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 $X$ is any $k$-scheme, then $X \rightarrow \text{Spec} \; k$ 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 $C$ is a category. We give some axioms that determine whether it is an additive category, or an abelian category.

Axiom 0: $C$ 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 $X$ and $Y$, the product and coproduct exists. (Hence $C$ has finite products and coproducts.)

Axiom 2: By the universal property of product and coproduct, we therefore have a map $X \coprod Y \rightarrow X \times Y$ 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 $X$ to $Y$. And adding the zero morphism (the morphism that factors through the 0 object) does nothing under this operation. You can check that this gives $Hom(X,Y)$ 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 $f$ and $g$ from $X$ to $Y$, and one map $h: Y \rightarrow Z$, then $h \circ (f+g) = h \circ f + h \circ g$). It is fun seeing how some of these work out.

We don’t yet know that $Hom(X,Y)$ 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 $X$ and $Y$, this operation + gives $Hom(X,Y)$ the structure of an abelian group.

At this point we have defined the notion of additive category. Note that we don’t require that $Hom(X,Y)$ have the structure of an abelian group (which I used to think of as an additional structure); we have the property that $Hom(X,Y)$ 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 $C$ and $D$, and you have a functor $C \rightarrow D$, 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.

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).

Suppose $0 \rightarrow F \rightarrow G \rightarrow H \rightarrow 0$ is a short exact sequence of quasicoherent sheaves.     If two out of three of $\{ F, G, H \}$ are locally free, what can we say about the third?

If $F$ and $G$ are locally free, then $H$ need not be.    This is a fundamental example.

If $F$ and $H$ are locally free, then $G$ is too.    This is important and not too hard.

If $G$ and $H$ are locally free and finite rank, then $F$ is too.  This is also useful.

But if $G$ and $H$ are locally free, and infinite rank, then $F$ need not be locally free.  The purpose of this post is to give Daniel Litts explanation to me of a simple example of this.  I thought this example was not right to include in the Rising Sea (it would be distracting), but I wanted to preserve it for posterity (and for myself).

I will describe a surjective map of free modules whose kernel is not locally free. Viewed geometrically, this provides a surjective map of trivial (infinite-dimensional) vector bundles whose kernel is not a vector bundle. Of course, the kernel is projective.

Let $M$ be a projective module over a ring $R$ which is not locally free.  (Such things exist, see for example the Stacks Project tag 05WG.  To be explicit, there $R = \prod_{i \in {\mathbb Z}^+} {\mathbb F}_2$ and $M$ is the ideal $( e_n : e_n = (1,1, \cdots ,1,0,0, \cdots) )$ (where there are $n$ 1’s).)

Then $M$ is a direct summand of a free module, say $M \oplus M' = R^{\oplus I}$.  Writing

$R^{ \oplus I'} = (M\oplus M') \oplus (M \oplus M') \oplus \cdots =M \oplus (M' \oplus M) \oplus (M' \oplus M) \oplus \cdots =M \oplus R^{\oplus J'}$

gives $M$ as a direct summand of a free module with free complement. Then there is a short exact sequence

$0 \rightarrow M \rightarrow R^{\oplus I'} \rightarrow R^{\oplus J'} \rightarrow 0$

exhibiting $M$ as the kernel of a surjective map of free modules, where the map $R^{\oplus I'} \rightarrow R^{\oplus J'}$ is the projection.

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).

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

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:

(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.

Update July 19, 2022: this later post contains a draft chapter incorporating this argument, so you can see it in context.

I want to share a proof of the Cohomology and Base Change theorem that really makes it much much more clearer to me than it had been before.  There might be some uncertainty as to its origin, so I’ll give my take on it.  I heard it from Eric Larson a couple of years ago.    We had discussed the fact that it should be some easy-to-understand fact about maps of free modules over Noetherian rings.   (Many people have known this — Max Lieblich and I had discussed it before; it is alluded to in Nitsure’s excellent presentation in FGA Explained; and I am certain similar discussions have happened many times in the past.)

Eric went home and figured it out, and the next day sent me a short and sweet argument in a pdf file (see below).   He had thought he had heard it before, but the only plausible sources he could have heard it from (Joe Harris, or the argument in Eisenbud and Harris) don’t have the argument.    But he had certainly heard an argument for Grauert’s Theorem (where the base is reduced).  So until I learn otherwise, I suspect he just thought he had heard a full argument of Cohomology and Base Change, and then tried to reconstruct how it should go, and just figured it out.   In other words, for now, I believe the argument is original to him.  But I would have expected that such a simple argument (especially with its central insight) would have been independently discovered earlier, perhaps many times. However, I am not aware of it in the literature anywhere.  Can anyone tell me where it has appeared before?  It is such a simple argument that, if I had not been consciously looking for it earlier, I would have believed that I had surely known it myself. So if someone says “I knew that, I just never wrote it down”, I’m not going to value that too highly. On the other hand, there are a number of great explanations of things in algebraic geometry that Dennis Gaitsgory told undergraduates when he was at Harvard; and Nitsure’s argument is presumably written down somewhere;  so I know there are things I haven’t seen.

