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June 2017 version

Posted by ravivakil under

2015-16 course,

Actual notes
[35] Comments
The June 4, 2017 version is in **the usual place**.

As always, I have a big list of emails, responses, etc. that I want to respond to. Please continue to send suggestions and corrections! I’m continuing to accelerate, but my responses are still not keeping up with your comments. (But please keep them coming!)

July 15, 2017 at 4:48 pm

Awesome! I really appreciate that you put these notes up for free and keep them regularly updated.

December 2, 2017 at 11:41 am

Thanks for the kind words, Bartek!

July 15, 2017 at 5:13 pm

The cancellation theorem for projectivity holds without assuming the target being quasi-compact. This result is proved in EGA II 5.5.5.v, and Stacks Project 0C4Q. But in 17.3.6 of the book, this extra condition is emphasized.

More details:

https://www.sharelatex.com/project/5925f31c5df9d6926723cbb6

July 19, 2017 at 1:09 pm

Just before Proposition 6.6.5. on page 190, it says “An integral finite type scheme is said to be rational if it is birational to for some Shouldn’t the last word/letter of this sentence be “n”?

August 2, 2017 at 5:03 pm

Yes! Now fixed!

July 21, 2017 at 6:24 am

In diagram in the hint of Exercise 10.1.N the morphism $\pi^\{\text{red}}$ is pointing to the wrong object.

July 21, 2017 at 6:32 am

I should have probably solved the exercise before commenting… I believe the arrow pointing to $X$ should just not be labeled.

August 2, 2017 at 5:01 pm

Actually, I think you were also right the first time! I’ve now fixed it — thanks!

August 3, 2017 at 11:47 am

In problem 24.4.L, the scheme X is defined twice. For the notation to be consistent with paragraph 24.4.13, what is labeled Y should probably be Z, and the second X should be Y.

August 3, 2017 at 6:22 pm

I agree with you! Now fixed.

August 4, 2017 at 11:31 am

Dear professor Vakil,

(1)In Unimportant Remark. 1.6.16 you said that the double dual of a vector space is right exact. But it seems that it is exact. Please see

https://math.stackexchange.com/questions/223280/induced-exact-sequence-of-dual-spaces

Taking the dual twice still gives us an exact sequence.

(2)In the same paragraph: “but does not commute with infinite direct sums”, but you used the infinite direct product notation instead of the infinite direct sum. I am not sure what you mean in here.

August 6, 2017 at 1:14 pm

Thank you Hao Xing. Right now I am not sure what I was thinking when I added this! Indeed, right now it seems to me that, assuming the axiom of choice, “dual” is an exact contravariant functor on vector spaces. (If U is a sub-vector-space of V, then any functional on U can be extended to a functional on V.) I will think about it a bit more, and perhaps just cut this remark… But I would definitely like another example showing this behavior (of a right-exact functor not commuting with colimits)…

I hope to write more here after thinking about this a bit.

December 2, 2017 at 11:43 am

Update: Regarding (1), you are right — it is exact, but for the purposes of this example we only care about its right-exactness. I have added a parenthetical comment. And in (2), you are also right, and I have fixed the notation!

August 11, 2017 at 9:30 pm

There is a slight typo.

On page 49, 1.6.4, the last sentence of the first paragraph:

“…holds in any abelian category.)”

I think the right bracket here doesn’t match any left bracket…

August 29, 2017 at 1:05 am

Here’s another one: in paragraph 25.1.4, it says “– that maps a different regular variety to a each point of Y –“. I believe the ‘a’ is unnecessary.

September 13, 2017 at 6:06 pm

Another: in the statement of Theorem 26.1.2, the subscripts of the x’s are a little off.

Also, in Theorem 26.1.3.(i), there is a unclosed parenthesis “(i.e., N has finite length, –“

October 1, 2017 at 9:54 am

Thanks, both fixed! (And to be sure I understand the first: I’ve changed the “r” to an “n”.)

October 1, 2017 at 9:52 am

Thanks, fixed!

