发表机构
School of Mathematics, Shanghai University of Finance and Economics; School of Mathematical Sciences, Fudan University（上海财经大学数学学院; 复旦大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究通常分次 AS-正则代数作为 AS-Gorenstein 孤立奇点的非交换消解，证明其存在性等价于平衡 CM 孤立奇点上 cluster tilting 模的存在性，并在维数 2 和 3 中建立了 Bondal-Orlov 猜想的非交换类比，同时给出三个实例。
AI 中文摘要
Li--Shen--Wu 研究了 AS-Gorenstein 孤立奇点的非交换消解。然而，即使这些消解存在，建立其存在性并构造此类消解通常也是困难的。本文研究了通常分次的 AS-正则代数作为 AS-Gorenstein 孤立奇点的非交换消解的条件。我们研究了诺特通常分次 AS-正则代数上的投射模，其自同态环可由底层的正则代数给出消解。这引出了平衡 Cohen--Macaulay 孤立奇点的非交换消解的更一般定义。我们证明了此类消解的存在性等价于平衡 CM 孤立奇点上 cluster tilting 模的存在性。相应的 Bondal-Orlov 猜想的非交换类比在维数 $2$ 和 $3$ 中得以建立。作为应用，我们研究了通常分次 AS-Gorenstein 代数上的 Hopf 作用，并探讨了不变环的非交换消解。我们给出了三个非交换消解的实例，其中一个是非连通分次的非交换孤立奇点。
英文摘要
Noncommutative resolutions of AS-Gorenstein isolated singularities are investigated by Li--Shen--Wu. However, establishing their existence and constructing such resolutions are generally difficult, even when they exist. In this paper, we study conditions under which a commonly graded AS-regular algebra serves as a noncommutative resolution of an AS-Gorenstein isolated singularity. We investigate projective modules over a noetherian commonly graded AS-regular algebra whose endomorphism rings admit resolutions by the underlying regular algebra. This leads to a more general definition of noncommutative resolutions of balanced Cohen--Macaulay isolated singularities. We show that the existence of such resolutions is equivalent to the existence of cluster tilting modules over balanced CM isolated singularities. The corresponding noncommutative analogue of the Bondal-Orlov conjecture is established in dimensions $2$ and $3$. As an application, we study Hopf actions on commonly graded AS-Gorenstein algebras and investigate noncommutative resolutions of invariant rings. We present three examples of noncommutative resolutions, including one in which the noncommutative isolated singularity is not connected graded.
Comments47 pages
