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.
January 2, 2023 at 2:53 am
Hello Professor Vakil,
first, i wish you a happy new year, with a lot of great mathematical discoveries. Then, i wonder if you could make some videos of your lectures on the parts II and III of your book on algebraic geometry.
With my best wishes, JJacques Brahim. ________________________________
January 5, 2023 at 5:58 am
Thank you for the kind words, and I wish you all the best as well! I think I didn’t do a great job with AGITTOC (I didn’t have much time to prepare anything, and everything was crazy in the pandemic), but my goal was to do something rather than nothing, and with the bar set so low, I was happy with the outcome. Very possibly at some point I might try again (with later parts of the notes), but I would want to spend more time planning and polishing. There is a freedom in giving a class at a blackboard, unrecorded, to a small audience — I feel like I have more permission to try to do a good job, without needing to do a perfect job….
January 5, 2023 at 6:18 am
from Supravat Sarkar. (I’ve made the same mistake I keep making! I’ll have to take a look when I get back from the Joint Math Meetings.)
—
Dear Ravi,
I looked at the new version of notes of “Foundations of Algebraic Geometry”. You modified the proof of Chevalley’s theorem. But I am still not convinced. In page 243 you wrote pi(Zbar)=pi(Z) union FL(Zbar\Z). This may not be true. If q€pi(Zbar) but not in pi(Z), then pi^-1(q) is contained in X\Z, but it may not be entirely contained in Zbar\Z.
Regards,
Supravat Sarkar
January 5, 2023 at 12:34 pm
A few comments, just on the first 18 chapters for now, as that is what you said is closest to being done. Apologies in advance if I am incorrect about anything.
1. On page 14, your sentence starting “And Chapter 29 on completions…” is still using the old chapter numbers.
2. I believe Exercise 6.5.O and Proposition 6.5.26 need Noetherian hypotheses.
3. In 8.3.V.b, you may want to require the Spec B_i to cover the space Y (or at least the closure of the image of X->Y). I could not find a way to show that the map is locally finitely presented in neighborhoods of points that are in the closure of the image, but not in some Spec B_i (though maybe this can be done).
4. At the beginning of Chapter 16, your statement refers to 4 sections of the chapter, but there are only 3 now.
5. In the proof of Proposition 17.6.2, in the step c” implies a, right at the end, you conclude because you have a locally closed embedding with closed image. It was not obvious to me why a locally closed embedding with closed image is a closed embedding; I do not think this was something stated explicitly anywhere. There is a simple proof though: it is a closed embedding above an open subset containing the image, and it is a closed embedding above the compliment of the image, using that the image is closed, so because closed embeddings are local on the target, it is indeed a closed embedding.
6. In section 18.4, the only usage (that I could see) of material from the previous sections Chapter 18 is that if C is projective over k, and C’->C is finite, then C’->k is projective. But this is also implied by 17.6.G as well. (I guess all this is to say that you could hypothetically place this section in the previous chapter.)
7. I do not believe it is described how to obtain a closed subscheme of a product of projective spaces from a bi-homogenous polynomial. You use these closed subschemes in 13.4.1 and 19.4.7. I do not think it is too hard to figure out how to do this affine locally, but originally I had been hoping for a “short cut” to this. For example, the closed subscheme in 13.4.1 can easily be described as pullback from the Segre embedding of a hyperplane.
Thank you again for the notes. I have more comments that I will write out later on the later chapters.
January 10, 2023 at 11:15 am
Thanks, this was really helpful! I have one follow-up question for you, in #3 below, (as well as a bunch of responses).
1. Good catch! There were other spots there too. Now fixed.
2. I’d intended these to be implicit. But I’ve now made these hypotheses explicit.
3. I can’t remember precisely what I had in mind, and haven’t thought it through, so I’d like to ask you: with the change you propose, was the problem eminently gettable? Also, was that series of problems (on local finite presentation) “comfortable” to work through for you? That development is new, and I would like it to be a reasonable (and not excessively painful) walk-through of the ideas.
4. Now I’m less sure where the hardness and the easiness of those sections is (in those three sections), so I’ve removed that sentence completely.
5. I’m going to make this change, although I’ve not decided precisely how. This issue could come up earlier (and does in Hartshorne, although not in these notes): a morphism is separated if the image of the diagonal is closed. Why? Because it is a locally closed embedding, with closed image, and hence (by the point you make) a closed embedding. Another place it comes up: to show the Grassmannian is projective (cheaply), you show that it is a locally closed embedding into projective space, and then you show (without having worked hard to prove any valuative criterion) that the image is closed. (I think that argumet is in the notes.) So this is a small point that can trip people up, so it seems worth pointing this out, e.g. with the nice short argument you give.
6. I was actually repeatedly thinking of moving this section back, and you’ve definitely pushed me over the edge. I also hadn’t realized that 17.6.G did the job — so I’ll move it after 17.6.
7. Like your point #5, I think this is worth stating, and I’ll do so, probably in the “closed subscheme” chapter as a short exercise or observation (and then I can refer back from those two sections you mention).
So hopefully the next version will deal with everything you mentioned. (And I owe some others responses too; I’m now doing pretty well with responding, and will keep going at this rate.)
January 14, 2023 at 7:26 pm
Yes, with the change I propose, the exercise becomes straightforward.
And the locally of finite presentation exercises were indeed comfortable. They make a good addition to 8.3.
March 14, 2023 at 7:04 am
Thanks a lot!
