The canonical module of a Cox ring

TL;DR

This paper characterizes the graded canonical module of a Cox ring, showing it is a free module of rank one shifted by the canonical divisor, with two different proofs provided.

Contribution

It provides a detailed description of the graded canonical module of Cox rings, including two proofs, one using equivariant twisted inverse functor and one more elementary.

Findings

The graded canonical module of the Cox ring is a free module of rank one.

The shift of the canonical module is given by the canonical divisor $K_X$.

Two proofs are provided: one using equivariant twisted inverse functor, one avoiding it.

Abstract

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.