I am trying to analyze Macaulay2 package "ToricVectorBundles". The package deals with equivariant reflexive sheaves on complete toric varieties. Such a sheaf is described by a set of filtrations of a fixed vector space $V$, each one corresponding to a ray of the fan (Klyachko description). The sheaf is locally free if and only if the filtrations satisfy a certain compatibility condition.
Suppose $\mathcal{F}$ is a reflexive sheaf on a toric variety $X$. If the sheaf is locally free, the package can compute its cohomologies $H^i(X,\mathcal{F})$. The computation is done as follows: 1) function deltaE computes the characters $\chi$ of the torus, for which the cohomology can be non-zero, 2) each group $H^i(X,\mathcal{F})_\chi$ is computed for $\chi$ in the range given by deltaE ,3) we take the direct sum of all $H^i(X, \mathcal{F})_\chi$ for $\chi$ in the range computed by deltaE.
However, if the sheaf is not locally free, the computation fails. The failure is in the function deltaE. It works only for locally free sheaves.
My goal is to find another way to estimate the locus $\{ \chi | H^i(X,\mathcal{F})_\chi \neq 0\}$. Can this be done for reflexive sheaves with Klyachko description?
