Late February 2024 version

A new version of the notes is available at the usual place (the February 21 2024 version). Everything is in potentially nearly final form for an “official version”, except for typesetting/formatting and the index. 

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(a) Significant change, but not substantive:  The figures are now basically done (meaning:  redone). Because I think people should be comfortable making their own sketches in real time as they figure things out, I’ve deliberately gone all in on a hand-drawn aesthetic. This is atypical, even amateurish, for serious mathematics books. So maybe I will reconsider.

(b) About composition of projective morphisms (the old Exercise 17.3.B), and more generally 17.3:  I realize now what the complication was in the old 17.3.B.  People were stuck at many steps, but the real issue was only the last one, to get to the quasicompact target case once you already had the target line bundle. There was indeed a gap there. Given what is done in the notes (and not even by this point in the version previously posted), we can show it only in the Noetherian case.  So the new version has 17.3 seriously rearranged in a number of ways.  In particular, now the old 17.3.B is a bit later, and the argument is for when the final target is either affine or Noetherian.  (Even this requires as a black box something that will only be proved in the cohomology chapter, which is Grothendieck’s coherence theorem for projective morphisms.)    I think this is now rigorous and complete. Please let me know if there are issues.  (Some of David Speyer’s ideas also in retrospect guided me on how to improve it.)  I’ve moved all the double-starred bits to the end (and I hope the reader ignores them all).  There is a single-starred section that is where the trouble lies, and that’s going to be hard going for those readers working through it.

(c) In the chapter on the 27 lines (you know which chapter number it is), I was always unhappy about needing Castelnuovo’s criterion.  János Kollár pointed an explicit workaround that I like a lot.  (I know that, roughly, doing it in this hands-on way, is very classical; but it is hard to do it rigorously without hand-waving, and you’ll notice this hand-waving in some expositions you may have seen.)  This is now Proof 2 of Proposition 27.4.1.  His explanation to me was direct and to the point; I’ve muddied it a little to fit the narrative, so the “worsening” is due to me.  (Most mathematicians have their own particular kind of thinking, which they are best at.  Kollár has this rather amazing ability of understanding very abstract things and very concrete things, both as well as anyone else, and those two things seem to be connected directly in his head in a way that they are not for most algebraic geometers. When I was in grad school, we secretly called him “The Mighty Kollár”, and even now it doesn’t seem an inappropriate name, although I would never say it to his face.) 

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General philosophical point:  I’ve noticed that there is a tension between the kinds of requests people have of the notes.     Roughly, on one hand, there is my desire to try to make it possible to cover some central core of the material in a year (for at least some people), which requires rather severe compromises.  I’m trying to make different compromises than most people have made in the past (in particular, I’d like to expect less from the reader in terms of background, but then I need to expect more from the reader in other ways).  The things I need to push back against are things like “This topic really needs to be included”, “This topic needs to be fleshed out more completely”, “This topic is not done in sufficient generality”, “This topic needs to be done more rigorously”, “The presentation of this topic is well tuned to me as a reader”, “I coudn’t solve this exercise”.  In all of these cases, the suggestions are good ones, but at some point the manuscript will sink under the weight of items loaded onto it.  Usually those making suggestions are happy to suggest what other things should be cut, but you might not be surprised to hear that these suggestions contradict each other.    I’ve tried to help by starring and double-starring some topics, but I can’t seem to stop some readers from not skipping them (and then getting boggeddown).  I’ve had a number of suggestions of things that “really are needed in such a work” that I’ve had to repeatedly decline.  Even the ones I have said yes to have let it creep up to 850 pages even after the I passed a secret “no new material” line in my mind.  (Some recent additions:  on top of the ones mentioned above:  a brief mention of projective normality; definition of Cartier divisor in generality; and more.  But even these are things I think the reader can quickly read on their own from their web having read these notes, and needn’t be here.)

One form of the compromises I’ve made is:  “If we need it, I don’t want to black-box it, and I want you to understand it, but I only need you to understand it well enough to use it and move on”, and “if we don’t need it, no matter how wonderful it is, we should just skip it, and you can learn it on your own later” (so if included, those things are starred or double-starred).  I’ve had a hard time maintaining this position consistently, and over time a lot of things have slipped by my defenses.

At this point I am still entertaining all sorts of suggestions, but am going to try to stick to things that particularly deal with mathematical errors (often leaving imperfections and imprecisions — and many of these were actually deliberate choices), or really affect the understanding of a significant portion of readers (which I have some broader sense of given comments over the years) and not just you personally.

