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.

**Smaller changes**

- 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.

**Challenge problems**

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 26, 2012 at 7:22 am

So, I think that in the more recent literature there *is* a consensus about what an unramified morphism is. Namely, it is a morphism which is locally of finite type with vanishing relative differentials. This is how Raynaud defined it in his book on henselian rings. This definition has the advantage that a closed immersion is always unramified. Moreover, an unramified morphism can be characterized as one which is \’etale locally (on source and target) a closed immersion.

When I first wrote the section on unramified morphisms for the stacks project, I used the EGA definition (using locally of finite presentation). This is still in the stacks project (as G-unramified morphisms) because we never remove material from the stacks project (although we can move it to the obsolete chapter). Although “G-unramified” fits well with the notions of syntomic, smooth, and etale morphisms, it turns out that virtually all the later results for G-unramified morphisms, are valid for unramified morphisms. On advice of David Rydh, I then changed it to the current definition. I really urge you to reconsider your choice, or make an even bigger disclaimer in the text.

September 26, 2012 at 3:16 pm

Hi Johan,

Great, I’m convinced. I’ll make this change, and have a caveat in the reverse direction (that EGA uses local finite presentation, but that the current consensus appears to be locally finite type). This slightly messes up the nice trichotomy of the EGA definition, and perhaps makes unramified morphisms in more out of place compared to smooth/etale. But that’s life.

September 26, 2012 at 8:05 am

In the stacks project there are likely many places where the argument given isn’t optimal. I’m very keen on finding improvements in exposition. I’ve looked at the exposition on smooth morphisms a while back and most of the arguments related to smoothness seem to be relatively slick, except for perhaps the proof of

Lemma Tag 00TF: “a flat finitely presented ring map with smooth fibre rings is smooth” (see the blog post entitled “Smooth ring maps” on the stacks project blog[). However, now I think we can improve/shorten/simplify the proof of this lemma, using the argument of the proof of 26.2.4 in FOAG. Somehow this argument is missing completely from the stacks project (why?). In factHere is the link.-R]Lemma Tag 07REis a very weak version of it but its proof doesn’t use the correct argument. There may also be a simplification in the section on lci maps. I’ll try to get to this next week sometime.PS: Lemma Tag 00TF as formulated only assumes flatness on the level of local rings, so some version of openness of flatness may be required there.

September 26, 2012 at 3:20 pm

I agree that the stacks development is slick. It’s been some time since I looked at the proof of 00TF; I don’t remember being bothered by it, but I always found this the key sticking point, so my standards would have been low (and my memory is hazy).

I should have mentioned that in the development in the notes, in the non-Noetherian case, I need a stronger local criterion for flatness than I proved (and I just give a reference to the stacks project

tag 046Z). I have just added this to the post above.September 26, 2012 at 4:34 pm

See this commit for the first change related to this discussion. More to come.

September 26, 2012 at 4:46 pm

Actually, this change also fixes the proof of 00TF as the first thing that proof does is use Lemma Tag 00SY whose proof I just fixed. The rest of the argument is straightforward. It isn’t as short as the proof in FOAG simply because we rely on the material on relative global complete intersections which I think is fine for the stacks project.

September 26, 2012 at 4:49 pm

Excellent! I’ve not yet done more than skim it, but I’ll read it in detail soon. I find this quite straightforward now. I’m unaware of seeing it in such a straightforward way before (except perhaps in conversation with you around 2000).

September 27, 2012 at 6:20 pm

Scattered remarks about the current (Sept 25) treatment of smoothness:

First, I was too hasty in dismissing the proof of [GW, Lemma 6.26]. It relies on the usual deep results on regular rings, as in [M-CA], including the regularity of A^n_k, but that’s ALL you need to accept. IGNORE the ref. to Example 6.5. The y is just some maximal ideal inthe polynomial ring mod ideal I gen. by the f_i. Mod out y to get a field K, so m_y/(m_y)^2 is a K-vector space. The local ring at y is regular of dim n. The Jacobian det at y is nonzero “in” K. If K-linear independence failed, would have a sum of polynomials g_i.f_i in (m_y)^2 (always mod I), with not all g_i in m_y. Now take all the partial derivatives of this sum and find a contradiction. [It looks to me, from Th. 19.5 of [M-CA], as if the argument even goes through with any regular ring in place of k. I’m no expert, and do not yet understand why there is so much focus on smoothness over fields.]

