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Another small post: a newer version

Posted by ravivakil under

Actual notes
[49] Comments
A new version is now posted at **the usual place** (the June 11, 2013 version). The main reason for this post is to have a reasonably current version on the web. Thanks to many people for helpful comments — most recently, a number of comments from János Kollár, Jeremy Booher, Shotaro (Macky) Makisumi, Zeyu Guo, Shishir Agrawal, Bjorn Poonen, and Brian Lawrence (as well as many people posting here, whose names I thus needn’t list).

There are a large number of very small improvements, and I’ll list only a few in detail. I’ve replaced the proof of the Fundamental Theorem of Elimination Theory (Theorem 7.4.7). (This new proof is much more memorable for me. It’s also shorter and faster, and generally better. It was the proof I was trying to remember, which I heard in graduate school. I couldn’t find it in any of the standard sources, and reproduced it from memory. But it must be in a standard source, because it is certainly not original with me!) Jeremy Booher’s comments have led to the completion discussion being improved a lot.

I had earlier called , where is the subring of , the “knotted plane”, to suggest the picture that it was a plane, with the origin somehow “knotted” or “pinched”. János Kollár pointed out that “knotted” is misleading, because it is not in any obvious way knotted. I’ve changed this to “crumpled plane”, but this isn’t great either. I’m now thinking about “pinched plane”. Does anyone have a good suggestion for a name for this important example?

**Still to do: ** To repeat my comments from **the previous post**, as usual, the figures, index, and formatting have not yet been thought about. My to-do list is quite short, so please complain about anything and everything (except figures, index, and formatting). (Perhaps in the next posted version, I may have “TODO” appearing in the text indicating where there is still work in progress.)

Finally, just for fun, here is a picture of the two rulings on the quadric surface, from the Kobe skyline. (It is clear why algebraic geometry is so strong in Japan!)

Kobe port (click to enlarge)

June 11, 2013 at 6:21 pm

“Pinched plane” is clearly the right name. If you take a big pinch of a piece of fabric, you identify two points (on a surface, no less). So a little pinch identifies two points that are already adjacent, that is, it maps a double point to a point.

The creased plane is Spec C[y,x^2,x^3]. Less sure about crumpled.

June 12, 2013 at 9:07 am

I am convinced!

June 13, 2013 at 1:53 pm

Now changed.

June 11, 2013 at 6:59 pm

Here http://ubuntuone.com/6FwWxpdUTloj931CepXsLX

is a pure HTML version of FOAGjun1113public, which is generated by pdf2htmlEX https://github.com/coolwanglu/pdf2htmlEX/blob/master/README.md

June 11, 2013 at 7:12 pm

Please use firefox or chrome for the link http://ubuntuone.com/6FwWxpdUTloj931CepXsLX

It may be slow due to the network (27M).

June 12, 2013 at 9:09 am

This is incredibly cool! It works very well. As the README page says, regarding how good pdf2htmlEX is, “A beautiful demo is worth a thousand words.”

June 14, 2013 at 9:28 am

I am glad that you like pdf2htmlEX 🙂 Hope you can recommend the open-sourced free tool (GPL3) to other academics, such as arXiv, AMS, or other teachers/students.

June 15, 2013 at 4:44 am

I rarely recommend things, but this seems worth recommending to me. (I just mentioned it at the meeting of the

CEIC, a standing committee of theInternational Mathematical Union.) I could imagine anarXivoverlay: a bot checks all articles with hyperlinks, and makes an html version available; this seems within the realm of possibility.June 13, 2013 at 1:38 pm

Speaking of rulings, you might have mentioned Shukhov Tower as well: http://en.wikipedia.org/wiki/Shukhov_Tower

June 13, 2013 at 1:53 pm

It is beautiful! If I understand it correctly, we can see the rulings of a quadric, and the quadric has a very narrow neck, and the structure goes just above the neck.

It was also worth reading about Shukhov.

I hope they don’t tear this down! And I hope they let tourists visit it too.

June 22, 2013 at 2:44 am

Hi Ravi,

I have a question.

Is Exercise 15.4.H so easy to prove for average students without assumption or assumption is a polynomial ring? Hartshorne and Liu contains such an assumption. Or I am misunderstanding something?

June 23, 2013 at 8:34 pm

Hi Ravi,

Question 2.

