Reduced \v{C}ech complexes and computing higher direct images under toric maps

This paper has three main goals : (1) To give an axiomatic formulation of the construction of "reduced \v{C}ech complexes", complexes using fewer than the usual number of intersections but still computing cohomology of an appropriate class of sheaves; (2) To give a construction of such a reduced \v{C}ech complex for every semi-proper toric variety $X$, where every open used in the complex is torus stable, and such that the cell complex governing the reduced \v{C}ech complex has dimension the cohomological dimension of $X$; and (3) to give an algorithm to compute the higher direct images of line bundles relative to a toric fibration between smooth proper toric varieties.