Spectral sequence for composition of global sections and tensor product of sheaves

Hi all,

on the forum page http://www.groupsrv.com/science/about506648.html one can read the following (i cut out nonimportant parts):

Question: Does anyone know any condition (non trivial) that ensure that the global sections of the tensor product of two sheaves is the tensor product of the global sections?

Answer: If X is not affine then, under sufficiently strong finiteness assumptions on the (co)homological dimensions of everything involved, you can approach this problem using a composite functor spectral sequence (which unfortunately will involve the left derived functors of tensor product and the right derived functors of global sections, so you will need some kind of finite-dimensionality assumption on Tor and H^* for quasicoherent sheaves over this scheme to construct the spectral sequence at all).

Can anybody tell me which is the mentioned spectral sequence?

Thanks!

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Grothendieck spectral sequences deal with the derived functor of a composition. If you're familiar with derived categories, then it takes a very simple form. It says that under the reasonable conditions that make sure that all derived functors in question exist, the (total) derived functor of the composition is the composition of the derived functors.

In the language of spectral sequences this will translate to a spectral sequence starting at $E_2$ with the $(p,q)$ term being the $p^\text{th}$ derived functor of the outside functor applied to the value of the $q^\text{th}$ derived functor of the inside functor and the statement is that this abuts to the $(p+q)^\text{th}$ derived functor of the composition.

The wikipedia link I included above is only for left derived functors, but it is not too hard to formulate it with right derived functors. In that case you end up with negative $p$s or $q$s, but formally it is similar.

There is a section on Grothendieck spectral sequences in Weibel's intro book to homological algebra. He is only dealing with right-right and left-left compositions, but you can do the crossover, too, but you need stronger finiteness conditions.

In the case you are asking, you can probably just write $H^*$ as the left derived functors of $H^n$ and then you have a left-left composition and you can use the formalism from Weibel's book.

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Thanks Sandor! I recently learned about the Grothendieck SS as in Weibel in my homological algebra course. I didnt imagine it would work for right-left compostions too. However, there is one thing very unclear to me, i hope you can help with this. Lets write $\otimes_{Sh}$ for tensor product of sheaves of modules on a variety and $\otimes_{Mod}$ for tensor product of modules. I am considering the functor $\Gamma \circ \otimes_{Sh}$ from sheaves of modules to vectorspaces and am interested in the value of $L^i(\Gamma) \circ \otimes_{Sh}$ on certain line bundles. – Joachim Feb 27 '13 at 1:12
(By the way, i guess we need to write $(-) \otimes_{Sh} \mathcal{L}$ for a fixed line bundle to make it into a functor instead of a bifunctor, excuse this sloppyness) My problem is the following two things: 1. The question was about relating values of $\Gamma \circ \otimes_{Sh}$ to those of $\otimes_{Mod} \circ \Gamma$, which is some sort of commutation rule. But this is not at all what the Grothendieck SS does, or am i wrong? – Joachim Feb 27 '13 at 1:16
2. If i imagine applying the Grothendieck SS to $\Gamma \circ \otimes_{Sh}$ we will need derived functors of $\otimes_{Sh}$ but as invertible sheaves on a variety does not in general have enough projectives, we run into a problem, right? – Joachim Feb 27 '13 at 1:18
Joachim, everything you write makes sense. To be honest, I wondered about the suggestion of using GSS (or any SS) for this question. The main issue is that in order to use derived functors for this you would need something to hold on the zero level or the composition level, but I don't see how that happens at all. – Sándor Kovács Feb 27 '13 at 1:43
Considering the actual question you are interested in, I would say that you have little chance to get anything meaningful. You obviously know what happens on an affine, but on for instance projective varieties this will rarely happen. Just think about the case of taking powers of a line bundle. It happens frequently that a line bundle has no global sections, but some power of it does. Similarly for any coherent sheaf: if you have an ample line bundle, then the tensor product with a sufficiently high power of the ample line bundle will make your coherent sheaf be globally generated. – Sándor Kovács Feb 27 '13 at 1:47