Given any commutative Noetherian ring R and an element x in R, we consider the full subcategory $\mathsf{C}(x)$ of its singularity category consisting of objects for which the morphism that is given by the multiplication by x is zero. Our main observation is that we can establish a relation between $\mathsf{C}(x), \mathsf{C}(y)$ and $\mathsf{C}(xy)$ for any two ring elements x and y. Utilizing this observation, we obtain a decomposition of the singularity category and consequently an upper bound on the dimension of the singularity category.

中文翻译：

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奇点范畴的歼灭子和分解

给定任意交换诺特环右和一个元素X在右，我们考虑完整的子类别 $\mathsf{C}(x)$ 其奇点范畴由对象组成，其态射由乘法给出X为零。我们的主要观察是我们可以在之间建立关系 $\mathsf{C}(x), \mathsf{C}(y)$ 和 $\mathsf{C}(xy)$ 对于任意两个环元素X和y。利用这一观察，我们获得了奇点类别的分解，从而获得了奇点类别维度的上限。