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June 2017 version

Posted by ravivakil under

2015-16 course,

Actual notes
[12] 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.

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.

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…