The twenty-seventh post is the February 19, 2013 version in **the usual place.**

The early days of scheme theory

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.

A formal function

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.