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

中文翻译：

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一维格伦斯坦环的倾斜理论

在表示论、交换代数和代数几何中，如何理解三角范畴$\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)=\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z} }R$允许倾斜（分别是淤积）物体$\mathbb{Z}$梯度交换 Gorenstein 环$R=\bigoplus_{i\geqslant 0}R_{i}$. 这里$\mathsf{D}_{\运算符名{sg}}^{\mathbb{Z}}(R)$是奇点类别，并且$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$是稳定的范畴$\mathbb{Z}$分级科恩-麦考莱 (CM)$R$-modules，在所有非极大素理想下都是局部自由的$R$.在本文中，我们在以下情况下给出了这个问题的完整答案$\暗 R=1$和$R_{0}$是一个字段。我们证明$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$总是承认有淤泥的物体，而且$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$当且仅当$R$是常规的或$a$-不变量$R$是非负的。我们的淤泥/倾斜对象将被明确给出。我们还表明，如果$R$是减少的和非常规的，那么它的$a$-invariant 是非负的，并且上面的倾斜对象在$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R=\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}R $.