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.

*Challenge Problems:*

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?

October 21, 2012 at 10:09 am

I just read/skimmed the new Chapter 13. Here are two things that I don’t think you say, and I think you should:

(1) One reason nonsingularity is important is because it is the only concept which makes sense in mixed characteristic. Z[x,y]/(xy-2) is a regular ring, but it isn’t smooth over anything (in particular, it is not smooth over Z). If you want to really get into this example, you can point out that it is analogous to k[x,y,t]/(xy-t), which is smooth over k but not over k[t]; in the number theory set up, there is nothing to play the role of k, so we have to talk about nonsingularity. (By the way, have you explained somewhere why you chose “nonsingular” over “regular”?)

(2) I would include the following exercise in 13.1 or 13.2. I realize that it can be done more slickly using differentials, but it is doable by hand, and it answers some questions good students will have.

Let A be a finitely generated k algebra, with generating set x_1, x_2, …, x_n. Let A=k[x_1, …, x_n]/I, and let f_1, …, f_r be a generating set for the ideal I. Let P be a prime ideal of A

(i) Let g be an element of the ideal I. Show that appending the column (\partial g/\partial x_1, \partial g/\partial x_2, …, \partial g/\partial x_n) to the Jacobian matrix doesn’t change the rank at P.

(ii) Show that the rank of the Jacobian matrix does not depend on the choice of generators for I.

(iii) Let y be an element of the ring A. Choose a polynomial q in k[x_1, …, x_n] such that y=q[x_1, …, x_n] and let h be the polynomial y-q[x_1, …, x_n] in k[x_1, …, x_n, y]. Show that the Jacobian matrix of (f_1, f_2, …, f_r, h) with respect to the variables (x_1, x_2, …, x_n, y) has the same rank at P as the Jacobian matrix of (f_1, …, f_r) with respect to (x_1, …, x_n, y).

(iv) Show that the rank of the Jacobian matrix at P is independent of the choice of generators for A.

I also feel that the different pieces of Chapter 13 don’t fit together well. Sections 1,2,3 and 7 form a natural sequence, 5 and 8 form another, and 4 and 6 feel like they should be one section called “difficult properties of regular rings”. I’m not sure where 9 fits in. I haven’t gone to the effort of figuring out all the dependencies between these sections, but I suspect that some other order would work better.

October 22, 2012 at 9:25 pm

You asked: “By the way, have you explained somewhere why you chose “nonsingular” over “regular”?” Answer: no, because I don’t have a good answer. I feel like “nonsingular” is more often used for this notion, but I’m not sure if that is because I hang with a bad crowd. On one hand, “regular” is an overused word in mathematics. On the other hand, it is often used with very roughly the same sense. “Nonsingular” has the disadvantage that it is defined as the absence of singularity. I would like to think of “smoothness” (by which I mean this notion — I am trying to avoid prejudicing the discussion by using “regularity”) to be the right notion, not “not-smoothness”.

Opinions?

October 23, 2012 at 7:24 am

The word “rank” should be “corank” in everything I wrote here.

October 24, 2012 at 10:46 am

Additional response in

comment 4 below.October 22, 2012 at 8:01 am

A proposed proof that smooth implies nonsingular. It’s long, and I would understand why you might not want to do it, but I think it fits into your approach, and it proves some nice things along the way.

New exercise 1, goes somewhere in 13.2: Smooth of dimension 0 over a field.

(1.i) If k is an algebraically closed field and A is smooth of dimension 0 over k, then A is a reduced disjoint union of copies of k.

(1.ii) If k is a field and A is smooth of dimension 0 over k, then A is a reduced disjoint union of separable extensions of k. (Hint: Exercise 13.2.F.)

(1.iii) Show that, if A is a reduced disjoint union of separable extensions of k, then A is smooth over k. (Hint: Separable field extensions have primitive elements which obey separable polynomials.)

New exercise 2, goes somewhere in 13.7: Etale implies “unramified as it is defined in number theory”.

Let Spec A –> Spec B be etale. Let Q in Spec A and let P in Spec B be the image of Q.

(2.i) Show that A_Q/(P A_Q) is etale over B_P/P. (Hint: Use 13.7.C twice.)

(2.ii) Show that P A_Q = Q A_Q. (Hint: New exercise 1)

New exercise 3, comes after new exercise 2: Let X –> Y be etale and Y be nonsingular; show that X is nonsingular. (Hint: New exercise 2)

New exercise 4, goes after defn of etale: Suppose that B[x_1, …, x_{n+r}]/(f_1, …, f_r) is as in the defn of smoothness. Show that W –> Spec B[x_{r+1}, x_{r+2},…, x_n] is etale.

