##
**The canonical module of a Cox ring.**
*(English)*
Zbl 1246.14012

In 1995 Cox studied the total coordinate ring (called Cox ring) of toric varieties.

Cox rings of normal projective varieties have become such important and interesting objects and many mathematicians try to study their ring-theoretic properties as generators, relations, syzygies, homological properties.

In this paper the authors describe the graded canonical module of a Noetherian multisection ring of a normal projective variety and give a contribution to the study of syzygies of the Cox ring.

If \(X\) is a \(d-\)dimensional normal projective variety over a field \(k\) and \(D_1, \dots, D_s\) are Weil divisors on \(X\), then the ring \(R(X;D_1, \dots, D_s)\) is defined as \[ \bigoplus_{(n_1, \dots, n_s) \in \mathbb{Z}^s} H^0\left(X, {\mathcal{O}}_X(\sum_i n_iD_i) \right)t_1^{n_1}\cdots t_s^{n_s}\subset k(X)[t_1^{\pm 1}, \dots, t_s^{\pm 1}]. \] In the case where Cl\((X)\) is freely generated by \(\overline{D_1}, \dots, \overline{D_S}\), the ring \(R(X;D_1, \dots, D_s)\) is called the Cox ring of \(X\).

For a Weil divisor \(F\) we set \[ M_F=\bigoplus_{(n_1, \dots, n_s) \in \mathbb{Z}^s} H^0\left(X, {\mathcal{O}}_X(\sum_i n_iD_i+F) \right)t_1^{n_1}\cdots t_s^{n_s}\subset k(X)[t_1^{\pm 1}, \dots, t_s^{\pm 1}]. \]

The main result in the paper is the following theorem.

Theorem 1.2. Let \(X\) be a normal projective variety over a field \(k\) such that \(H^0(X,{\mathcal{O}}_X)=k\). Assume that \(D_1, \dots, D_s\) are Weil divisors on \(X\) which satisfy the following three conditions. (1) \(\overline{D_1}, \dots, \overline{D_S}\) are linearly independent over \(\mathbb{Z}\) in the divisor class group Cl\((X)\).

(2) The lattice \(\mathbb{Z}D_1+\cdots +\mathbb{Z}D_s\) contains an ample Cartier divisor.

(3) The ring \(R(X;D_1, \dots, D_s)\) is Noetherian. Then, \(R(X;D_1, \dots, D_s)\) is a local \(\mathbb{Z}^s-\)graded \(k-\)domain and we have an isomorphism \[ \omega_{R(X;D_1, \dots, D_s)} \simeq M_{K_X} \] of \(\mathbb{Z}^s-\)graded \(R(X;D_1, \dots, D_s)-\)modules. The authors give two kinds of proofs of the previous theorem. The first one is based on the equivariant twisted inverse functor developed by M. Hashimoto. The second proof avoids the twisted inverse functor, but some additional assumptions are required.

Using Theorem 1.2 the authors give a necessary and sufficient condition for the canonical module to be a free module.

Corollary 1.3. Suppose that the assumptions in Theorem 1.2 are satisfied. Then, \(\omega_{R(X;D_1, \dots, D_s)}\) is a free \(R(X;D_1, \dots, D_s)-\)module if and only if \[ \overline{K_X}\in \mathbb{Z}\overline{D_1}+\cdots +\mathbb{Z}\overline{D_s} \] in Cl\((X)\). Since the Cox ring is a unique factorization domain, its graded canonical module is a free module. The authors prove that the graded canonical module of the Cox ring of a normal projective variety \(X\) is a graded free module of rank one with the shift of degree \(K_X\).

Corollary 1.5. Let \(X\) be a normal projective variety over a field. Assume that Cl\((X)\) is a finitely generated free abelian group and the Cox ring of \(X\) is Noetherian. Then, the canonical module of the Cox ring is a rank one free Cl\((X)-\)graded module whose generator is of degree \(-\overline{K_X}\in \)Cl\((X)\).

Cox rings of normal projective varieties have become such important and interesting objects and many mathematicians try to study their ring-theoretic properties as generators, relations, syzygies, homological properties.

In this paper the authors describe the graded canonical module of a Noetherian multisection ring of a normal projective variety and give a contribution to the study of syzygies of the Cox ring.

