sheaf co-comology of toric variety, representatives?

[Friedemann Groh]

Hi,

I like the HH^i(X,L)-command very much. (X a toric variety and L a sheaf)

Are there additional M2-commands to decompose this co-homologies with respect to the characters m, as in §9.1. in the book "Toric Varieties" by Cox, Little,Schenck? Obtaining representatives of these classes in the Cech complex would also be very nice.

Experimentally, I have determined such representatives in Matlab, but in the future I would prefer to use M2.

Many thanks for any advice,

Friedemann

[Greg]

Dear Friedemann,

Currently, there are no methods for decomposing the cohomology groups with respect to characters within the NormalToricVarieties package.  However, it wouldn't be very difficult to implement this option (it is essentially already done behind the scenes).  

What format do you think would be most useful?  For example, one could (optionally) output a vector space graded by the character group (i.e. an M-graded QQ-module).  Alternatively, one could have a (co-)homology method that also took a character as input and returned just the corresponding isotypical component.

All the best,

Greg.

[Friedemann Groh]

Dear Greg,

Thank you very much for your reply. For me, the first option, as a m-graded QQ vector space, would be very helpful: I would like to loop through all non-trivial m-components and then get the degree m and a basis of the corresponding QQ-vector space in each pass. Would you return QQ-vectors with coordinates q_[i0...i1] of co-chains in the Cech complex padded with zeros if module F(U0 \cap ... \cap Un) vanish at the character m?

Please let me know, how I can be of any help to get this feature?

best regards

Friedemann