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.

**What next?**

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.

February 19, 2013 at 2:29 pm

I wanted to mention one more thing: I’ve enjoyed Paolo Aluffi’s

“Algebra Chapter 0”— he really takes the same perspective in introducing algebra at the graduate level. He is unapologetic about the fact that thinking categorically can help, while also being clear that category theory should come naturally and organically with new concepts, and not just as dry formalism. His introduction is appealing; he is quite clear about what he is intending to do, and he concentrates on a few important aims (and succeeds).I’ve added his book to the bibliography for this reason.

February 19, 2013 at 8:01 pm

1) Okay, I don’t get the Bond reference.

2) For applications of Zariski’s Main Theorem, my favorite is that a in flat proper family over a normal base, if a general fiber is irreducible then special fibers are equidimensional and connected in codim 1.

3) Do you not also want to assume, for your connectoStein morphisms, that higher direct images vanish?

February 20, 2013 at 9:34 am

Hi Allen,

1) Look at the license plate! (I admit that I am especially attuned to the number of 216 now. 216A is the full course number of the fall installment of the class.)

As Bond and Grothendieck were both heroically active in the 1960’s, it is reasonable to expect that they would have met. Given Grothendieck’s pacifism, they may not have been on good terms. I could imagine that Grothendieck would be an ideal Bond villain. “Monsieur Bond, we meet again, but this time the advantage is mine! Let’s see how you well you like a generic point — of a nonreduced non-Noetherian scheme — in characteristic p! Bwa-ha-ha-ha!”

2) What’s the argument/reference?

3) That’s a reasonable question — the class of morphisms you describe is natural and important. (For others: as an example, a singularity of a characteristic 0 variety is rational if for one, or equivalently every, resolution of singularities, the resolution is of the form Allen describes.) But in this case, I really want just the pushforward. It’s what comes up in Zariski’s Connectedness Lemma, and Stein morphisms; and I want families of geometrically connected, geometrically redueced curves to have this property.

You were actually the person I thought most likely to come up with a good name, so probably there’s no great name! (Maybe the version you mention in 3), where many higher direct images vanish, should be multiStein, which would make the definition I want “EinStein morphisms”. Or because a number of different hypotheses, including proper and locally Noetherian, are sutured into the definition, these are “FrankenStein morphisms”…

February 20, 2013 at 5:47 pm

It seems more versatile to simply give names to the conditions on the structure sheaf (such as -connected for EinStein or -acyclic for multiStein) and then add whatever decorating adjectives (proper, (locally) noetherian, etc) when necessary…

February 20, 2013 at 7:43 pm

-connected is a very clever idea (unlike my facetious ones). (What about -isomorphic?) Other opinions?

February 22, 2013 at 7:01 pm

-isomorphic is good too. I slightly prefer -connected as it leaves open the option of defining (the admittedly ugly looking) –-connected for any integer , much like in algebraic topology.

February 26, 2013 at 11:35 am

I continue to lean toward your proposal of -connected. It suggests: “better than just connected”. I want to be very careful when messing with established conventions, so I have asked about this again in the

next postwhere I ask for advice on a number of things. (Others should still weigh in in advance if they have opinions!)I’m glad to have other options on the table too.

February 21, 2013 at 1:43 pm

I believe it would be a good idea, given that you’re talking about completions of rings and modules, to give the context of uniform spaces (or just the analogy with Cauchy sequences and the construction of the reals, like my AG lecturer did to me) and how every topological module carries a natural uniform structure (I think). My teacher also said that the bad thing in AG is that you can’t use the Implicit Function Theorem and local parametrisations like in differential geometry and this is a way to remedy that.

Just my two cents.

February 26, 2013 at 11:39 am

Your point that we should meet new concepts not because they are just the next topic in some syllabus, but because we feel some need for them, is a really good one. I intend to act on this (not in time for the next version, but for the one after that), likely giving an exercise giving an idea of how this allows us to think about the implicit function theorem.

I hadn’t heard of uniform spaces before. (This is a good example of how the accidents of our histories and educations give us a skewed perspective on the subject, unbeknownst to us!) I’d assumed it was something from some quite different part of mathematics, but have found (thanks to google) that the notion was introduced by Weil! Because this story is less familiar to me, and thus I’ll be less able to be convincing, I may not take this particular route.

(Update Feb. 27, 2013: this isn’t done in the Feb. 25 2013 version, but should be done in the next one.)February 26, 2013 at 3:52 pm

[…] (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 […]

March 25, 2013 at 3:13 pm

[…] to criticize it), given a little more geometric motivation for completions following the advice of Andrei, and responded to more suggestions sent by […]