Many of the recent comments (including some still for me to think about) are here on this website.  Some excellent ones have come to me by email, from a group of students in Poland, by way of Joachim Jelisiejew.   I want mention them here, and I also look forward to seeing what these students go on to do mathematically in a few years’ time, because they are clearly very talented.

 

February 2024 version: content now complete, including index

A new version of the notes is available here. (It, and all older versions, are available at the usual place.) I don’t have much to report, but I just wanted to post the current version as it continues to converge. Please do continue sending in comments, particularly in the next month. There is a good chance that fairly soon a “snapshot” will be taken for an official publishable version (although that doesn’t mean that it will then be frozen).

In some more detail: I’ve now implemented the vast majority of the ideas from suggestions people have given, although there are some still to go. I’m redoing the figures in a consistent style, and about 2/3 of them are redone so far. The index will get some attention later, but the raw material is there, and I welcome suggestions and corrections. The only things I’m not interested in are latex issues (margins, etc.; but typos and errors in spacing are fair game). I am now ready for comments on figures, although there are still 1/3 of them that I haven’t re-done. 

I’ll very be busy with other duties for the next four years, so I will be able to spend much less time and attention on this. I suspect it will be enough of a break that I won’t be able to return afterwards with everything as fully in my head as it is now, so this will likely be my last chance for me to really make this as good as it can be before letting it go off into the world on its own. This really feels like nearing the endgame.

So thank you all for accompanying me on this stage of what has been a most interesting and rewarding journey for me.

 

July 2023 version: content now complete, including index

A new version of the notes is available here. (It, and all older versions, are available at the usual place.) The main change from the previous version: The index is potentially done.

Now the editing I will do will be primarily in response to comments and suggestions from others. I am very interested in any suggestions and corrections you may have (including things you told me before that I have forgotten). The only things I’m not interested in are: latex issues (margins, etc.; but typos and errors in spacing are fair game) and ugliness of figures (but content of figures is fair game).

 

June 2023 version: potentially finalized content

A new version of the notes is available here. (It, and all older versions, are available at the usual place.) For the first time, I would say that the content and editing is “potentially done”. All content is potentially polished. (Not done: beautification issues e.g. fighting with latex over line breaks. And the index is very rough.) I am thus very interested in any suggestions and corrections you may have, including things you told me before. (I think I’ve implemented all the edits from my to-do list from comments you have made through the years, by email or here on this site.)

In more detail: This is a fairly substantial revision. I also taught from the notes in the last two quarters, and the excellent comments of the excellent people in the class helped tremendously.    I may not do much more before I declare this project “done”.

New arguments added recently (perhaps in the last few revisions):

• a scheme with no closed points, and a little more about coproducts
• fixed proof of Serre duality (long in coming)
• a glimpse of the Koszul complex, and proof of the Hilbert Syzygy Theorem (I learned how to think about this from Michael Kemeny)
• improved exposition of proof of formal function theorem (unlike other cases, really the same proof)
• all the important flatness facts are now done much more easily
• and many more things I can’t remember right now.

I’m definitely looking for any small remaining issues.  And also any mathematical mistakes or omissions.   

A random cool thing I wanted to share: I have always wanted a way to order mathematical notes, in perhaps a semi-public way or a private way, that would be nonlinear and easy and robust. Wikis are good but have some imperfections. The “back end” of the stacks project (gerby) is fantastic for public presentation of huge amounts of interconnected material, but less suited to person note-taking because it is somewhat fragile, and has significant start-up time to use properly. (For more on gerby, see: https://gerby-project.github.io/ .) I stumbled on Jon Sterling’s “forest”, and I can do nothing better than just point you here: https://forest.jonmsterling.com/index.xml and recommend that you take a look and explore. He has done a lot of thinking on “tools for mathematical thought”, which is precisely the the sort of thing I was wanting to think through more myself. It is something akin to a manifesto. (As is unfortunately common as many of you know, I owe him an email.) I would like to explore this further (despite extreme lack of time), and I wanted to advertise this, in case others would like to try it out too!

 

April 2023 version of the notes

A new version of the notes is available here. (It, and all older versions, are available at the usual place.)

Supravat Sarkar kindly pointed out that my Chevalley’s Theorem proof was still flawed at the very last step. I have now, I think (again!) fixed it. See Section 8.4.4 for that.

