TILTING THEORY FOR GORENSTEIN RINGS IN DIMENSION ONE

Abstract

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$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R=\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}R$.