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 $B$?

One thought I had this morning: the name “principal open set” seems a better (more descriptive) name than “distinguished open set”. I then googled the phrase to see if it had been used in some different way, and found to my surprise that it actually had been used in precisely in this way; I’m not sure where this usage originated.

But I’m inclined to switch over completely to this phrase. The clearest downside is that the notation $D(f)$ refers to “Distinguished”. But I already prefer to think of it as the “Doesn’t-vanish” set.

This might suggest a better name for the following two “topologies”: the topology on ${\rm {Spec}} A$ consisting of principal/distinguished open subsets; and the “topology” on a scheme $X$ where the allowed open subsets are affine open subsets, and the allowed open morphisms are “principal/distinguished” inclusions. I’ve been calling the latter the “Distinguished Affine Topology”, and I’m not sure if I gave a name to the former. Are there better names for these?

Separately, now that AGITTOC (at least the first incarnation) is over, and I might write more here, as a world-readable (and world-commentable) notebook.

I’ve not posted a new version of the notes in a long time, because parts of it have been “closed for construction”. But I might resume soon. It might be easiest to post just a few chapters from the beginning, and gradually move forward.

As always, there are many comments here that I’ve not responded to. You might be surprised to find that I’ve actually read them, and made changes in response to many, and have intended changes in response to others.

Hi everyone,

I was afraid this would happen — I forgot to set my alarms last night, and just woke up, and won’t have things sufficiently ready for today’s scheduled pseudolecture. So I’ll have to postpone it.

There are still two more to go. The next one will not be next week — it will be the week after next (Saturday October 3). There is a new seminar, approximately monthly, by Dawei Chen and Qile Chen at Boston College, with two talks , and the second will conflict with AGGITOC’s regular time: https://sites.google.com/bc.edu/map/home . There will be some people interested in both events. (Next week’s second speaker is Hannah Larson, who is definitely worth catching, incidentally.)

I’ve posted this on zulip and the Algebraic Geometry Discord, so I hope this reaches everyone!

Once we wrapped our heads around what morphisms of schemes are like, I jumped ahead to chapter 9 to show that fibered products exist. I did this for a few reasons. I wanted to make clear that there was nothing stopping us from immediately understanding fibered products. (The reason I left it after chapter 8 in the notes is that it can take some time to digest it the first time you see it, and there was lower-hanging fruit to pick. Also, once you begin to think about the fibered product, you are led to consider many other things, so it is a substantive topic in its own right.

What I most want you to do is to listen to my exhortations about how the existence is, understood properly, “easy” (in the technical sense — it is conceptual, although you have to train your mind in order to make it natural). So watch and read and digest. Once you have digested it, you are free to read more about Yoneda’s Lemma, and “Zariski sheaves” and Grothendieck topologies — but only if you are at the stage where these are easy reads, and not when they are entrancing but opaque.

I would then recommend trying a bunch of explicit problems in section 9.2, which we haven’t discussed yet, but will let you see that you can really work with fibered products in practice. For this, you need to know something about tensor product — but you’ll find out how little there is to actually know, and how everything follows from these few facts. Section 9.3 is just about interpreting “pullbacks” and “fibers” in terms of fibered product. (Example 9.3.4, on a double-cover of the line, is super-enlightening, and I discussed it in the last pseudolecture.) And from there you can easily see why various properties are preserved by base change/pullback/fibered product (section 9.4). (Please skip 9.5, even if it would be otherwise very interesting to you — it is in the process of being rewritten.) So at this point you can plausibly be done most of chapter 9 (except for 9.5 which I asked you to skip, and 9.6, which isn’t hard, but which I’ve not yet talked about in a pseudolecture).

Here are some problems from chapter 9 that are worth trying.

If you are new to a lot of this, you can try Exercise 9.1.A, which doesn’t build on lots of other things, so you get a chance to just understand something without having to remember a huge superstructure beneath it. 9.1.B is the key connection that gets us from algebra to geometry (the “local model” of the fibered product).

On the other hand, if you are a fancy person, you can do the exercises to understand the existence of fibered products in terms of representable sheaves.