September 14, 2017 at 6:07 am

Another typo.

On p. 115, Exercise 3.4.C,

“…that products are associative, i.e., (I_1I_2)I_3)=I_1(I_2I_3).”

should be

“…that products are associative, i.e., (I_1I_2)I_3=I_1(I_2I_3).”

October 1, 2017 at 9:46 am

I had to look closely to see what you meant — that one is subtle! Now fixed, thanks.

October 1, 2017 at 9:48 am

Thanks! You have eagle eyes! Now fixed.

September 11, 2017 at 9:51 am

In 8.3.9, should “On the affine open set of , if the set corresponds to the radical ideal ” have the last equation read ?

November 23, 2017 at 9:52 am

You’re right — now fixed! (In the version not yet posted…)

September 21, 2017 at 11:53 am

I’m assuming the diagram on (9.1.5.1) should be the map , not ?

October 1, 2017 at 9:45 am

Yes, thanks for catching that — now fixed!

October 1, 2017 at 11:39 am

Just a heads up–on the beginning of page 249 you have the word “with” repeated twice.

October 5, 2017 at 11:14 am

Thanks, now fixed! (To others: that was in the 2nd last paragraph of 9.1.5.)

October 5, 2017 at 6:32 pm

I’m confused about the Noetherian hypotheses in section 28.2. Key Theorem 28.2.1. doesn’t explicitly mention Noetherian hypotheses (on the ring B), but Lemma 28.2.2. does mention, and also seems to use them in the proof, namely when you choose finitely many generators for D^{m+1}, which is a submodule of the finitely generated module K^{m+1}.

Then in “Proof of Key Theorem 28.2.1”, you mention the projective case of coherence of higher pushforwards. I assume that this means Theorem 18.8.1.(d), which has the assumption that O_Y is coherent. But since you’re anyway applying Lemma 28.2.2, don’t you need the ring B to be Noetherian?

In Exercise 28.2.E, you give two conditions, either Y being locally Noetherian, or Y being reduced. But don’t we already need Y to be locally Noetherian to be able to construct the complex K?

Finally, in Exercise 28.2.F, you say that Noetherian hypotheses won’t be needed, although the original statement of Grauert’s Theorem assumes that the base Y is locally Noetherian.

Thanks, and sorry for the long post.

Ps. Right before Exercise 28.2.H., “- part of of part (i) of the theorem.” has ‘of’ twice.

November 11, 2017 at 8:07 pm

Hi Tuomas, You are completely right on all counts! The changes are either implemented, or en route to being implemented.

October 15, 2017 at 4:46 am

Is there a reason why two of the entries in the table on page 46 are blank? (It looks like they could be filled with Mod_A, Mod_A.)

November 19, 2017 at 1:19 pm

Frankly, I have no idea, and I am mystified! I have now fixed this.

October 16, 2017 at 7:35 pm

In the proof of Theorem 29.6.1(b), it says “then alpha(X_0 \ U) is a closed subset of Y’ “. Should it be alpha(X \ U) instead?

November 19, 2017 at 8:52 pm

Yes, thanks, fixed!

November 15, 2017 at 3:59 pm

Page 68, the last line: “the row p+r, from which…” Should that be “q+r” instead?

Page 68, the second line from the bottom, should the second superscripts be (p-1, q+1) instead of (p+1, q-1)?

Page 67, Exercise 1.7.I Second line: For r>>0, interpret S^{p,q}_{r} /dS^{p+(r−1)−1,q−(r−1)}_{r-1}… as the cycles… I am not sure if this is ahint for how the filtration goes or you actually ask for the interpretation of S^{p,q}_{r} /dS^{p+(r−1)−1,q−(r−1)}_{r-1}+S^{p−1,q+1}_{r-1} as the cycles…?

It would be great if you can add something more in Exercise 1.7.I.

November 15, 2017 at 4:04 pm

“a hint”, “It” would be great…. sorry for my own typos in the comment

[I’ve patched both! – R.]