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!



Spectral sequence for composition of global sections and tensor product of sheaves
[Grothendieck spectral sequences](http://en.wikipedia.org/wiki/Grothendieck_spectral_sequence) 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.


