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!
July 1, 2023 at 9:49 am
Very cool!
Thanks, Ravi
July 1, 2023 at 10:50 am
p. 16 Section 0.2, first paragraph, last sentence, typo: “firt quarter”.
July 1, 2023 at 10:53 am
That was fast! But now fixed!
July 1, 2023 at 11:29 am
p. 53 Section 1.6.7 indices are still backwards for the definition of chain homotopy (should be C^{i+1} -> D^i).
July 1, 2023 at 12:37 pm
Right again (now fixed) — I don’t think I had that in my to-do list for some reason. (I feel a little guilty about using the word “homology” but because my indices as “rising” (superscripts) I could/should call it “cohomology”. But it sounds like people aren’t getting confused, so long as I don’t get things like this^^^ wrong.
July 5, 2023 at 2:08 am
Thanks for the notes – it’s a joy to work with them!
Some small remarks:
– Exercise 14.6.D.: Typo in description: Pullbacck instead of Pullback
– Exercise 15.1.D.: I feel like there should be pi^*O(m) = O_{A^{n+1} \ 0} instead of pi^*O = O_{A^{n+1}} \ 0.
– p. 424: “The language of base points … readily
applies to of the linear series.” This seems to be missing some words.
July 30, 2023 at 5:09 pm
You are right on all three counts, and they are all patched. Thanks!
July 7, 2023 at 6:41 am
Dear Ravi,
I love your book and hope to see it in print soon!
Two small comments (if I am not mistaken):
In Exercise 6.6.D the exact sequence is not necessarily left exact, i.e. multiplication by x is not injective.
In Exercise 6.6.N the Noetherian hypothesis is not needed for the first assertion, i.e. that A_p non-reduced is equivalent to the p is in the support of the nilradical. It is only needed for the second assertion about the openness of the reduced locus.
July 31, 2023 at 11:36 am
Thank you Jan! These are all right, and I’ve edited both appropriately.
July 7, 2023 at 11:12 am
Thank you, Ravi!
July 9, 2023 at 12:41 am
Another comment on remark 6.6.30:
As the ring k[x]/(x^2) is not reduced, this is maybe not a strong counterexample that the converse of 6.6.R does not hold.
July 17, 2023 at 4:38 am
Please ignore my comment above: I misread remark 6.6.30.
July 30, 2023 at 4:59 pm
No worries; I value having people reading as closely as you!
July 13, 2023 at 2:03 am
Dear professor Vakil,
Is it possible to add the word `Gorenstein’ somewhere in your book, in case this can be done easily?
Thanks for your wonderful work!
I appreciated you book very much, and hope to see it in print soon!
July 30, 2023 at 4:59 pm
That’s a good idea – it is easy to say given what the reader knows, and an important concept. Likely I’ll add it in the next week. (And thank you for the kind words.)
July 17, 2023 at 8:19 am
p585. penultimate line, typo, “An element of the relative (co)tangent ‘spaceis’ called a relative (co)tangent vector.”
July 30, 2023 at 5:06 pm
Thanks, now fixed! (Unfortunately I will have added a lot of errors like this, while making the index, where spaces have accidentally disappeared.)
July 21, 2023 at 7:04 am
In exercise 14.3.O, it appears the symbol \times should be replaced with \otimes.
July 30, 2023 at 4:57 pm
Thanks! I see what you mean, and it definitely could be. The one with times maps to the one with the otimes, which maps to the right side of that map in the notes. So we’re both right. I want to write it like this so the two vector vector bundles (the two sides of the “\times”) are most visible.
(And thanks for your other suggestions and corrections you sent by email; I think I owe you a response, and if so, hope to get to it soon.)
July 23, 2023 at 5:02 pm
Small typo: question 9.3.Q “show that the P(1,1,n) is isomorphic” probably missing “space” or something.
July 30, 2023 at 11:18 am
Another typo in Def. 10.7.7. …, we say \rho: X^{red}\to X is “the” reduction of *X if*, missing a space between “X” and “if”
July 30, 2023 at 4:55 pm
Thanks, now fixed! I’m now properly doing the index, which causes spacing errors like this when I’m not careful, and these are *really* hard to catch.
July 30, 2023 at 4:56 pm
Thanks, now fixed!
August 15, 2023 at 4:54 am
Hi Ravi and friends,
Thanks for your interest in my Forest and related ideas! I sadly broke the link here, and now the correct URL is: https://www.jonmsterling.com/. (At some point I may figure out how to install redirects, but that is for another day.)
Best wishes,
Jon
February 4, 2024 at 7:44 am
For a long time, I’ve been meaning to reply to this, and have not had time. I realize I may not have time to properly reply, so instead, I would like to quote one emails Jon sent me because it will be of interest to others. I have not asked him permission to post it (which is bad form), but I *think* he would not mind.
“I thought you might be interested to know that I have created a tutorial for building forests using Forester this week: https://www.jonmsterling.com/jms-0052.xml (“Build your own Stacks Project in 10 minutes”). Incidentally, I have reconstructed the Forester tool from scratch since we last corresponded and as a result it is much more efficient and requires fewer external tools.”
December 10, 2023 at 11:28 pm
On p. 130 after (4.1.3.1), you say that “signs are involved” in the right hand map, and specifically that the map $A_{f_j} \to A_{f_i f_j}$ is the negative of the localization map. How do these signs enter naturally? The negative of the localization map isn’t even a ring homomorphism. (Nor is the first map $0 \to A$ for that matter.) I wonder if framing the category-theoretic definition of sheaf in terms of exact sequences (which is suggestive but not really rigorous) rather than purely in terms of equalizers (which seems to be the standard definition) is more of a hindrance than a help.
December 16, 2023 at 3:47 pm
Hmmm, never mind; I see what you mean here. I guess I’m not yet one of the “experts” you mention at the start of the sentence 😛
December 19, 2023 at 8:47 pm
I will say, though, that I don’t think the condition $i \neq j$ is needed in the rightmost product of (4.1.3.1)
January 22, 2024 at 5:29 pm
True! But that will only open another kettle of fish (which I am carefully not mentioning).
December 15, 2023 at 3:48 pm
On p. 131, 4 lines before Ex. 4.1.C, “which restricts to a_i” should be “which restricts to a_i/g_i”, I think.
January 22, 2024 at 5:27 pm
Fixed, thanks! But this makes me wonder if I should have also written a_z/g_z two lines earlier, rather than a_z / f_z^{l_z}. (On general principle, I’m not going to make this additional change, unless you tell me it is a good idea.)
January 22, 2024 at 6:32 pm
No, in fact I think my original suggestion probably should have been to change a_i to a_i/f_i^l_i, since technically you haven’t actually defined g_z.
January 22, 2024 at 6:46 pm
I think I see your point (and in any case I made the change). I “sort of” defined it on the previous page, but I don’t like “sort of” defining things, and prefer the cleanliness of what you propose.
December 16, 2023 at 3:48 pm
p. 130, two lines before (4.1.3.1), “equalizerexact” is missing a space.
January 22, 2024 at 5:30 pm
Thanks, fixed! And also more generally, thanks for your many comments which are really useful, and which I am now working through!