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Nineteenth post: revisions

Posted by ravivakil under

2011-12 course,

Actual notes
[6] Comments
A revised version is now posted **at the usual place**. There are no new sections. I’ll be posting revisions over the next academic year, in response to comments from **the course**, comments here (both new and old), and a large number of emails I’m gradually going through.

I won’t bother adding a new post in the future when there are only relatively minor revisions. But if you want to be informed, just let me know. And if people would prefer that I announce each revision, I’ll do so.

October 9, 2011 at 3:44 am

Minor typo: In Exercise 3.5.G, there’s a line where E and F should be switched.

October 9, 2011 at 5:33 am

Oh sorry, I missed the place on the homework where you said you were already aware of this.

October 11, 2011 at 10:52 am

Thanks Sam! If you catch more, please let me know… it’s amazing how many things get missed…

October 17, 2011 at 9:10 am

Hi Ravi,

I have two minor remarks/questions (on the Oct 5, 11 version):

1. In 5.5.3 the definition of graded rings over A says that they are graded over integers. In 5.5.5 a projective A-scheme is defined as the Proj S for a finitely generated graded ring S over A; but here don’t you want the grading over non-negative integers, to make sure that Proj S is indeed a projective scheme over A?

2. In the same spirit, in Section 7.4.2, in the definition of the n-th Veronese subring of S the direct sum should be over all integers (i.e. not only over non-negative integers) unless the grading is assumed to be over non-negative integers.

Thanks!

October 18, 2011 at 4:38 pm

Oops, I just realized that Proj S = Proj S’, where S’ is the graded subring of S generated by homogeneous elements of non-negative degree. So I retract my earlier comments.

May be you can add it as an exercise ðŸ™‚

October 20, 2011 at 11:36 am

Thanks for pointing this out. In correctly trying to fold in Z-graded rings, I’ve caused ripples of problems. I think they are now patched, and a new version will be posted within a week (hopefully tomorrow).

(Damn you David Speyer! ðŸ™‚ )