In section 9.2, I would recommend all the exercises that are the gateways through which algebra becomes geometry: 9.2.A, 9.2.B, 9.2.F

Then you can understand how to change “base fields” in this language, to for example relate things over $\mathbb{Q}$ to things over $\overline{\mathbb{Q}}$ to thinks over $\mathbb{C}$. Exercises 9.2.H to 9.2.J deal with this.

If you are a fancy person, you can try 9.2.E, which includes a ring that Jonathan Wise mentioned a few pseudolectures ago — $\overline{\mathbb{Q}} \otimes_{\mathbb{Q}} \overline{\mathbb{Q}}$.

And Exercise 9.2.K is not important in any way, but it is entertaining!

In Section 9.3, 9.3.A will give you some insight into fibered products — it works well for topological spaces.

In Section 9.4, if you do a few parts of problem 9.4.B, showing that various properties of morphisms are preserved by base change, then you’ll see how to do this in general.

That’s all the time I have for today — tomorrow I’ll hopefully write down a bit about classes of morphisms of schemes. I should really have done that before telling you to do Exercise 9.4.B.

I am leaning more and more toward having some “office hours” on zulip, so I can actually answer some questions you may have — I might pick a time and be there, and then I would come back to it periodically, so people can ask questions asynchronously. But only if it would be useful to enough of you (at least a few)…

As I mentioned at the start of pseudolecture 11, my intent is for this “edition” of AGITTOC to end in two or three weeks. Then I will pause for a month or so, and I might then begin a new edition, starting very roughly where we left off, but with presumably a somewhat different audience.

In this post, I’d like to go over the material we are talking about in these few pseudolectures, to give some sort of written overview, and to suggest problems to do.

Let’s resume our story at the start of part III of the notes, at the start of chapter 6. What do we mean by maps of geometric spaces, now that we have some idea of how we want to think of the geometric spaces by themselves? Certainly we will have to understand maps of points, open sets, and functions, and we will want to understand how our “local models” map. (If you are not seeing this for the first time, you may enjoy trying to do this simultaneously in several categories — varieties over an algebraically closed field; schemes; and complex analytic spaces — to really see what is essential about the constructions, and what is specific.

We are quickly led to the notion of a morphism of ringed spaces. If we are careful, we are led to the notion of a morphism of locally ringed spaces. More specifically, at the very start of our journey, we were expecting that the locus where functions vanish should be a closed subset (this came out in questions and comments on zulip and in the first couple of pseudolectures). This drove our definition of the Zariski topology, and furthermore made us realize that the stalks of our spaces were local rings (thus handing us the definition of locally ringed spaces). With locally ringed spaces, we had the notion of the value of a function at a point as well.

So all of this helps motivate how a map of locally ringed space should be correctly defined — it is a map of ringed spaces such that “the pullback of the locus where a function vanishes should be the locus where the pullback of the function vanishes” (and hence for “doesn’t vanish” as well).

So, to make friends with this, try Problem 6.2.A (morphisms of ringed spaces glue) and 6.3.A (morphisms of locally ringed spaces glue).

If you are quite new to this, try 6.2.B and 6.2.C, which might be more tractable, and are also important.

Everyone should do 6.2.D — it provides the local model of or morphism.

Question everyone has: why do we need locally ringed spaces? What’s an example of a morphism of ringed spaces that is not a morphism of locally ringed spaces? What’s a morphism $\text{Spec} \; A \rightarrow \text{Spec} \; B$ as ringed spaces that doesn’t correspond to a map $B \rightarrow A$ as rings? For this, you should do Exercise 6.2.E. We won’t really need it, but it is good to see.

Important exercise 6.3.C will show you how thinking in terms of morphisms of locally ringed spaces forces Spec’s to map the way you want them too.

Exercise 6.3.E is enlightening if you want to see how things you might already understand (about how to think about projective space) translates into our new language.

Exericse 6.3.F — that maps to an affine scheme are the same as ring maps in the opposite direction — is crucial, and a must-do — it has already been used multiple times by the time I write this.