 

December 2022 version of the notes

Happy New Year! The picture is of an amazing artwork by Gabriel Dorfsman-Hopkins and Daniel Rostamloo of 27 lines on a cubic surface. Here is a link to the webpage for the sculpture.

The latest (December 31, 2022 version) of the notes are posted here, at the usual place. In terms of content, it remains (and will remain) very similar to the previous version. But it has been substantially edited and polished.

In more detail: the first 18 chapters are much more polished than before, and are nearly in “potentially final form”. Most of the later chapters are still waiting for that level of tender loving care. Please continue to let me know a$bout any suggestions, corrections, typos, etc., in any chapter, no matter how small. The only exceptions are stylistic things (margins, fonts, low-quality figures).

Here are some changes I find interesting or worth mentioning.

Chapter 8: The Chevalley’s Theorem proof still had a small flaw, and is now fixed. (Thanks to Hikari Iwasaki for comments on this and many other things near there.) It is now one that I find so natural that I can’t forget it, and can easily explain while walking across the quad.

Chapter 10: The starred section, and doubled-starred proofs, on things like “geometrically reduced/integral/etc.” have been awaiting patching for a very long time. I finally did it. If you are curious, or want to solidify your understanding of those issues, please take a look. I hope that it has become very comprehensible. The main black box (that I punted to the chapter on flatness) is that if is any -scheme, then is universally open.

Chapter 12: The chapter on dimension is substantially rewritten, and I hope it is a cleaner way through the topic. There are some explanations directly or indirectly from Mel Hochster and Qing Liu. (I highly recommend Qing Liu’s book; I love how he thinks in a very “clean” way.)

Chapter 13: I had been meaning to rethink the older version of the chapter on regularity and smoothness for some time. I’ve now done so, and fixed a lot of problems. I know from experience that I have undoubtedly introduced new ones. From the new point of view, the central players here is that if you want to get at notions of regularity and smoothness, you are led to the notion of the Zariski tangent space and the Jacobian criterion, and everything revolves around them.

Chapter 15: If there is a single central fulcrum on which everything balances, it is at the almost literal center of the notes: the link between line bundles and divisors. This is a tricky topic the first time you see it, and section 15.2 has always been trickier for readers than I want it to be. I hope it is better now.

Also, Pol van Hoften told me the definition of abelian category which he prefers, and I was very surprised to find it much clearer and enlightening than anything I had known before. How could this be, given how old and important a notion it is? Pol’s definition is from Jacob Lurie’s Higher Algebra, although his description of it is his own. I have now included it in the notes. I’ll tell you it here, because it is something directly memorable enough that (unlike the previous definition) I can remember it without trying. Also, I learned something new: the notion of abelian category is intrinsic to the category itself — there is no additional structure required. That was so unclear from other definitions that I did not realize it despite using this notion for years! Here is Pol’s description. We are motivated by the abelian categories we already know and love. Suppose is a category. We give some axioms that determine whether it is an additive category, or an abelian category.

Axiom 0: has a zero object (an object that is both final and initial). Of course this axiom should be numbered zero. And of course abelian categories need a zero object, so this is easy to remember.

For the next axioms, we note that abelian categories have finite products and coproducts, and they are the same. Let’s axiomatize this.

Axiom 1: For any two objects and , the product and coproduct exists. (Hence has finite products and coproducts.)

Axiom 2: By the universal property of product and coproduct, we therefore have a map for all pairs of objects. We require that this map be an isomorphism.

Now here is the insight I didn’t realize: with this information, we can discover how to “add” two morphisms from to . And adding the zero morphism (the morphism that factors through the 0 object) does nothing under this operation. You can check that this gives intrinsically the structure of a commutative monoid (it is associative, has identity). And it even automatically distributes in the way we’d like (e.g., if we have two maps and from to , and one map , then ). It is fun seeing how some of these work out.

We don’t yet know that it is an abelian group, because we don’t know that + has an inverse. So that is our next axiom.

Axiom 3. For any two objects and , this operation + gives the structure of an abelian group.

At this point we have defined the notion of additive category. Note that we don’t require that have the structure of an abelian group (which I used to think of as an additional structure); we have the property that with its pre-existing operation + is such that + has an inverse.

Axiom 4 and 5 are just the two axioms making additive categories into abelian categories. Axiom 4: kernels and cokernels exist. Axiom 5: some property (you can have several choices here, leading to the same definition) that kernels and cokernels have to satisfy. Usually it is “some property and its dual”. Once again, there is no additional structure required; just a condition.