Proof that smooth over a field implies nonsingular: Let X be smooth of relative dimension n over k. Passing to local charts, New Exercise 4 shows that we may assume W is etale over A_k^n. By Exercise 13.4.C, A_k^n is nonsingular. By New Exercise 3, this means that X is nonsingular. QED

October 22, 2012 at 11:16 am

Thanks very much for these thoughts, David! I am very much convinced by the thrust of what you write, and I will soon fully digest the details. I in the process of seriously rearranging Chapter 13 in response to what you wrote. (The currently public version looks quite organic because my understanding of nonsingularity and smoothness has evolved over time. This is not a defense — any manifesto such as this about how one should think about the subject should not be a wildly raucous rainforest, but instead a carefully constructed Japanese garden.)

In the next comment below are some comments from Peter Johnson who makes a case that I may not implement, but that I wanted to pass on to you all (especially in case some of you have opinions).

Something else which has had a significant effect on the chapter: I needed the fact that regular local rings are integral domains, and the arguments I had learned was (basically) the one in Atiyah-Macdonald, which is quite long (even though it develops useful ideas en route). Most references I’d seen either give the argument, or just refer to another source with that argument. But Qing Liu’s classic book “Algebraic geometry and algebraic curves” gives a short and devastatingly elegant argument. Like all

“proofs from the book”, after reading it, I don’t see how one could prove it in any other way. (I very much like Liu’s book. I think certain things are very important in such a book, and Liu does everything very well.) This saves me a section.The current sectioning (in the draft-in-progress):

13 Nonsingularity and smoothness [not just “nonsingularity”]

13.1 The Zariski tangent space

13.2 Nonsingularity, and smoothness over a field

(includes the fact that regular local rings are integral domains;

ideally this would be a bit later, but this is the last “general theory” section, and in any case it is short)

13.3 Examples (moved into a new section just because there were more than 26 exercises)

(includes the fact that affine space over a field is nonsingular, at the end)

13.4 Bertini’s Theorem

13.5 DVRs

13.6 Smooth (and etale) morphisms (first definition)

This ends the “important part” of the chapter.

The remaining sections are in progress, and not fully placed (and all starred). I may want to indicate to the reader that 13.1-6 are the most essential.

13.7 Valuative criteria for separatedness and properness

13.8 More sophisticated facts

(i.e. things I don’t want to prove, and won’t use, but are worth seeing)

13.9 Filtered rings and modules, and the Artin-Rees Lemma

(needed in 13.5, through the fact that \cap \mathfrak{m}^i = 0)

(I actually want people to read this at some point as they work through

the notes; it is short and easy, and will get used again later. It is only

in this chapter by accident, because it first gets used here.)

David, I intend to add a bunch of your suggestions, and will do so after they have been digested. (Burp!)

October 24, 2012 at 10:46 am

Additional response in

comment 4 below.October 22, 2012 at 11:19 am

Two messages from Peter Johnson are below. I will respond at some later point when I have time.

Message 1 (accompanied by an email that descent should be done in chapter 13)

13.2.F How to descend (spoiler alert):

Last gasp of presentation-dependent def. of smoothness. (Could reduce now to

case l alg. closed for a less dependent way, but very soon it won’t matter.)

Given: variety X = Spec A, presentation A = k[x_1, .., x_n]/(f_1 ,.., f_r),

extension X_l l-smooth everywhere (relative to presentation), x a point of X.

Want to see x k-smooth in X. Use a largest Jacobian matrix with det a unit

in stalk at x. Can suppose used f_1, .., f_m. Done if m = r. Avoidable, but

seems easiest to assume det is a unit in A. (If not, shrink X: add new var z,

relation z.det-1 just after f_m. det(new Jacobian matrix) = det^2. Unused

f_i are still unusable. X_l was shrunk; remains smooth rel. new relations.)

Now work in polynomial ring over extension field l. Given an f_i, i > m, and

any point (prime ideal) p of X_l, from study of 6.26 and 6.17, etc. of [GW],

can see there is some g not in p with g.f_i in ideal I =(f_1, .., f_m).

Thus 1 is in the ideal {g: g.f_i \in I}. Use a k-basis of l (with a little

care, can avoid AxC) to see that, over k, f_i is in (f_1, .. , f_m). Done!

Message 2:

Although early descent is unexpectedly easy, it bothers me that something as

fundamental as smoothness is not presented in the natural way it deserves.

It (for general morphisms) is intrinsic and highly local, which to me demands

a “correct” kind of def. At the level of stalks, want one local ring A f.p.

over another B (residue field k), with fiber product object a .. regular k-alg?

No, is unstable under base change, so instead require geometrically regular.

Study of geometrical aspects can motivate/explain the def. All (constant rel.

dim. ..) leaks out to some affine opens. NOT good to base def. on them, but

is soon harmless to work in any selected one.

The Jacobian criterion is only a computational device, maybe with limitations,

not really appropriate to include in definitions. It gives a useful test for

when there exists a generating set of relations that (using linear parts) is

l.i. in things like m/m^2, m the maximal ideal of A. Or work less locally.