Exercises 6.3.J and 6.3.K came up in the question period in the eleventh pseudolecture — maps from an affine scheme that is “local” are very understandable.

Section 6.4 is about projective geometry. It is worth reading, and the one exercise to do here above all else is 6.4.A, which tells you the extent to which maps of graded rings give you maps of projective schemes. The point of the exercise is to learn why the statement is not surprising.

Section 6.5 is about rational maps, which are really “mostly-defined morphisms”. First and foremost, you should understand (precisely and intuitively) what a rational map is, and why the notion makes sense. Ignore the proof of 6.5.5, which is botched. (I’ve revised it, but not yet made the updated version public.) To get a feel for why rational maps are a useful notion, I would recommend 6.5.D (which connects it to field theory), and the many examples starting with pythagorean triples in 6.5.8. You may find it interesting that diophantine questions end up oddly paralleling questions over “function fields”.

We did not discuss representable functors and group schemes (section 6.6), but if you wish, you should read it. It requires more mathematical maturity, but you know everything you need to know to read it. Similarly, section 6.7 explains how to define a very useful classical object (the Grassmannian) which generalizes projective space. It becomes cleaner and easier later with more perspective.

That’s all the time I have right now, so I will just post this. In the next couple of days, I hope to write more about things we’ve covered, notably, the affine communication lemma, various properties of morphisms, and fibered products.

I’m also intending to spend time on zulip, and in particular answer questions and discuss things. Perhaps it would be helpful to set up a zulip channel that would be “office hours” for the week, and have a chunk of time where I could answer questions?

As promised, I want to now give you some problems to think about, mainly in Chapter 4 and Chapter 5.

But first, a quote for the topologists among us:

It was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry.

— Solomon Lefschetz (A Page of Mathematical Autobiography, Bulletin of the American Mathematical Society, Volume 74, Number 5, 1968)

The end of Chapter 3

If you are happy with 3.7.C and 3.7.D, then you are happy with the theory of this section. If you can do 3.7.B and 3.7.G, then you can work with these ideas.

The structure sheaf.

Our goal here is to understand the sheaf of functions on $\text{Spec} \; A$ (or $\text{mSpec} \; A$) as simply as possible. We want really to know that it is a sheaf, and to be able to work with it. The idea to keep in mind is that we will understand it only through the distinguished open subsets $D(f)$ (where $f$ Doesn’t vanish), and will take the functions on $D(f)$ to be $A_f$, except to make sure this is well-defined (i.e, “if $D(f) = D(g)$, then $A_f = A_g$“) we define the functions in a way depending only on the open set itself (Definition 4.1.1).

So Exercise 4.1.A will make sure you are completely comfortable with the second trick. And Exercise 4.1.B and Exercise 4.1.C will make sure you are comfortable with the first.

If you are already familiar with exact sequences, then Remark 4.1.4 will be a clarifying perspective.

The recurring counterexamples in 4.1.6 are good to keep in mind. You might have noticed that an example from pseudolecture 9 was a variant of the last one — $\text{Spec} \; ( \overline{ \mathbb{Q} } \otimes_{\mathbb{Q}} \overline{\mathbb{Q}}$ is a non-Noetherian scheme for which all the stalks are Noetherian (in fact, they are fields).

Section 4.2 is all about drawing pictures. It should be fun.

In Section 4.3 there are a lot of things to get used to, so a lot of exercises worth doing. You can see if you get confused by 4.3.A. If 4.3.B is straightforward, then you’ll know you are comfortable with these ringed spacdes. 4.3.E(b) is our first (provable) example of a scheme that is not affine!

Exercise 4.3.G will finally make clear that we have defined our locally ringed space with the properties we want.

People often complain that algebraic geometry is sometimes done so “formally” that there are no examples. I agree that it is hard to really understand something without really playing with actual examples. So in 4.4, do what it takes to become comfortable with these three examples. Exercise 4.4.A is one we will use later, for example, when we defined “Proj” in pseudolecture 9. You should do 4.4.D (and then distill it to make it as painless as possible), and then you’ll really know you can compute stuff on a variety, using its cover.

