferred to the last section of the chapter). But other than that, the rest of the chapter is surprisingly (to me) straightforward.
In 29.5, Zariski’s Connectedness Lemma and Stein Factorization are proved. As a sample application, we show that you can resolve curve singularities by blowing up.
In 29.6, Zariski’s Main Theorem is proved, and some applications are given. For example, we finally show that a morphism of locally Noetherian schemes is finite iff it is affine and proper iff it is proper and quasifinite.
In 29.7, we prove Castelnuovo’s Criterion (paying off a debt from the chapter on 27 lines), and discuss elementary transformations of ruled surfaces, and minimal surfaces.
Finally, in 29.8, the Theorem of Formal Functions is proved. I am following Brian Conrad‘s excellent explanation (and I thank him for this, as well as for a whole lot more). It applies in the proper setting (not just projective), and is surprisingly comprehensible; it builds on a number of themes we’ve seen before (including Artin-Rees, and graded modules). I think Brian told me that he was explaining Serre’s argument. The proof is double-starred, but I hope someone tries to read it, and makes sure that I have not mutilated Brian’s exposition.
I’m going to continue to work on the many loose ends, and to fix things that people continue to catch. There are also a number of issues on which I want to get advice (on references, notation, etc.). I think it makes sense to ask all at once, rather than having the questions come out in dribs and drabs (as in that case people may read the first few, but then stop paying attention). So I intend to do this in the next post, and likely within a month.
Here is an example of the sort of thing I will ask, that is relevant for the chapter just released. There is a kind of morphism that comes up a lot, and thus deserves a name. Suppose is a proper morphism of locally Noetherian schemes (Noetherian hypotheses just for safety), such that the natural map is an isomorphism. Can anyone think of a great name for such a morphism? I’d initially thought about using “Stein morphism”, but that’s terrible (as pointed out by Sándor Kovács), because it suggests something else (from complex geometry). Sándor has suggested “connected morphism”, and Burt Totaro correctly points out that this is the “right” version of “connected fibers”, but this seems imperfect because it suggests something slightly wrong. I think that “contraction” would be good, but that’s already used in higher-dimensional geometry (more precisely, the contractions there are these types of morphisms). Someone (my apologies, I can’t remember who) suggested “algebraic contraction”, which seems somehow better. But for now, I’ve not called it anything, and perhaps it is better that way.
December 17, 2012
The twenty-sixth post is the December 17, 2012 version in the usual place. A large number of small improvements have been made, and the exposition has converged substantially, although there isn’t much big to report.
The (small) changes:
The terminology “local complete intersection” is changed to “regular embedding” (the more usual language, along with “regular immersion” — note that I have gone with “embedding” rather than “immersion” throughout). This was because the notation I was using would cause confusion because of the existence of an importance class of morphisms called “local complete intersection morphisms” (“lci morphisms”).
I am more careful about distinguishing the canonical bundle of a smooth projective variety (the determinant of the cotangent bundle) from the dualizing sheaf, before they are identified in the last chapter, to avoid anyone being confused about what is being invoked when.
A proof of the classification of vector bundles on the projective line (sometimes known as Grothendieck’s Theorem) is given in 19.5.5. (A proof is possible by painful algebra, and another proof is possible using Ext’s. Because of what else is discussed in the notes, I take a middle road: an easy proof without Ext’s, which relies on an easy calculation with 2 by 2 matrices. Note: I have not done the important fact that classifies extensions, which has tied my hands a little.) This required popping out “a first glimpse of Serre duality” into its own section (19.5).
Some discussion relating to Poncelet’s Porism is added in 20.10.7.
Still to do.
The introduction and the Zariski’s Main Theorem / Formal Functions chapter are still to be written. (And of course things like the index, figures, and formatting won’t be dealt with until the end.) The only other substantive things still to be written are a brief discussion of radicial morphisms (for arithmetic folks) and a brief proof and discussion of the Hodge Index Theorem (for geometric folks). Other than that, I expect essentially no other new material to be added.
I’ve caught up with almost all of the corrections and suggestions people have given me, except for a double-digit number of pages from both Peter Johnson and Jason Ferguson. (Thanks in particular to the comments of the reading group at Stanford for useful comments: Macky, Brian, Zeb, Michael, Evan, and Lynnelle!)
My to-do list still has 212 things on it. But that is a drastic improvement; the notes are converging rapidly.
I will not make any further progress until January. I hope to have a draft of the introduction done in January, and the final substantive chapter in February.