How to decide about smoothness at a point? Use generators/relations for A (cf.

computations with differentials), here to be done via the Jacobian criterion.

[k-corank is a local invariant. Obvious details: from finite set of generators,

calculate corank from generating set of relations; is unchanged under adding

a redundant generator and a defining relation. So can merge two presentations

without affecting the corank. May well save proof for later or omit, but

NOTE: no need for “usual” requirements like closed points or separability.]

Under field extension, corank is stable, although ring structures may suffer.

Harder part is sufficiency, where a descent argument may be unavoidable.

A motivation that may be visible only much later: as I see it (extrapolating,

since I have not yet consulted good references), a practical aspect of formal

smoothness, related to but better than the Jacobian criterion, is to provide

inverse functions (in a completion) as power series, obtained term by term via

a Hensel-like idea that should be algorithmic when using reasonably nice rings.

Here I cannot yet imagine what can be done in this area without imposing f.p.

October 23, 2012 at 9:15 pm

Some supplementary comments on the above:

The post is (on my screen) in a font that does not distinguish I (capital i) from l (ell), and unfortunately I was using both. The last (ell) denotes a field extension of k, but I now think it prudent, maybe necessary, to consider only algebraic extensions in my earlier messages, to avoid problems with dimensions.

The proposed stalk-level def. of smoothness (of a general scheme morphism) at a point could of course be phrased to say the k-algebra from the fiber is k-smooth. Or we could require regularity only after extending to an algebraic closure of k, as that’s where to use corank (then field descent) to easily show equivalence with more “standard” definitions. (Get pure dimensionality, as the k-alg is a local ring and must be regular, hence an integral domain. Then, roughly, use that dimensions stay the same under algebraic field extension. Etc., etc.) This general approach may also simplify some hard parts of 26.2.

I would be interested to know what aspects of such a plan seem unlikely to work well, and could try to sketch more there. No doubt the “etale” approach over one field k, as the Notes may well present, also has strong points in its favor.

I liked David Speyer´s examples (and much else), with morphisms smooth except at one point. One sees from them how descent can fail when not over fields.

October 23, 2012 at 10:01 pm

Oops, I was overly cautious (consulted 12.1G, not 12.2.I).

General field extensions are not a problem here.

October 24, 2012 at 10:46 am

Additional response in

comment 4 below.October 24, 2012 at 10:38 am

This is a response to both David and Peter. A new version (dated October 23, 2012) is now posted at

the usual place. Here are some of the changes.In response to David’s first comment: I like your comment (1), and it is now implemented, in some sense, in 13.2.8. I went with an example of DVRs instead, as they will be more directly relevant. And technically, everything is smooth over itself (including of course your example). This sounds a bit silly, but I started by leading into your comment by pointing out that Z is not smooth over anything, but realized that this comment would confuse people rather than help them.

I like (2) too, and it is now implemented in 13.2.E. As I type this, I have just realized that it belongs just before 13.1.H instead. I will move it there in the next public version. (The process of making a hyperlinked pdf is surprisingly time-consuming.)

Your last paragraph, on how the pieces of the Chapter didn’t fit well together, was convincing to me as well. I have reordered the sections, and rewritten the Chapter 13 introduction (before 13.1) in a way intended to signal clearly to the reader how they should think of this material. I am also making clear that this chapter is now very much about both nonsingularity and smoothness (not what it was before, which was nonsingularity, with a helping of smoothness on the side). I am of course very happy to hear how these signposts could or should be improved.

About the later comments from David and Peter: I’d like to summarize this as your pointing out that the Smoothness-Nonsingularity Comparison Theorem (now 13.2.10) can indeed be proved with the technology of Chapter 13, rather than waiting for Chapters 22 and 26. I agree, and I am convinced. But I have tentatively decided not to give these arguments in Chapter 13. I am open to further discussion, and indeed interested in following up. Here are my reasons.

The notions of nonsingularity and smoothness take a long time to digest the first time you see them. Along with dimension, they are significantly harder than the earlier chapters, as measured by the questions and discussions that arise when I work with people learning this material. It is easy to forget this, because after the fact we develop a good intuition for it. That’s why Part IV (“Harder properties of schemes”) consists of just these two chapters. In this case, Theorem 13.2.10 can be proved, but with effort that will be comparable to the hardest results already in this (very full) chapter, and without saving much later effort in chapters 22 and 26.

You both implicitly make the good point that the reader will think of Theorem 13.2.10 as ferociously hard because we are waiting so long to complete the proof. I’ve tried to ameliorate this by stating explicitly right after Theorem 13.2.10 that the proofs *could* be done now, and by sketching (as an example) how the proof of (b) (given by David) goes. I would be interested in hearing whether you think this does the job of putting the reader in the right frame of mind. (I’m even more interested in hearing from people thinking about this for the first time — what do *you* think?)

