  We combine work of Cox on the total coordinate ring of a toric variety and
results of Eisenbud-Mustata-Stillman and Mustata on cohomology of toric and
monomial ideals to obtain a formula for computing the Euler characteristic of a
Weil divisor D on a complete simplicial toric variety in terms of graded pieces
of the Cox ring and Stanley-Reisner ring. The main point is to use Alexander
duality to pass from the toric irrelevant ideal, which appears in the
computation of the Euler characteristic of D, to the Stanley-Reisner ideal of
the fan, which is used in defining the Chow ring. The formula also follows from
work of Maclagan-Smith.
