In this paper, we describe the graded canonical module of a Noetherian multisection 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 . We 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, which avoids the twisted inverse functor, but some additional assumptions are required in this proof.
"The canonical module of a Cox ring." Kyoto J. Math. 51 (4) 855 - 874, Winter 2011. https://doi.org/10.1215/21562261-1424884