Annihilators and decompositions of singularity categories

Pub Date : 2024-04-23 DOI:10.1017/s001309152400018x

Özgür Esentepe, Ryo Takahashi

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Abstract

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.

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奇点范畴的湮没器和分解

给定任何交换诺特环 R 和 R 中的元素 x，我们考虑其奇异性范畴的全子范畴 $\mathsf{C}(x)$ ，这个子范畴由 x 乘以的态量为零的对象组成。我们的主要观察结果是，对于任意两个环元素 x 和 y，我们可以在 $\mathsf{C}(x), \mathsf{C}(y)$ 和 $\mathsf{C}(xy)$ 之间建立一种关系。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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