Happy holidays everyone!
October 10, 2012
The twenty-fifth post is the October 10, 2012 version in the usual place. (Update Oct. 24: a newer version, dated October 23, 2012, is posted there now. Some of the changes are discussed in the fourth comment below.) The discussion of smooth, etale, and unramified morphisms has been moved around significantly. Johan de Jong pointed out that “unramified” should best have “locally finite type” hypotheses, thereby making its link with the other two notions more tenuous; and Peter Johnson pointed out that one could give the definition of smoothness much earlier, at the cost of initially giving an imperfect definition (a trade-off I will happily take).
I am very interested in having these changes field-tested. (Most of the rest of the notes are now quite robust thanks to the intense scrutiny they have been subjected to.) I know that when something is revised, the revisions are looked at much less. But I am hoping that someone hoping to learn about smoothness, or solidify their understanding, will give this a shot in the next couple of months. I know that in the course of doing this, my understanding of these ideas has been radically improved. Because the actual algebra was elided in most of the “standard sources”, I hadn’t realized what was important and what was unimportant, and what didn’t need to be hard and what needed to be hard. So I can at least make a promise to many readers that they might learn something new.
Here are the changes, along with suggestions of what to read (for those who have read earlier versions).
Chapter 13: Nonsingularity
13.2.8 The Smoothness-Nonsingularity Theorem is an important player. (a) If k is perfect, every nonsingular finite type k-scheme is smooth. (b) Every smooth k-scheme is nonsingular. This gets stated early, but proved late. To read: the statement of the Theorem. (To experts: Am I missing easy proofs? I think it has to be as hard as it is. Update Oct. 24, 2012: David Speyer and Peter Johnson have outlined proofs in the comments below, using just the technology of Chapter 13.)
In 13.4, I had a bad exercise, which stated that if l/k is a field extension, and X is a finite type k-scheme, then X is smooth if and only if its base change to l is smooth. One direction is easy, but I’m not even sure how to do the other direction at this point in the notes. This converse direction is now 22.2.W, which I’ll discuss bellow. To read: nothing.
13.7 is the new section on smooth morphisms, including a little motivation. Everything is easy, except showing that this definition of smooth morphisms correctly specializes to the older definition of smoothness over a field. (Notational clash that I have not resolved: the “relative dimension” of a smooth morphism is n in this section, but was d earlier. There are reasons why I couldn’t change the n to d and vice versa. I don’t think this will be confusing. (But in general I have tried hard to be consistent with notation.) To read: these 3 1/2 pages.
Chapter 22: Differentials
22.2.28-30 (a very short bit): Here a second (third?) definition of smoothness over a field is given (as we can now discuss differentials) — this was in the older version. The second definition allows us to check smoothness on any open cover, for the first time, which in turn allows us to more easily check (in 22.2.W) that smoothness of a finite type k-scheme is equivalent to smoothness after any given base field extension. This in turn allows us to establish an important fact in 22.2.X: a variety over a perfect field is smooth if and only if it is nonsingular at its closed points. This had early been in Chapter 13, but relied on 22.2.W. This also establishes part of the Smoothness-Nonsingularity Comparison Theorem. To read: 22.2.W and X (very short). Update Oct. 24, 2012: in the Oct. 23 version, this is now made into a new section, 22.3, which also includes generic smoothness. 22.2.W and X are now 22.3.C and D.
22.5: Unramified morphisms are now discussed here. This section is easy. To read: 1.5 pages. (Update October 24, 2012: the new section 22.3 bounces this section forward to 22.6 in the Oct. 23 version.)
Chapter 26: Smooth, etale, and unramified morphisms revisited
This chapter is notably shrunk. 26.1 still has motivation, but now the definitions I used to give are now just “Desired Alternate Definitions”. 26.2 now discusses “Different characterizations of smooth and etale morphisms, and their consequences”. The central (hard) result is Theorem 26.2.2, which gives a bunch of equivalent characterizations of smoothness. Before proving it, a number of applications are given. The statement of Theorem 26.2.2 is rearranged, the proof is the same. To read: skim 26.1 and read 26.2 up until the (and not including) the proof of 26.2.2: 4 light pages.
There are some problems I would still like to see worked out by real people.
26.2.E: I moved this from the “unramified” section, but should probably move it back, as I think it can be done with what people know there.
26.2.F: Is this gettable?
September 25, 2012
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)
September 5, 2012
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.
- 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).
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.