Then in the next section, “Proj” is a machine to make varieties/schemes from many more open sets, but using just one ring. You may want to Exercise 4.5.B to see how homogeneous polynomials “cut out” a scheme in projective space in a hands-on way, so you’ll have a good feel for it when we think about Proj in generality.

Exercise 4.5.D will put graded rings in general into this context. Definitely do Exercise 4.5.E, and take your time with it. (Feel free to solve it in any way you want, even if it involves rearranging some of the development of the material in this section.) The following few exercises fill out the construction of Proj; try a sampling (or all of them!) to convince yourself that there is nothing tricky here. If you prefer compatible germs, do Exercise 4.5.M.

If you have seen these things before, you may want to try 4.5.Q, not because it is fancy, but because it can be confusing, and it relates to a cause of continuing confusion. Grothendieck often thinks of projectivizations of a vector space not as one-dimensional subspaces, but as one-dimensional quotients. So whenever you see “projectivize” in any algebraic geometry paper (or anything very near algebraic geometry), you have to be careful. In fact, there is method behind Grothendieck’s madness (at least this particular madness) — since he is thinking of geometric things in terms of functions on them, he is thinking of the vector space in terms of the linear functions on them (i.e., the dual vector space), and one-dimensional subspaces of a vector spaces indeed correspond to one-dimensional quotients of the dual vector space.

Okay, enough philosophizing! In chapter 5, the first section is just extending topological notions to these new things we have created. The easy exercises are important in that they make sure you can toss these notions around without thinking. I’m somewhat torn about how important 5.1.E really is — but it is worth doing!

I’d like Exercise 5.1.H to be easy, but I fear it is not. Someone should try it and let me know!

In section 5.2, the key exercise to do is Exercise 5.2.F (integral = reduced + irreducible). Exercise 5.2.H involves an important concept as well. Exercise 5.2.I is why some people start with irreducible varieties, in order to make the sheafy issues less scary.

Section 5.3 is home to the Affine Communication Lemma, which we will use repeatedly, and with great effect. You should understand the proof, and why there isn’t much there! Then the exercises give you lots of opportunities to practice with it. You should try some of the exercises involving the new notions (such as Noetherianness of schemes), but the ones I’ll most suggest you do is 5.3.H and 5.3.I.

Logically (and in pseudolecture 9), section 5.4 could come after 5.2, as it involves more “stalk-local” properties. There are some exercises to make sure you see how the theory fits together (such as 5.4.F), but the really fun ones deal with actual explicit rings (5.4.G to 5.4.L; 5.H is much more useful than it looks).

Then skip the rest of Chapter 5 (I’m in the process of completely rewriting the part on associated points and associated primes), and we will discuss morphisms of schemes in pseudolecture 10!

Plan for these problem.

You can think about these problems for the next couple of weeks. But if by next Wednesday (September 2) you can send your shepherd (or indeed any shepherd) an update on (a) what you have been working on, and (b) how it is going, that will help me figure out what to say next. Also please tell them (c) what is the most confusing thing, (d) what is the coolest thing, (e) what exercise you most want to see an answer to, and (f) what exercise you liked best.

And lastly, more inspired by the question of why the sheaf of functions is called $\mathcal{O}$, from Tomas Prochazka.

Hi Ravi,

I think you mentioned the linguistic origin of O? I read last year something about the origin of some notation in math (I don’t know to what extent is it reliable but it kind of makes sense):

O … holomorphic from Italian (where they don’t use H, not even in this word) 🙂 it’s quite funny if it is true

k … for field from German Körper (body, that’s how field is called I think in some languages, including Czech)

Z … for integers, in German Zahlen – numbers

e … for unit, again German from einheit

U,V … for open sets, German and French Umgebung (neighbourhood) and voisinage (I don’t know French)

F,G … in topology, ferme, Gebiet

finally they also said that Klein bottle was not a bottle but a surface (in German Flasche vs Fläche) but somehow there was a typo or misunderstanding and they started translating it as a bottle.