About the specific proofs you gave: David, your proof is indeed the proof I gave later. Most of what you say is already in Chapter 13. The heart of your argument is the fact that if is an etale morphism, and is nonsingular, then is nonsingular. That is Exercise 26.2.D, and I find it much easier to do there (because we can use flatness). I didn’t fully follow your argument (in particular, on how to get at the Zariski tangent space), but found it really great that it is (should be?) possible to do this already in Chapter 13.

Peter: I also like what you wrote, and agree that we can do all this (and particular, show chart-independence, and descent of smoothness for field extensions) with the machinery already developed. I now try to say this better in 13.2.7. I find the kind of argument involved, which is also similar in spirit and difficulty to the descent-type arguments arising in (double-starred) 13.2.16 and the (starred) proof of 13.6.3, to be notably more difficult than most of the other arguments in Chapter 13. Given how much effort the readers will be spending on getting comfortable with the basic topics, I don’t want to needlessly distract them with these harder (if admittedly powerful) facts. (I should admit that these harder arguments are not fully field-tested. A number of students have solved 13.2.16’s exercises, but 13.6.3 was added after I taught the course this past year.)

David and Peter, I realize you were not necessarily advocating that these arguments should be included, and that you were responding to my question about whether it was possible to get at the Smoothness-Nonsingularity Comparison Theorem by elementary means. So I realize you don’t necessarily disagree with me.

This is now a long comment, but I want to add one more unrelated point: Qing Liu’s slick argument that regular local rings are integral domains is in 13.2.15. (It is redone in the philosophy of the rest of these notes, so if the proof has gotten mangled or inelegant, the fault is mine, not Liu’s.)

November 2, 2012 at 11:37 pm

Smoothness is an interesting topic, and good presentations seem rare.

What sources are general (more than loc. Noetherian), not overly long, and

well-written? So far I know only of [GW]. It is more detailed than the

Notes has space for. Very readable, often illuminating, almost error-free.

I found much of Ch. 13 hard (less so than Ch. 10), and ended up consulting

[GW] a lot while continuing to think through details, some not properly

understood earler. My aim is always to analyze, understand, simplify.

I believe difficulties could in part be alleviated by taking slightly

different approaches in ways to be indicated fully at some later point.

Since Ravi invites criticism and comments, I (hopefully one of many) will

send them, concentrating on mathematics and exposition, trying not to go on

too much about matters of taste where full agreement cannot be expected.

Here, extracted from longer notes, are a few more postable comments:

13.2.C: At some point you may want something stronger; harder but not long,

via ideas in [GW, Lemma 6.27] (cf. similar method in 13.2.L of Notes).

There is a slight defect in their proof of Lemma 6.27: “By Lemma 6.26”

implicitly assumes x is closed in Y. Instead, could argue to show

O_Y,x is at most d-dimensional, then slightly change what follows.

6.27 really proves: ANY point x with corank x = dim O_X,x is smooth.

Also (via reference to 6.17 at the end), it is not hard to extract a

fundamental fact on local redundancy of excess “smooth” relations.

13.2.F: Wherever the converse (field descent) is done, the last idea may

help remove doubt that there is a fairly short straightforward proof:

just translate all into pure algebra, then apply suitable methods,

giving a proof most of which is in a previous post.

13.2.E: Here or a little later, it is highly desirable to add similar easy result

that corank is unchanged under localization. So, by Affine Comparison,

the corank at p is independent of the open affine, say one in a cover.

You could save some space: almost all of 13.2.7 (the apology) can go.

Best of all, one sees at once that corank is a (very) local invariant.

Few like having to accept things on trust until MUCH later, a strategy

to use only in rare compelling cases. 13.2.10 may be one; this isn’t.

13.2.8: “.. only concept that makes sense” ?? For examples not over a field,

the relevant concept is relative smoothness. It ALWAYS makes sense.

What is so bad about mixed characteristic, say (if I understood right)

Spec GF(p) -> Spec Z, even Spec Z -> Spec Z, further examples?

26.1.4: To me, the defect in the first kind of example is not so much that the

morphism is not flat but more that it won’t be of finite type.

November 3, 2012 at 10:38 pm

I now see that 13.2.8 is based on an earlier post of David Speyer, with a good observation on “smooth” sets of relations, somehow not put into the Notes. A relevant example is Spec Z[t]/(t^2-1) -> Spec Z. Still, this belongs after a def. of relative smoothness. Where it is, smoothness had been defined only for k-schemes.

November 3, 2012 at 10:31 pm

Taking a risk (freshly written!), here is the key part of one proposed simplification.

