TILTING THEORY FOR GORENSTEIN RINGS IN DIMENSION ONE

In representation theory, commutative algebra and algebraic geometry, it is an important problem to understand when the triangulated category $\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)=\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ admits a tilting (respectively, silting) object for a $\mathbb{Z}$ -graded commutative Gorenstein ring $R=\bigoplus _{i\geqslant 0}R_{i}$ . Here $\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)$ is the singularity category, and $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ is the stable category of $\mathbb{Z}$ -graded Cohen–Macaulay (CM) $R$ -modules, which are locally free at all nonmaximal prime ideals of $R$ . In this paper, we give a complete answer to this problem in the case where $\dim R=1$ and $R_{0}$ is a field. We prove that $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ always admits a silting object, and that $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ admits a tilting object if and only if either $R$ is regular or the $a$ -invariant of $R$ is nonnegative. Our silting/tilting object will be given explicitly. We also show that if $R$ is reduced and nonregular, then its $a$ -invariant is nonnegative and the above tilting object gives a full strong exceptional collection in $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R=\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}R$ .