### Actual notes

I am going to attempt to put up a version every month, in order to ensure that I continue to spend a little time tidying things up every so often.  So the January version is now  posted at the usual place (the January 29, 2015 version, also in e-reader format).

The list of things to do has shrunk, but there is a good deal still to do.  I have had many useful detailed comments to later digest and then include from Gurbir Dhillon, Tony Feng, Lisa Sauermann, and Jesse Silliman; and Benjamin Ljundberg, Chandrasekhar Raju, and Scott Zhang.

The last version posted was in 2013. I wanted to get the next version out in 2014, and time is running out!  So here it is, posted at the usual place (the Dec. 30, 2014 version).  (There is also a version suited for an e-reader — thanks to Jack Sherk asking for it, and explaining to me how to make it.)

In this 2014 post, I can’t help mentioning the passing of Grothendieck.   I was more moved than I expected to hear the news and feel the ripples in the mathematical community.  It feels strange that he is now a historical figure, even though he had walked away from his unfinished cathedral long before I was even aware of its existence.

The notes have continued to evolve around the edges, although the material is stable. Please continue to give me corrections and suggestions!  There are few sections that need tender loving care; I mention them on the front.  There are many other things on my to-do list as well, including many comments you’ve made.

The notes now have a tentative title (The Rising Sea:  Foundations of Algebraic Geometry).  The phrase is due to Grothendieck, translated by Colin McLarty; it is the title of Daniel Murfet’s wonderful blog.

The index is in the process of being made.  (No need to give me specific comments until it has converged!)

Finally, if you would like notes on commutative algebra that are very much from the same point of view as these notes, you may enjoy Andy McLennan’s notes, available here.  It also contains an (explicated) English translation of Serre’s epochal FAC, and all the algebra needed as background.

A new version is now posted at the usual place (the June 11, 2013 version).   The main reason for this post is to have a reasonably current version on the web.  Thanks to many people for helpful comments — most recently, a number of comments from János Kollár, Jeremy Booher, Shotaro (Macky) Makisumi, Zeyu Guo, Shishir Agrawal, Bjorn Poonen, and Brian Lawrence (as well as many people posting here, whose names I thus needn’t list).

There are a large number of very small improvements, and I’ll list only a few in detail.  I’ve replaced the proof of the Fundamental Theorem of Elimination Theory (Theorem 7.4.7).  (This new proof is much more memorable for me.  It’s also shorter and faster, and generally better.  It was the proof I was trying to remember, which I heard in graduate school.  I couldn’t find it in any of the standard sources, and reproduced it from memory.  But it must be in a standard source, because it is certainly not original with me!)   Jeremy Booher’s comments have led to the completion discussion being improved a lot.

I had earlier called $\rm{Spec} A$, where $A$ is the subring $k[x^3, x^2, xy, y]$ of $k[x,y]$, the “knotted plane”, to suggest the picture that it was a plane, with the origin somehow “knotted” or “pinched”.  János Kollár pointed out that “knotted” is misleading, because it is not in any obvious way knotted.  I’ve changed this to “crumpled plane”, but this isn’t great either.  I’m now thinking about “pinched plane”.  Does anyone have a good suggestion for a name for this important example?

Still to do:  To repeat my comments from the previous post, as usual, the figures, index, and formatting have not yet been thought about.  My to-do list is quite short, so please complain about anything and everything (except figures, index, and formatting).  (Perhaps in the next posted version, I may have “TODO” appearing in the text indicating where there is still work in progress.)

Finally, just for fun, here is a picture of the two rulings on the quadric surface, from the Kobe skyline.  (It is clear why algebraic geometry is so strong in Japan!)

Kobe port (click to enlarge)

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.

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 $\pi: X \rightarrow Y$ is a proper morphism of locally Noetherian schemes (Noetherian hypotheses  just for safety), such that the natural map $\mathcal{O}_Y \rightarrow \pi_* \mathcal{O}_X$ 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.

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 $Ext^1$ 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!

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?

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