The view taken here: smoothness is nicer than flatness (a weaker condition studied only later) and is highly local. We want to see some nontrivial equivalences not mentioning flatness, and with no unnecessary complications in arguments. In particular, will simplify and strengthen the hardest part of 26.2.2. All is done locally, working more with rings than with schemes. Smoothness of morphisms does not imply control over inverse images of points or affine opens, so we use only the relevant parts of fibers.

First, fields k. For k-schemes X (handled locally via k-algebras), adopt an absolute, visibly local and invariant def. of a k-smooth point p: its local ring is geometrically regular, which can be taken to mean that its extension to an algebra over some algebraic closure of k is a regular ring. (cf. [Liu, 3.28]; stated there for algebraic k-varieties). This is stronger than p being regular (nonsingular). [So any quasicompact separated k-smooth X (smooth at all points) is a k-variety.] Know that k-smooth X are standard k-smooth (local presentations with all given relations used in a Jacobian).

Now consider a relative and completely general setting. Let \phi: X -> S be a morphism of schemes, p a point of X with image q in S. Let A, B denote the local rings at p, q, and use the local ring map B -> A, tensoring A with the residue field k of B to get a k-algebra A_k (cf. base change to a whole fiber).

My def.: \phi is smooth at p if B is reduced, A finitely presented over B, A_k k-smooth.

Claim: This is equivalent to \phi having a standard R-smooth restriction U -> V between affine opens, with p in U, V = Spec R, \phi smooth at all points of U.

Sketch: Start by choosing generators x_1, .., x_n for A over B, with generating set f_1, …, f_m of relations (polynomials over B), chosen to have images over the residue field k where the first r of these give a standard k-smooth presentation of A_k, with a suitable r by r Jacobian submatrix invertible in B. Use that to adjust any f_i, i > r (redundant for A_k): can subtract some sum g_j.f_j (j = 1,..,r) to make f_i 0 in A_k. Properties of germs give a neighbourhood U of p, mapping to V = Spec R with R_p = B where all of the above holds at all points of U, with coefficients of the f_i now in R. If i > r and p’ is any prime of R, every coefficient of f_i must lie in p’. All such coefficients are 0, as B is reduced. So might as well use only r relations. The other direction is routine.

November 4, 2012 at 9:57 am

Assuming the above has no fatal flaw (typo: R_q not R_p), one can replace “finitely presented” by “finitely generated”. Thus a seemingly new notion of smoothness for points coincides with the usual ones when X -> Y is locally of finite type and Y is reduced.

Could use terms “smooth at a point” to mean A_k (as above) is k-smooth, and “smooth around a point” to mean standard smooth in some neighbourhood, as usual. See [GW, Def. 6.14], [where nothing except convenience would be lost by taking the open immersion j, obviously meant to be defined from f, to be an isomorphism]. Def. 13.6.2 in Notes is the same, after a minor correction (“invertible in B” <– no, not B).

November 4, 2012 at 10:53 pm

Having reflected further, I still think the suggested route to a hard result is promising, although it will give less than first thought. There is an obstacle (exemplified by Spec Z[t]/(2,t) -> Spec Z), but it does not seem overwhelming. Avoiding it calls for SOME extra hypothesis on \phi — openness? Only as a last resort would I want to impose flatness.

Sorry to have posted this so soon. It will need more time to mature. Even if a route like this is not adopted, it is very instructive to study it.

If anyone is still interested, the crux of the method is in the adjustment process, which was barely hinted at. It involves inverting matrices (NOT the Jacobian) to find a solution for a system of linear equations with the coefficents of the g_i (notation as before, degrees bounded) as unknowns. Except in “bad” cases that need more analysis, a solution in some A_k will extend to one on an affine open in R. A proof is expected; then can fine-tune the hypotheses.

November 27, 2012 at 3:19 pm

Dear Peter,

This is intended as a response to your last few posts. Thanks for them! I’ve found them enlightening. They also help keep me convinced that I do not want to get fancier in chapter 13; waters get deep very quickly, and this is already a very hard chapter.

I will continue to think about the things you have said.

November 20, 2012 at 10:30 am

Random typo: page 560, second line below (23.3.2.1), there’s a missing close parenthesis after Y.

November 21, 2012 at 4:16 pm

Thanks Charles! Now fixed (in the next version to be posted). Happy thanksgiving!

November 21, 2012 at 10:02 pm

Dear Prof. Vakil,

I am a reader of your notes, currently the version October 23, 2012. I have a suggestion on submitting typos. Could you please create an online submitting system, for example, using Google Form.

I create a demo Google Form

https://docs.google.com/spreadsheet/viewform?fromEmail=true&formkey=dDNzdTZwN2lWNjc4X2hDUHh1bkM4b2c6MQ

The response can be found at

https://docs.google.com/spreadsheet/ccc?key=0AjXZ9NPvIUZOdDNzdTZwN2lWNjc4X2hDUHh1bkM4b2c#gid=0

This is just a demo and suggestion. I hope you can create your own google form and publish the link on your course webpage. After a few years, there might be few or even no typos in your notes.