If \(X\) is a \(d-\)dimensional normal projective variety over a field \(k\) and \(D_1, \dots, D_s\) are Weil divisors on \(X\), then the ring \(R(X;D_1, \dots, D_s)\) is defined as \[ \bigoplus_{(n_1, \dots, n_s) \in \mathbb{Z}^s} H^0\left(X, {\mathcal{O}}_X(\sum_i n_iD_i) \right)t_1^{n_1}\cdots t_s^{n_s}\subset k(X)[t_1^{\pm 1}, \dots, t_s^{\pm 1}]. \] In the case where Cl\((X)\) is freely generated by \(\overline{D_1}, \dots, \overline{D_S}\), the ring \(R(X;D_1, \dots, D_s)\) is called the Cox ring of \(X\).

For a Weil divisor \(F\) we set \[ M_F=\bigoplus_{(n_1, \dots, n_s) \in \mathbb{Z}^s} H^0\left(X, {\mathcal{O}}_X(\sum_i n_iD_i+F) \right)t_1^{n_1}\cdots t_s^{n_s}\subset k(X)[t_1^{\pm 1}, \dots, t_s^{\pm 1}]. \]

The main result in the paper is the following theorem.

Theorem 1.2. Let \(X\) be a normal projective variety over a field \(k\) such that \(H^0(X,{\mathcal{O}}_X)=k\). Assume that \(D_1, \dots, D_s\) are Weil divisors on \(X\) which satisfy the following three conditions. (1) \(\overline{D_1}, \dots, \overline{D_S}\) are linearly independent over \(\mathbb{Z}\) in the divisor class group Cl\((X)\).

(2) The lattice \(\mathbb{Z}D_1+\cdots +\mathbb{Z}D_s\) contains an ample Cartier divisor.

(3) The ring \(R(X;D_1, \dots, D_s)\) is Noetherian. Then, \(R(X;D_1, \dots, D_s)\) is a local \(\mathbb{Z}^s-\)graded \(k-\)domain and we have an isomorphism \[ \omega_{R(X;D_1, \dots, D_s)} \simeq M_{K_X} \] of \(\mathbb{Z}^s-\)graded \(R(X;D_1, \dots, D_s)-\)modules. The authors give two kinds of proofs of the previous theorem. The first one is based on the equivariant twisted inverse functor developed by M. Hashimoto. The second proof avoids the twisted inverse functor, but some additional assumptions are required.

Using Theorem 1.2 the authors give a necessary and sufficient condition for the canonical module to be a free module.

Corollary 1.3. Suppose that the assumptions in Theorem 1.2 are satisfied. Then, \(\omega_{R(X;D_1, \dots, D_s)}\) is a free \(R(X;D_1, \dots, D_s)-\)module if and only if \[ \overline{K_X}\in \mathbb{Z}\overline{D_1}+\cdots +\mathbb{Z}\overline{D_s} \] in Cl\((X)\). Since the Cox ring is a unique factorization domain, its graded canonical module is a free module. The authors prove that the graded canonical module of the Cox ring of a normal projective variety \(X\) is a graded free module of rank one with the shift of degree \(K_X\).

Corollary 1.5. Let \(X\) be a normal projective variety over a field. Assume that Cl\((X)\) is a finitely generated free abelian group and the Cox ring of \(X\) is Noetherian. Then, the canonical module of the Cox ring is a rank one free Cl\((X)-\)graded module whose generator is of degree \(-\overline{K_X}\in \)Cl\((X)\).

Reviewer: Cristiano Bocci (Grosseto)

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\textit{M. Hashimoto} and \textit{K. Kurano}, Kyoto J. Math. 51, No. 4, 855--874 (2011; Zbl 1246.14012)

### References:

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[7] | M. Hashimoto and M. Ohtani, “Local cohomology on diagrams of schemes” in Special Volume in Honor of Melvin Hochster , Michigan Math. J. 57 , Univ. Michigan Math. Dept., Ann Arbor, 2008, 383-425. · Zbl 1197.14002 · doi:10.1307/mmj/1220879415 |

[8] | M. Hashimoto and M. Ohtani, Equivariant Matlis and the local duality , J. Algebra 324 (2010), 1447-1470. · Zbl 1213.13036 · doi:10.1016/j.jalgebra.2010.05.017 |

[9] | Y. Kamoi, Noetherian rings graded by an abelian group , Tokyo J. Math. 18 (1995), 31-48. · Zbl 0834.13009 · doi:10.3836/tjm/1270043606 |

[10] | O. N. Popov, The Cox ring of a Del Pezzo surface has rational singularities , preprint, |

[11] | K.-i. Watanabe, Some remarks concerning Demazure’s construction of normal graded rings , Nagoya Math. J. 83 (1981), 203-211. · Zbl 0518.13003 |

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