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Apologies

Posted by ravivakil under

Uncategorized
[8] Comments
I apologize in advance for all of the following.

- I won’t have a chance to follow comments as frequently as I would like to. At this point, I expect to check in roughly once per week, with some big gaps during busy periods (conferences; fall recommendation periods; family illnesses; etc.).
- I hope to be able to respond to most comments deserving a response, but I fear I won’t be able to.
- If you would like to respond to something privately, feel free to email me. As others who have tried to email can tell you, I get enough email that I’m unable to respond to more than a fraction (and even that not in a timely manner), but I certainly try.
- Posts will be retroactively edited to fix problems.
- The many comments people make will lead to improvements of the notes (or at least changes!). I apologize in advance for not properly acknowledging everyone individually for their help.

Having gotten all of that off my chest, I feel much better. I hope you do too!

July 3, 2010 at 10:00 pm

I hope you will be able to post pdfs as well. I think wordpress does not support all the LaTeX stuff. For example I think it does not support xypic.

[I definitely will! – R.]July 12, 2010 at 5:21 pm

To push it a bit more, is it possible to have some virtual office hours, for instance like Kiran Kedlaya here:

http://www-math.mit.edu/~kedlaya/18.726-spr05/virtual.html

in which you will answer questions?

[I’ll try to answer at least some of the questions posted in the comments. Others may answer them too. Most likely it will be more useful to have the questions (and answers) in the comment section associated to a particular topic rather than having a single location for all questions on all topics. In short: I’ll try to be useful, within the varying numerous constraints on my time. -R.]

July 29, 2010 at 6:18 am

Ask and ye shall…

‘found a typo for you!

Search in page “functors are are done later, because”

[Thanks, fixed! – R.]

August 13, 2010 at 7:25 pm

Hey Ravi,

A non-urgent question about the most recent version of the notes: Can you clarify what you mean to ask in Exercise 26.3.B? It reads, “show that sheafHom(F, I) is acyclic for the functors Hom and sheafHom”, where F, I are O-modules and I is injective.

That seems to be asking to show that Ext^i(G, sheafHom(F, I)) and sheafExt^i(G, sheafHom(F,I)) both vanish for i>0. We certainly need the first of these (in case G = O is the structure sheaf) to apply the Grothendieck spectral sequence to deduce Exercise 26.3.J. However that case is easy to do directly (a la Grothendieck in “Tohoku”) by verifying that sheafHom(F, I) is flasque, by the injectivity of I.

To solve 26.3.B, you suggest to use the previous exercise 26.3.A, an easy one which shows sheafHom(-,I) is exact. My only idea for how to do this works only in the special case where whatever category of O-modules we care about has enough locally frees of finite rank. When E is locally free, with E* denoting its dual, we have Hom(F,sheafHom(E,I))= Hom(F \otimes E*, I), by the sheafHom, tensor product adjunction. Since E* is locally free and I is injective, this is visibly exact as a functor in F, so sheafHom(E,I) is injective. Thus a resolution E –> F of F by locally frees of finite rank induces an injective resolution sheafHom(F,I) -> sheafHom(E, I), and so the groups we are interested in

[sheaf]Ext^i(G,sheafHom(F,I))

can be computed as the cohomology of the complex

[sheaf]Hom(G,sheafHom(E,I))

= [sheaf]Hom(G \otimes E*, I),

which is easily seen to be acyclic

(again since the complex E* of dual vector bundles is a flat resolution of F, and since I is injective, invoking 26.3.A at the very end in the sheafy case).

This argument is kind of nice, but uses the existence of resolutions by vector bundles in an essential way, so its application seems entirely restricted to coherent sheaves on projective schemes. Granted, that’s the case we really care about. But in 26.3.B you mention no such restrictions. So do you have an alternative proof in mind, perhaps a generalization of Grothendieck’s argument for the flasqueness of sheafHom(F,I)?

I hope you’re having a good time in India.

Regards,

Sam

August 13, 2010 at 7:34 pm

Actually my argument is bogus, even with the restrictions I imposed, since G \otimes E* is not acyclic. (Its cohomology computes the Tor_i(G,F), of course.)

So I guess I’m back to square 0.

August 20, 2010 at 2:32 pm

I’ll sort this out by the time we get there (if not sooner). I’ll have to think through that exercise sequence, and it’s not clear from the wording what I was asking! It sounds like (from your offline comments) only something much less is true, and gets used.

May 3, 2012 at 9:42 am

Update May 3, 2012: the problem Sam is referring to is now 30.3.I in the

May 2, 2012 versionof the notes. I notice now that it has been updated to show that the sheaf-Hom in question is flasque and hence acyclic for the global section functor . I’ve not yet checked whether this sorts out Sam’s question, but I’ll find out in the next weeks, when I reach the proof of Serre duality in this year’s incarnation of the class.May 18, 2012 at 11:01 am

Almost 2 years later: I’ve mentioned this to Sam in person, but wanted to post it here: in the current version of the notes, this is now fixed — this exercise is now removed, as it’s not used at all. (If you’re going to read the exposition on Serre duality, you may want to wait a couple of weeks — I’m teaching it now, so it will be revised in time for the next posted version of the notes.)