Best wishes,

Bob

November 26, 2012 at 10:42 am

Dear Bob,

Thanks very much for this! (Sorry for being slow to respond to this and other comments; I’m in a period when I’ll be quite backlogged in email, and also in responding to the posts on the website.) This is good to know. I think for now, while the content is still in the last stages of converging, I’d prefer feedback on the blog, so people can see all the discussion in one place. But soon after that, I will likely have a stage where I’d want to seek out large number of corrections, and now that I know how this works, I may set it up in this way.

Thanks!

Also, thanks for those corrections; they have now been made (in the next version to be posted). Ziyu Zhang also sent me a number of related corrections in that section, which I’ve also just implemented.

November 24, 2012 at 5:11 am

Minor correction/question: the last line of page 584: only the columns are exact, right?

PS: your sections on spectral sequences are fantastic! The spectral sequences now seem much less daunting to me than couple of days ago.

November 24, 2012 at 5:14 am

Forgot to mention the details: it’s Section 24.3.6 (Oct 23 draft)

November 27, 2012 at 3:23 pm

Thanks for the comment on spectral sequences! I understood them much better when I realized that things I knew earlier could be understood as special cases. About your question: I’ve not yet thought it through (and I hope to in a week or so), but based on my recollection: is it not through that the I^{i,j} complex is exact in both directions, and after applying G, then you lose exactness of the rows?

November 29, 2012 at 2:00 am

Hi Ravi,

The complex of I^{i,j} comes from injective resolution of the complex of F(I^i) which already lost the exactness from applying F.

But (as far as I understand) your proof of Theorem 24.3.5 is correct, you do not use the exactness of rows of I^{i,j}’s anywhere.

Thanks!

December 1, 2012 at 3:45 pm

Thanks, that makes sense to me. I’ve now changed only “note that the rows and columns are both exact” to “note that the columns are exact”. I think this resolves the error, but of course if I’m wrong or if I’ve forgotten something, please let me know. (You have sharp eyes!)

November 25, 2012 at 7:42 pm

Dear Ravi,

Some minor typos in the Oct. 23 draft:

On p. 312, in the second line of Def. 13.2.6, I think it should read

“pure of dimension d”.

On p. 313, line 9 from bottom: “wish a some insight …” has an extraneous “a”.

On p. 329, last line, I think it should read “and is smooth over k”.

On p. 629, at the end of (ii) implies (iii) in 26.2.4, there is a missing right parenthesis.

Cheers,

Matt

P.S. Let me register a vote for regularity rather than non-singularity as the adjective for describing a scheme with everywhere regular local rings. (I would only use non-singular in the case of a scheme over a field, and actually I would probably use it synonymously with smooth, and hence my usage would be in disagreement with yours over a non-perfect field. Am I in a minority here?)

November 25, 2012 at 8:27 pm

Also, are all the numbers in Ex. 13.2.M correct?

November 25, 2012 at 8:48 pm

And, is there a typo in one of the numbers in part (b) of Ex. 27.3.A?

November 27, 2012 at 3:33 pm

Hi Matt,

Thanks very much! I’ve made all the corrections except for those relating to my comments below.

(i) About “pure of dimension d”: I’m not sure of the conventional grammar here. Should it be that a variety is pure dimension d? Or a variety is of pure dimension d? Or a variety is pure of dimension d? I had written the first, and the second also sounds fine to me (and I’ve currently switched to the second), and you are suggesting the third. I feel like “pure” should modify “dimension”. (Note to myself: This wording arose somewhere else, and I changed it there too. So if Matt convinces me, I’ll need to change it twice.)

(ii) About the numbers in Ex. 13.2.M: I

thinkthey are correct (any two varieties whose codimensions in projective -space add to at most $n$ must intersect), but possibly I’m doing something silly.(iii) About 27.3.A(b): yes, you’re right — the should be . Now fixed.

(iv) About regularity vs. nonsingularity: I am interested in what others think, and will ask around. (Readers: please tell me your opinion if you have one, either here or by email!) You’re one of the few people who I’m convinced by without corroborating evidence, so I’ve currently added it to my (short) to-do list. (The reason I like this: this is an important class of points, and I think of the property not as the absence of badness, but as the presence of goodness.) Because I have to be careful in making this change, it may not happen until the new year, giving time for people to stop me.

Thank you (as always)!

November 29, 2012 at 7:16 pm

Dear Ravi,

Actually, now that you’ve pointed it out, “pure of dimension d” isn’t right; I think in this situation I say “equidimensional of dimension d”, which seems unnecessarily verbose. Perhaps “of pure dimension d” is best? (Simply saying “a variety is pure dimension d” seems weird to me, but I’m not sure I trust my judgement on this.)

