  We use Cox's description for sheaves on toric varieties and results about the
local cohomology with respect to monomial ideals to give a characteristic free
approach to vanishing results on arbitrary toric varieties. As an application,
we give a proof of a strong form of Fujita's conjecture in the case of toric
varieties. We also prove that every sheaf on a toric variety corresponds to a
module over the homogeneous coordinate ring, generalizing Cox's result for the
simplicial case.
