  In this paper, we shall describe the graded canonical module of a Noetherian
multi-section ring of a normal projective variety. In particular, in the case
of the Cox ring, we prove that the graded canonical module is a graded free
module of rank one with the shift of degree $K_X$. We shall give two kinds of
proofs. The first one utilizes the equivariant twisted inverse functor
developed by the first author. The second proof is down-to-earth, that avoids
the twisted inverse functor.
