We study residues on a complete toric variety X, which are defined in
terms of the homogeneous coordinate ring of X. We first prove a global
transformation law for toric residues. When the fan of the toric vari
ety has a simplicial cone of maximal dimension, we can produce an elem
ent with toric residue equal to 1. We also show that in certain situat
ions, the toric residue is an isomorphism on an appropriate graded pie
ce of the quotient ring. When X is simplicial, we prove that the toric
residue is a sum of local residues. In the case of equal degrees, we
also show how to represent X as a quotient (Y\{0})/C such that the to
ric residue becomes the local residue at 0 in Y.