In Notes, why be ashamed of naive (locally standard) def. of smoothness, in fancy language loc. a certain \’etale cover (as mentioned, without def., in 26.2.5)? CAN define smooth in 13.2.4, over rings. Only much later will prove and use deeper equivalences. Probably best to start relatively: say maps are smooth when via some covers they have local form like that in 26.2.L. Until later, only use X -> Spec(field). Note: end of def. in 13.2.4 makes “loc. of finite type” redundant.

What seems to bother most about this is the choice of affine covers. But, given any suitable Spec A -> Spec B, it is easy to localize one if we also localize the other, keeping the rank condition. Roughly, given a non-nilpotent g in A, add a new variable z just after x_1, .. x_r, and a new relation f_{r+1} = g.z-1; matrix is larger, with same det as before. Then think about affine communication. In the “absolute” case X -> Spec(k), get a base of “nice” affine opens. One reason for preferring k-schemes is that localization clearly preserves pure dimension d. No need to wait for Ch. 26. Admittedly, it’s best to defer studying the choice of the f_i.

A matter of taste, but one could equally well give definitions at the level of stalks (local rings), with nonzero det in some k(p). As usual, this will then “leak out” to some affine neighborhood.

In 26.2.L, there seems to be NO need to have W distinct from Spec A (just shrink Spec A as above). Similarly, could simplify [GW, Def. 6.14] (where the relation between j and f was accidentally omitted): the open immersion j might as well be the identity. In 26.2.4(iv), the diagram just “says” a morphism is smooth in the naive (locally standard) sense.

October 3, 2012 at 5:58 am

Dear Peter,

My apologies for the typical slowness in my response, which will continue throughout the fall quarter. Let me address what I see as your central and most important point, in paragraph 3, about when one can address smoothness. There are arguments on both sides, and I could give some arguments against, in order to give other readers some sense of them. But for the sake of brevity, I won’t, and go straight to the point. I stated at the start of the current chapter that smoothness is introduced disappointingly late. You point out that one can give an imperfect definition far far earlier. I tend to like such tradeoffs a lot, and indeed I gave two earlier definitions of smoothness over a field before getting to the “right” one precisely because of this. My main concern when I first entertained this was whether it would be too hard (or impossible) to connect this to the naive definition of smoothness over a field that I used, as it would require some potentially tricky argument at least implicitly involving faithfully flat descent (in the “trivial” setting of field extensions), of the sort that is used in the hardest part of the hardest theorem in the smoothness chapter. I’ve since thought this out (while sitting in Berkeley traffic this Saturday), and I think it shouldn’t be bad at all.

So in short, I am currently quite taken by this idea, and want to see if it can work. (Translation: I will try to write it. I’ll moved “unramifiedness” out first, following Johan’s suggestion, and then try this.)

I’ll skip responding to the smaller suggestions in this comment and the next one (as I’ve only a minute now). The ones relating to the implementation of your radical movement of smoothness I may bypass, as my intended exposition may be different. The other comments I’ll read when I have a chance, and think through fairly soon. (I still have tens of pages from you of other emailed comments that will be addressed in the next few months!)

So thanks very much for this suggestion! It has actually (at least tentatively) significantly changed how I think smoothness should be introduced!

October 1, 2012 at 7:24 am

Supplementing my remarks just before, one can adjust the proof of [GW, Lemma 6.26] to sidestep an apparent use of Serre’s deep result, thus achieving a main aim of the Notes (naive k-smooth => nonsingular), in Ch. 13. Use that affine spaces over fields are nonsingular (done in Sec. 13.4). Then do all the relevant work within one such space, following PART of the proof in [GW] but ignoring the rest.

Work directly with a point x (smooth rel. to the scheme X) and corresponding prime ideal y in k[X1, .., Xn], not necessarily maximal. All still goes through as indicated before, using the field K = k(y) then later modding out an ideal of relations with very nice properties. The upshot is that x is a regular point of X, with local ring of dimension at most d (d if x is closed in X).

This is a good place to describe geometric aspects (roughly: lci, curvilinear coordinates, and their relation to the full local coordinate ring). Part would be much as in Shafarevich [BAG, Ch.2, 3.2].