So if you have two abelian categories and , and you have a functor , how do you know if it is a map of abelian categories? All you need to do is to check that it preserves products and coproducts (of two elements) and kernels and cokernels. All other parts of the structure come at no extra charge.

I will teach the second and third quarters of “Foundations of Algebraic Geometry” this winter and spring quarters (Pol taught the first one and by all accounts did a remarkable job). I hope to continue to edit the file during this time, and clean up more chapters.

 

August 2022 version of the notes

It has been a while, but finally, the next version of the notes are here. (For convenience, all earlier versions are posted at the usual place. Added Sept. 4, 2022: a version with an imperfect index is here.)

It took a while because I split the quasicoherent sheaves chapter in two (one is much earlier, and is quasicoherent sheaves ~ “modules over a ring”, and the later one is quasicoherent sheaves ~ “vector bundles”), which required a cascading series of changes in logic and ordering and references.

But it is in much better shape, and I intend to post updates periodically this academic year. People have given me a large number of very useful comments, and they are all in the process of being considered (and often implemented). I think comments made on this blog have been mostly responded (even if to say “it is now on the list”).

Please continue to let me know about any suggestions, corrections, typos, etc., no matter how small. The only exceptions are stylistic things (margins, fonts, low-quality figures).

At this point the figures are supposed to now be substantively correct (except of course where there are things on my “to do list” to fix). Everything before the chapter on dimension now has essentially no remaining known issues (except for a few pages on geometric fibers that I am still rewriting). Later chapters are in different stages of editing.

Here are some random notable changes that I can remember off the top of my head — the things mentioned in earlier posts (associated points; cohomology and base change; etc.) are now in. I like the proof of Chevalley’s Theorem much better (it feels now very easy modulo the fact that finite morphisms are closed).

 

Cohomology and base change: new exposition

A couple of posts ago, I gave the slick proof of cohomology and base change that I learned from Eric Larson (and that to the best of my knowledge should be credited to him). Here is the rewritten chapter on cohomology and base change, now only 17 pages long, which incorporates his argument (as well as various other corrections and suggestions people have sent through the years).

ch28jul1522Download

As always, I’m happy to hear of any errors/typos/suggestions!

 

An introduction to associated points (and associated primes): new exposition

I have rewritten the introduction to associated points. The new version is available here.

If you are interested in learning about associated points, or solidifying your understanding, or getting a more geometric view of them, please take a look — it is ten pages, and intended to be quite readable to those who have read the first five chapters or so of The Rising Sea.

The new exposition attempts to more directly follow the geometric point of view, set out well for example by Matthew Emerton (perhaps on this blog — if I can find the link I will add it here). I was trying to do something with my previous exposition, but did not succeed, and I think this works better. But I am very interested in hearing what people think, who are reading it right now.

This should be something that you can work through in an evening, and discuss with someone else. I am hoping there will be a couple of epiphanies in there. I am certain there will be typos!

The point of view is summarized here:

foagjul1522associatedpointssummaryDownload

(For some reason the letter “p” was deleted in the last line of the pdf above…)

I guess there is no harm in putting the entire thing here, in case someone feels like skimming through it on this page.

foagjul1522associatedpoints-1Download

 

November 2017 version

The November 2017 version is in the usual place.

The editing has started to move forward significantly. All of your suggestions are now in process — many are done; most others are sorted by chapter and being gradually dealt with; and there are a few that I still have to think through. There remain some sections that need serious (re)writing, but at least the content is collected (in my private notes). I may post a list of them at some point, so you aren’t alarmed by the poor state of some of the sections. There are also some very few topics I may still add. (Examples: the Koszul complex and Hilbert syzygy theorem, following Michael Kemeny; universal delta functors, following Ravi Fernando; a quick introduction to Grobner bases; gluing schemes, following Hannah Larson.)

The figures are now “complete” in the sense that they may not be beautiful, but they should be considered completely drafts, with all the content they should eventually have. In particular, please feel free to complain about them (or make suggestions).

If there issues in how things display (e.g. curved arrows), please let me know — and also which pdf viewer you are using.

The index, and formatting, both remain to be properly dealt with.

And once again I’m struck by how many people deserve thanks for their numerous comments and suggestions.  Any attempted list will leave out many people.  Most at least are obvious from this wordpress site.