In Ex 15.4.K, isn’t any assumption necessary for ring ?

In Hint, 15.4.K needs 15.4.D(b). But there is a field. My proof for 15.4.D(b) works only for Noetherian ring .

July 8, 2013 at 9:57 am

Hi Frank,

Sorry to be slow on this! I’ve been away, and now that I’m back, my limited time to work on the notes has been spent on the index (which I hope to be done with soon).

I promise to get back to you (about this and the 2nd question below) before long. I took a quick look, and was not able to immediately be led to a solution, which is a sign that you are right (as I try to set things up so that I am prompted to do the right thing by reading the notes to myself).

July 8, 2013 at 10:04 am

p.s. I tried also to email the address you gave when posting, but it didn’t work; feel free to give me your email address so you needn’t follow the posts (and feel free not to as well). I of course promise not to spam you!

June 23, 2013 at 12:55 pm

Hi Ravi,

I am really enjoying the notes!

Just a comment about the proof of Chevalley’s theorem. I feel like you should add the following exercise:

It suffices to prove the theorem when X and Y are reduced affine schemes.

You seem to use this in exercise 7.4.L when you are applying the Grothendieck generic freeness lemma. It also highlights two important ideas: reducing to the affine case and the point set part of a morphism does not care about nilpotents.

A happy reader,

Daniel

June 23, 2013 at 1:03 pm

woops I should have turned the page. Sorry!

July 8, 2013 at 9:58 am

No worries, and if you catch anything else, just let me know! I’m relieved that you felt the need for the same thing that I did, because it is a sign that I’m on the right track.

July 16, 2013 at 10:00 am

Hi Ravi,

First of all, thank you very much for the wonderful notes!

I have two minor questions.

1. In Exercise 13.7.G(b), is a Noetherian assumption necessary? Without it I don’t see why the support has to be finite (even though it’s clearly finite on every affine open). For example, we can glue an arbitrary number of P^1’s along P1\{infinity} and take the trivial coherent sheaf.

2. There appears to be a minor typo 8 lines from the bottom of page 421. There should be an ‘\in’ after (s1,s2) and “such that agree” should probably be “such that they agree”.

Thanks again,

Daniel

January 17, 2015 at 11:51 am

1. You’re right — sloppiness in my use of “curve” (as noted by others) is as fault. I have just added “quasicompact”.

2. Fixed!

Thanks for both!

August 7, 2013 at 2:05 pm

Dear Ravi Vakil,

I’m having quite some trouble trying to justify the reduction to the projective space in exercise 18.1A. How can we guarantee that the pushforward of a coherent sheaf on X is a coherent sheaf on P^n_{A} assuming only that A is a coherent ring? We cannot apply theorem 18.8.1(d) here, because Hilbert’s basis theorem isn’t true for coherent rings, so \mathcal{O}_{P^n_{A}} doesn’t need to be a coherent sheaf of rings. Also if f is a closed embedding, the corresponding pushforward doesn’t need to preserve coherence, unless the associated ideal sheaf is of finite type.

Regards,

Nuno.

[I still want to respond to this. -R]October 18, 2013 at 1:07 pm

First of all, sorry to be quite slow in responding to comments here, and to email! I’ve been working on the index — I’m trying to make it particularly good, but it is taking a long time and is really tedious, and I’m not sure if I am succeeding. I think I need another 10 hours or so more of work time before it will be done. But I intend to respond to all the loose ends! For now, to keep the conversation going:

i) Referees have suggested that the working title Foundations Of Algebraic Geometry isn’t great because Weil’s book had the same name. I am thinking about changing it to “The Rising Sea: Foundations of Algebraic Geometry”. I’ve liked that quote from Grothendieck, and first heard the name from Daniel Murfet’s

wonderful site. (I have his permission to use it.) He in turn got it from Colin McLarty’s translation.ii) Here is Daniel Barter’s argument proving Proposition 6.5.5 (which says that for reduced schemes, birationality is the same as having isomorphic dense open sets). My current proof sucks (as has been made clear to me by many people), and it will be changed.

Daniel’s argument is

here.October 18, 2013 at 1:26 pm

I agonized over the notation for the skyscraper sheaf, which is the pushforward by the inclusion of a point . I thought might confuse people, so I went with . But I like the first better (and Tom Graber does too, which has prompted this comment) — any opinions? (Feel free to reply by email…)

January 16, 2014 at 3:38 pm

Dear Ravi,

just a minor comment on Chapter 17.