Cheers, Matt

December 1, 2012 at 3:41 pm

Dear Matt,

I’ve asked around informally, and the consensus seems to be that regular is the way to go (as you recommend), and that “nonsingular” tends to be used for varieties. So I have attempted to make this change. I now use “regular” (but mention “nonsingular” for varieties). Instead of non-regular, I say singular; this is slightly awkward (and would slightly offend some people), and I can be convinced to change this. A singular point I call (as usual) a singularity. I’ve attempted to alert the reader to alternate usages. I’m fairly confident that I’ve made these changes imperfectly, and now there will be some errors in the notes.

About “of pure dimension d”: I’m happy for more people weigh in. My current sense is that what you are saying (“a variety is of pure dimension d”) is grammatically best, but that the phrase “a dimension four variety” might be in common use. (I care about these sorts of things, and there seems little harm in trying to get these things right.)

November 27, 2012 at 3:36 pm

Hi Matt: One more thing: it appears that you’ve looked at the part of Chapter 13 that I most hoped someone like you would look at, and that you haven’t blanched at the choices made. But if you have any suggestions or criticisms, I’d very much like to hear them. (And from everyone else too of course — the reading group at Stanford will reach this section in the next couple of weeks.)

December 1, 2012 at 7:17 am

I have some corrections (mostly minor typos) for part I (the other ones will follow). They all show up in the latest (23rd of October) version.

p 39/40: inconsistent notation ker_f and ker f

p 41: 8th line should be coker f is a cokernel of the kernel of g

p 41: these statements, e.g. 0->A->B exact => f mono don’t hold in arbitrary categories with 0 and kernels, the categories should be additive.

even if the chapter is on abelian categories, it is somewhat misleading here whether we consider exact sequences

only in additive or in more general categories, because they can be defined even if the category is not additive

p 69: 5th line from bottom: we are in [] additive category … + “an”

p 70: line 3, to [] other … + “the”

p 71: after 3.3.F: ‘Hence we can define terms such as …’ here cokernel presheaf appears again, although defined in the first line

p 72: 3.4.2 perhaps “s is a global section”?

p 75: 3.4.O: I found this very hard. Maybe give a little hint ‘use a skyscraper sheaf’?

p 78: 3.5.F: last sentence, ‘feel’ free

p 96: second line from bottom + ‘of’ before A_Q^1

also, the term ‘closed point’ is introduced nowhere (and we don’t yet have the topology)

p 99: 4.2.M I don’t understand at all what this exercise is about. What is there to show?

p 99: last text line: we have ‘a’/’the’ map

p 100: 4.2.O b) which map of a) do you mean? the one with I = J = 0? This is not clear. The meaning becomes clear while doing the exercise, though.

p 101: 4.2.9 ‘asperct’ in the Mumford quote

p 113: 4.6.P the first thing to prove after ‘Noetherian’ is the same as the first thing in the exercise…?

p 119: in d) in the picture it should probably say b_i instead of a_i

p 120: 5.1.4 ‘do not vanish outside of U’ should probably be ‘outside the complement of U’ or ‘in U’, in analogy to the definition with D(f)

p 121: There is a ‘)’ missing in line 5 after ‘Exercise 4.2.A’

p 125: 5.3.G a), last line: that -the- subset of X

p 130: 5.4.9 the roles of u and t are swapped w.r.t. 5.4.7; may be a little bit confusing at first

p 143: 6.2.D. I fell for this! It has nothing to do with the remark after 6.1.E. Maybe supply some little remark,

otherwise people may be thinking a lot about showing that the property “A_p is reduced” is an open property.

(there was at least one other person on math.SE who thought like this)

p 145: doubled “and only if” in third line from bottom

p 147: 6.3.7 projective varieties do not have to be of finite type?

p 148: 6.3.F: Finish (i) and (ii) of part *(b)*

also, (i) is already finished … ?

p 148: 6.3.9 c,ii) The denominators of the generators of A_{f_i} should probably have some other power than j

p 155: 6.5.C, f is a nonzero function on a reduced affine scheme (what is D(f) for arbitrary schemes?)

p 156: after 6.5.F, Thank*s* to Exercise 6.5.F, we **we** can …

p 156: second last paragraph, (v), ‘is the closure of **the** those’

p 159: 6.5.M “Prove that every p_i is an associated prime”. Do you mean of M? That doesn’t seem to be true.

Let R = k[x,y] for k a field,

and let M = (x,y)R. Then R is Noetherian and M is finitely generated over R, so the assumptions are satisfied.

For all f in M, we have Ann(f) = (0), so (0) is the only associated prime. But there does not seem to

exist a filtration with R the only factor.

As a concrete example that (0) does not need to be the only prime appearing,

I choose the filtration (0) \subset (x) \subset (x,y) = M with factors

(x) isomorphic to R/(0)=R as module and (y) \cong R/(x). The primes are (0) and (x).