(Incidentally, Ogus told me a fantastically elegant proof of the local criterion for flatness, which was in his first published paper I think, that I love.)

Of course, I’ve digested Eric’s argument into The Rising Sea (and I really wish I hadn’t moved around part of a chapter a couple of years ago, leaving the manuscript in some disarray; I hope to post the new version before too too long).  And I’ll eventually post my take on Ogus’ argument here, and I’m also digesting it into the Rising Sea.

One of Eric’s insights, for me, is this.      When we generalize the notion of “finite-dimensional vector space” to families, we get finite-rank vector bundles.  These have constant rank, but for a coherent sheaf to be a (finite rank) vector bundle, we need more than it be constant rank. Eric suggests that to generalize the notion of “map of finite-dimemnsional vector spaces” in a particularly good way, we want more than a map of vector bundles (as coherent sheaves) — and we want more than that it is of “constant rank”.  It is this stronger condition about maps of vector bundles that gives this strong condition of cohomology commuting with base change, which can be applied in different settings (including Grauert, and the Cohomology and Base Change Theorem).

Here is Eric’s argument.

I’m going to replace the proof of “spectral sequences for double complexes” in the introductory chapter with just a reference to the  following “picturebook”.

I find this explanation surprisingly comprehensible, and this is really the first time I feel like I understand why spectral sequences work, and why they are not mysterious or black magic.

I haven’t been sure what to do with this pdf document, so I’ve just been giving it to people as a “gift”.  So consider this to a gift to you too.

It is also posted here. (I am a fan of 3blue1brown.)

Although it won’t be evident from this site, I’ve been gradually editing the notes, mainly in response to suggestions and corrections you have sent me or told me. Because things are (very slightly) rearranged, there are lots of temporarily broken links, so I’ve not posted a version in a long time. I think I would like that to change. There are lots of little things I want to tweak — for example, adding Eric Larson’s “proof from the book” of Cohomology and Base Change, and Arthur Ogus’ proof (also “from the book”) of the local criterion for flatness.

But what is moving me to write this today is an email from Zihong Chen, asking about the flawed proof of the Kunneth formula in an earlier version of the notes (since removed, but it may have been after my last posting). I’d scribbled down notes on what I should have said, so I thought I should type it properly, and post it here. (I’m not sure if I will add it to the notes. The price is two pages, which is pretty steep at this point. But it fits squarely into the narrative of the Rising Sea.)

Happy new year all! Here is how I will celebrate the new year.

Jarod Alper will be teaching a course on “Introduction to stacks and moduli” at the University of Washington, and the course website is here: https://sites.math.washington.edu/~jarod/math582C.html

I’m going to attend, and I think this course could be as influential as Martin Olsson’s course on stacks back in the Beforetimes (which was central to his development of his wonderful book). Jarod is allowing me to advertise it here. He told me: “Right now I’ve posted a very lengthy introduction & motivation, which I’ll spend all of about one lecture on. I will be gradually posting (and revising) the notes during the quarter.”

The class meets Mon/Wed 11:30-12:50 (beginning Mon Jan 4, 2021), Pacific time. On the course website, he writes: “If you would like to participate informally in the class, please send me an email at jarod@uw.edu with (1) your name, (2) email address, (3) affiliation (if any), (4) status (e.g. 3rd yr PhD student, postdoc, …), and (5) a one sentence summary of your background in algebraic geometry.”

For Stanford folks: I am strongly recommending that most of my current students attend this (with some exceptions depending on their interests), and also recommending that many of my “maybe-they-are-my-students” consider taking it.

A standard and very reasonable question people have when first learning about schemes is the following: is it true that every affine open subset of an affine scheme is a distinguished/principal open subset? The answer is (as most of you know) is “no”, so the follow-up question is: “what is a simple example that is rigorously provable to someone at the stage where they ask this question”.

Nikolas Kuhn gave a slick answer to this question. Consider the cuspidal curve $y^2=x^3$ in the plane $\mathbf{A}^2_k$ (where $k$ is of course a field), which has normalization $\mathbf{A}^1_k = \text{Spec} \; k[t]$. Then if you remove $(1,1)$, you get an affine open subset (why?), but that set is not even set-theoretically cut out by a single equation (hint: pull the equation back to be a function of $t$, but notice that this equation doesn’t lie in the subring $k[x,y]/(y^2-x^3) = k[t^2, t^3] \subset k[t]$.

Entertaining follow-up question: this affine open set is $\text{Spec} \; B$ for some ring $B$. What is $B$?