In my view, definitions 17.1.4 and 17.2.3 are slightly at odds with each other. I think if you define tot F= Spec Sym F^\vee, then it is more natural to set P(F)= Proj Sym F^\vee.

It’s a matter of taste, of course, but the choice of duals here often causes a lot of confusion.

January 16, 2014 at 3:47 pm

Also, putting the dual in 17.2.3 would reconcile that definition with

definition 4.5.12 (projectivisation of a vector space).

January 25, 2014 at 4:13 am

Hi,

I think there might be a slight mistake after the statement of chow’s lemma 18.9.2. You say that we can conlude that every PROJECTVIE variety admits a birational morphism from a projective variety and I think what you want to say is that a PROPER variety admits …

I have wanted to say this for a while so this is the perfect opportunity : thank you very very much for those notes, they are amazing for anyone out there who tries to learn algebraic geometry.

January 17, 2015 at 11:45 am

Thanks, now fixed! (And thanks for the very kind words!)

February 23, 2014 at 3:37 am

Ravi,

First and foremost, thanks for the notes. There are many tiny comments I’d like to make at some point (when I have time to compile them), but there is one in particular that I thought might be helpful.

You obviously strive to make your notes optimally understandable/readable, and like giving exercises to aid in understanding. That said, there is one exercise that seems very instructive, and is missing from your notes (unless I’ve missed it somewhere–if so, I’m sorry!).

Namely, while it’s implicit in all the results following, the fact that (Weil) divisors on $\mathbb{P}^n_k$ are just formal sums of irreducible homogenous polynomials in $f\in k[x_0,\ldots,x_n]$ (up to constants), and that the associated valuations are what one would expect (how high a power of $f$ a polynomial is divisible).

From this it’s easy to see that if $\displaystyle D=\sum_i n_{f_i} V(f_i)$ is an effective divisor with degree (i.e. $\displaystyle \sum_i n_{f_i}\deg(f_i)$) $d$ then it is equivalent to $dH_i$ (where $H_i$ is any of the hyperplane classes) by just realizing that $D-dH$ is $\displaystyle \mathrm{div}\left(\frac{f_1^{n_{f_1}}\cdots f_m^{n_{f_m}}}{x_i^d}\right)$. With this it’s obvious that $\text{Cl}(\mathbb{P}^n_k)=\text{Pic}(\mathbb{P}^n_k)=\mathbb{Z}$.

As of now you deduce this last result from the exact sequence (excision sequence?) which, while the same in spirit, lacks the satisfying ‘oomph’ of the hands-on proof as above.

Just a thought!

Best,

Alex

March 3, 2014 at 3:15 am

in the hint to Exercise 13.1.I it says “cleverly apply exercise 13.1.I”. what exercise should it actually refer to?

Thank you,

Edo

January 17, 2015 at 11:47 am

Now removed — I have no idea what I was thinking! But clearly I was being a little too clever…

May 8, 2014 at 10:32 pm

Ravi,

There seems to be a mathematical typo on page 494. Below 19.1.2 you say “Once we know what ‘unramified’ means this will say that unramified+finite=closed embedding.” Unless I am being silly this is clearly false though–take an unramified extension of number fields, and consider the associated extension of number rings. This is finite and unramified, but does not correspond to a closed embedding. Of course this doesn’t contradict 19.1.2 since it’s not degree less than one at all points–just take a split prime.

I’m not sure how precisely unramified fits into the picture easily. If you were unramified, finite, injective on points, and never have any non-trivial extension of residue fields, then the conditions of 19.1.2 hold (at least for curves). But, besides that, I don’t see an easy fix.

I apologize in advance if I made an amateurish mistake in understanding!

Best,

Alex

May 9, 2014 at 6:21 am

Hi Alex,

You are right, thanks! I’m not sure what I was thinking (and bumped into this issue in the course this year) — I think what I was thinking was essentially what you said — unramified, finite, and degree at most one at all points implies closed embedding. But in any case, this isn’t a helpful comment in 19.1, so I’ll fix/remove it.

(The notes with numerous small changes should be reposted before long… but nothing big has changed…)

best,

Ravi

July 29, 2014 at 1:43 pm

Is there any good reason of using \varinjlim and \varprojlim instead of simply colim and lim?