February 14, 2023 at 6:46 am
On page 121, after exercise 3.6.N, in the paragraph “We will soon see …” is a typo: it says “irreducible closesubsets”
February 14, 2023 at 12:07 pm
Thanks, fixed!
February 14, 2023 at 12:13 pm
Here is a comment from Supravat Sarkar at Princeton. He is right, and I have to think about it, but I thought I’d post it here right away so you all could see it (and perhaps comment if you see fit). – R
Dear Ravi,
I looked at the new version of notes of “Foundations of Algebraic Geometry”. You modified the proof of Chevalley’s theorem. But I am still not convinced. In page 243 you wrote pi(Zbar)=pi(Z) union FL(Zbar\Z). This may not be true. If q€pi(Zbar) but not in pi(Z), then pi^-1(q) is contained in X\Z, but it may not be entirely contained in Zbar\Z.
Regards,
Supravat Sarkar
February 14, 2023 at 12:17 pm
Another comment I got by email, that I haven’t had a chance to think about yet, from Dhruv Goel:
Dear Prof. Vakil,
I’m an undergraduate at Harvard and I’ve been reading through TRS. Two quick things:
1. In the latest version of TRS (Dec 31, 2022 draft), did you intend for Exercise 7.3H to come before the parenthetical remark at the beginning of 7.3.7?
2. Is there a coordinate-free solution to Exercise 4.5.T? I’m trying to say that every homogenous maximal ideal of Sym V^* has as vanishing locus a one-dimensional linear subspace of V. I tried thinking about homogenous ideals in Sym V^* cutting out cones in V, but got stuck dealing with nonlinear polynomials. Somehow, we have to use the hypothesis that k is algebraically closed, but linear algebra doesn’t care about that (until you get to say eigenvalues). I asked Prof. Harris, and he didn’t have a clean solution; I thought I might as well write to you.
Thanks!
Best,
Dhruv.
February 17, 2023 at 4:55 am
On page 155, third paragraph, you say that we showed that A^n is irreducible. We did so if the base ring is an integral domain. I guess you wanted to write A_k^n ? (Because in the next exercise 5.1.A you also speak about P_k^n.)
February 21, 2023 at 11:07 am
Yes, that’s right, I’ve now clarified this.
February 18, 2023 at 6:10 pm
Possible typo in chapter 15. In the first sentence second paragraph of section 15.1.2 should it not be that the polynomial is of degree n in m space?
February 21, 2023 at 11:04 am
I think I didn’t word this well — what I meant is that the binomial coefficient is a polynomial in n, i.e. p(n), of degree m: it is the polynomial (n+1)(n+2)…(n+m) / m!. It is indeed confusing, because it is counting (the dimension of) the polynomials of degree n in our m+1 variables. Does that make more sense? I’ve changed this a bit to say “a polynomial in the variable n, of degree m,” in the hopes of being clearer.
February 24, 2023 at 6:45 am
I think so, maybe something like “the space of coefficients of degree n in m+1 variables”?
February 26, 2023 at 6:29 pm
Typo in exercise 22.2.U: should the exercise instead be to describe an isomorphism from $\Omega_{X\times_Z Y/Z}$ to $pr_1^*\Omega_{X/Z}\oplus pr_2^*\Omega_{Y/Z}$? (Instead of describing an isomorphism from $\Omega_{X\times_Z Y/Z}$ to $\alpha^*\Omega_{X/Z}\oplus \beta^*\Omega_{Y/Z}$)
Also, thank you for the notes! I’ve very much enjoyed them.
March 14, 2023 at 6:59 am
Thanks, now fixed!
March 25, 2023 at 1:06 pm
Minor suggestion: On Page 163 (5.4.1 Normality) at the very end you say “reducedness is a stalk-local property” and refer to Exercise 5.2.A. But I don’t think the exercise is very related. Reducedness is a stalk-local property *by definition* 5.2.1, so I thought it might be better to refer to that Definition.
April 2, 2023 at 11:38 am
Thanks for this! I had changed the definition of reduced to it being stalk-local, which led to this error, and I wouldn’t have caught it if you hadn’t pointed it out.
March 25, 2023 at 2:48 pm
On page 177 there is a holdover from the old organization of chapters: Exercise 6.3.B refers to “the definition given at the very beginning of the chapter”. But that stems from times where quasicoherent sheaves where one single chapter 13. The text the exercise refers to is now the beginning of chapter 14. So I guess you should change the reference, or actually rather move the exercise to the beginning of chapter 14?
April 2, 2023 at 11:35 am
Hm, this is a good point. Free sheaves are now not defined until Chapter 14. I’ve now eliminated the reference to “the beginning of the chapter”, and just put “free sheaves” in quotes, with the hope that the subsequent equation display will make clear what I mean.
March 30, 2023 at 2:14 am
Typo: Exercise 7.3.C (page 200) has a missing closing parenthesis after the first reference.
April 2, 2023 at 11:31 am
Thanks, now fixed!
March 30, 2023 at 4:27 am
I don’t know if this is inadvertently or intended, but Exercise 7.3.H (page 203) which asks you to show something about definition 7.3.7 is given before definition 7.3.7 itself is given. If one does the exercise 7.3.H, then one needs to read the future text 7.3.7 anyways, so I guess it would make more sense to put it after 7.3.7. Indeed I saw that a few versions ago it was given after 7.3.7; I don’t know why the order changed.
April 2, 2023 at 11:29 am
Thanks! I’m not sure exactly why I did that. I’ve moved 7.3.H, and also 7.3.J. (They have changed names of course.)