Vanishing theorems on toric varieties

we use Cox's description for sheaves on toric varieties and results about local cohomology with respect to monomial ideals to give a characteristic-free approach to vanishing results on toric varieties, As an application, we give a proof of a strong version 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.