July 29, 2014 at 3:52 pm

There isn’t a great reason. My main thinking is that the notation would be quite symmetric. It doesn’t help me remember which is which, to be honest — I remember that the cokernel is a colimit, and a kernel is a limit; the kernel (and more generally the limit) maps *to* the diagram, and the cokernel (and more generally the colimit) *receives* maps from the diagram.

August 14, 2014 at 6:27 pm

Well, yes, the mnemonic you give here (which is also in the text) was the thing that made me think switching notation would be beneficial. There’s no sensible way to figure out which arrow direction is which, unless you actually want to remember even more terminology.

September 13, 2014 at 12:55 pm

The way I remember this is to think of projective (inverse) and injective (direct) limits. If you draw your arrows in your sequence of objects from left to right (as most people do), then the inverse limit (a categorical limit) will be to the left and the direct limit (a categorical colimit) will be to the right. Thus, the arrow in \varprojlim is to the left and the arrow in \varinjlim is to the right.

August 15, 2014 at 5:33 am

I’m convinced. I’m convinced enough that I don’t know why I originally went with and . I will make this change. It may be some time (on my “to-do” list, which has been growing for some time, but will eventually start shrinking again), so if there is a defender of and , I would be interested in hearing a case for it.

August 21, 2014 at 1:33 am

Dear Ravi,

I don’t know how universal this is, but in my world, I have used \varinjlim when it is a direct limit, instead of a general colimit. That is how I remember to look more carefully at my surroundings when I’ve written in colim instead of \varinjlim in my notes.

Best,

Alex

October 1, 2014 at 3:24 pm

26.1.4 (computing the depth) currently reads: “Is there a zerodivisor x on M in m? If not, then depth M = 0”

Shouldn’t it be “Is there a non-zerodivisor x on M in m?”?

October 1, 2014 at 3:41 pm

Yes indeed — you are right, thanks!

November 4, 2014 at 11:51 pm

Any ideas when you’ll have time to update the current version? Thinking about ordering a hard copy through lulu.com for myself, but would feel very annoyed if I did it just before a new version is pushed out on the web. 🙂

Thanks for this great resource! These have really helped with my qual prep!

November 23, 2014 at 8:22 am

(I’m glad it has been helpful! That’s why I’ve enjoyed doing this.)

I think it’s time to post an update soon (as it has been evolving continuously since the previous public version, although the content is essentially the same). Thanks to your comment, I am going to aim to do this over the Christmas break (as early as possible), once exams are done.

December 2, 2014 at 7:25 pm

Hey Ravi,

I have another mathematical typo, I believe. It seems in your usage of the word ‘curve’ in 17.4.2, and in surrounding areas, is pretty loose. Namely, prior to chapter 19 where you fix curve to mean geometrically integral, smooth, projective, curve just means dimension 1 for you. Thus, for 17.4.2, you should include ‘separated’, otherwise the line with doubled origin is a counterexample.

Thanks!

Best,

Alex

December 16, 2014 at 12:34 pm

I think most of Section 17.4.4 has similar problems with hypotheses — maybe it’s just me?

January 17, 2015 at 11:37 am

I agree with both of you, and this section’s hypotheses will be cleaned up soon. Unfortunately, I expect (from bitter experience) to miss things, so if you happen to remember where in the surrounding areas hypotheses are missing, please let me know!

December 14, 2014 at 2:20 am

A minor typo: In Exercise 5.2.C on page 153, the first sentence should probably read, “Show that (k[x,y] / (y^2, xy))_x has non-zero nilpotent elements.”

Thank you for all the care and effort you are putting into this book.

December 16, 2014 at 6:10 am

I *think* that the current version is right — it may look like that scheme has nilpotents, but it actually doesn’t! (If I am wrong, please let me know… I am going to try hard to have another version public over Christmas break 2014.)

Thanks also for the kind words!

December 16, 2014 at 6:03 pm

Oops, you are right! The fuzz goes out along with the point it’s sitting on.

I am looking forward to the Christmas goodies!

December 16, 2014 at 12:31 pm

By localizing at you’re throwing out the embedded point at the origin, so I think this is okay and you end up with something naturally isomorphic to .

December 16, 2014 at 6:04 pm

True, thanks.