The thing here is that (x) is an associated prime of M/(x), but not of M.

p 160: 6.5.P b) you write Ass capitalized but ann not?

Thank you for doing such great work! It is really a pleasure reading your notes.

December 2, 2012 at 3:49 pm

Dear Gregor,

Many thanks for your comments — you have caught many subtle points!

I’ve updated the file in response to your comments; I’m not sure when I’ll post the next version, but it shouldn’t be too long. I have a few responses. (Any comment I don’t respond to I agree with completely, and there is nothing further for me to say.)

“p 41: these statements, e.g. 0->A->B exact => f mono don’t hold in arbitrary categories with 0 and kernels, the categories should be additive.

even if the chapter is on abelian categories, it is somewhat misleading here whether we consider exact sequences

only in additive or in more general categories, because they can be defined even if the category is not additive”

True. I now make clear that we are working purely in an abelian category, at the start of 2.6.5.

“p 75: 3.4.O: I found this very hard. Maybe give a little hint ‘use a skyscraper sheaf’?”

Good idea, now done. (I’ve responded to this in a comment because it may help others who are similarly stuck.)

“p 99: 4.2.M I don’t understand at all what this exercise is about. What is there to show?”

I don’t follow you. You might mean “this is obvious”. Or you might mean something else. Can you please say more?

“p 100: 4.2.O b) which map of a) do you mean? the one with I = J = 0? This is not clear. The meaning becomes clear while doing the exercise, though.”

In the end I decided not to make any change, as you were able to figure it out in the course of the problem. I fear that if I gave more direction, I may confuse more people than I would help. (But I am open to being convinced…)

“p 130: 5.4.9 the roles of u and t are swapped w.r.t. 5.4.7; may be a little bit confusing at first”

Good point. I’ve just switched the u and the t in the last line of 5.4.7. Could you please let me know if that resolves the confusion, or if I’ve missed something?

” p 147: 6.3.7 projective varieties do not have to be of finite type?”

Yes, they do, but that follows from the definition of projective variety, so I didn’t repeat it here. (Perhaps I should…)

“p 148: 6.3.9 c,ii) The denominators of the generators of A_{f_i} should probably have some other power than j”

The exponent is now k_j.

“p 159: 6.5.M “Prove that every p_i is an associated prime”. Do you mean of M? That doesn’t seem to be true.”

You are right! Now fixed. (It was also caught by the reading group this year at Stanford, in the last week or so.)

” p 160: 6.5.P b) you write Ass capitalized but ann not?”

There was no reason for this. I’ve now fixed the inconsistency.

Thank you very much for these detailed comments! If you find anything else (no matter how small), please send it on!

December 3, 2012 at 7:00 am

Dear Ravi,

here’s just a quick follow-up to address your remarks.

Concerning 4.2.M:

“I don’t follow you. You might mean “this is obvious”. Or you might mean something else. Can you please say more?”

I thought that it seemed obvious. After all, I had to think this through while doing 4.2.I and 4.2.J, so I read 4.2.M as a “reality check” remark, but then noticed it said “exercise”. But rereading it now I can see that one may have to do a quick check, depending on how well 4.2.I/4.2.J was understood. So I guess it’s fine.

“Good point. I’ve just switched the u and the t in the last line of 5.4.7. Could you please let me know if that resolves the confusion, or if I’ve missed something?”

That resolves it pretty well, I think.

Concerning projective varieties:

“Yes, they do, but that follows from the definition of projective variety, so I didn’t repeat it here. (Perhaps I should…)”

No, you’re right, I can see it follows. I missed this, but it’s obvious if you check with the definition of projective scheme. I guess for me it will still take a while to internalize all the definitions…

I am now reading part II, but it will take a while. Thank you in advance for the fun lecture. When I’m done, I will point out my findings – it’s probably better than nitpicking every second day in a single comment about a missing letter.

December 3, 2012 at 7:01 am

Ah, I should have replied. Sorry.

[No big deal! — R.]December 3, 2012 at 8:59 am

Dear Gregor,

Thanks! In 4.2.M, I’ll add “(reality check)” to help make clearer what I want. And about nitpicking: please feel free to send things along whenever is convenient for you!

October 25, 2017 at 7:13 pm

I’ve been meaning to add this for a long time! Peter Johnson, who has given me millions of useful suggestions, has given a low-tech “Introduction to smooth morphisms of schemes” in

arXiv:1602.06569v2.November 3, 2017 at 1:55 pm

I’ve been meaning to mention this for a long time — as many of you know, Peter Johnson has given me lots of great advice on these notes. One of the things he discussed with me was smoothness, and how to think about it. So I wanted to refer you to a preprint of his: “Smooth scheme morphisms: a fresh view”, on the arXiv: http://front.math.ucdavis.edu/1602.06569