Pre-resolutions of noncommutative isolated singularities

Noncommutative cyclic isolated singularities

The question of whether a noncommutative graded quotient singularity $A^G$ is isolated depends on a subtle invariant of the $G$-action on $A$, called the pertinency. We prove a partial dichotomy theorem for isolatedness, which applies to a family of noncommutative quotient singularities arising from a graded cyclic action on the $(-1)$-skew polynomial ring. Our results generalize and extend some results of Bao, He and the third-named author and results of Gaddis, Kirkman, Moore and Won.

Noncommutative Knörrer periodicity and noncommutative Kleinian singularities

We establish a version of Knörrer's Periodicity Theorem in the context of noncommutative invariant theory. Namely, let $A$ be a left noetherian AS-regular algebra, let $f$ be a normal and regular element of $A$ of positive degree, and take $B=A/(f)$. Then there exists a bijection between the set of isomorphism classes of indecomposable non-free maximal Cohen-Macaulay modules over $B$ and those over (a noncommutative analog of) its second double branched cover $(B^\#)^\#$. Our results use and extend the study of twisted matrix factorizations, which was introduced by the first three authors with…

孤立奇点、光滑序与Auslander正则性

Isolated singularities, smooth orders and Auslander regularity

本文证明了满足弦理论中奇点条件的逆几何工程的光滑序（在任何紧化维数上）是Auslander正则的。此外，我们还对孤立中心奇异点上光滑阶的稳定局部结构进行了分类。

Some invariant subalgebras are graded isolated singularities

In this note, we prove that the invariant subalgebra of the (-1)-skew polynomial algebra under a permutation action is a graded isolated singularity, and thus a conjecture of Chan-Young-Zhang is true.

非交换商奇点上分次极大Cohen-Macaulay模的稳定范畴

Stable Categories of Graded Maximal Cohen-Macaulay Modules over Noncommutative Quotient Singularities

倾斜对象在三角范畴研究中起着关键作用。Iyama和Takahashi的一个著名结果是商奇点上分次极大Cohen-Macaulay模的稳定范畴有倾斜对象。本文利用非交换代数几何证明了Iyama和Takahashi定理的一个非交换推广。也就是说，如果$S$是Noether AS-正则Koszul代数，且$G$是作用在$S$上的有限群，使得$S^G$是Gorenstein孤立奇点，则分次极大Cohen-Macaulay模的稳定范畴${\underline {\operatorname {CM}}}^{\Bbb Z}(S^G)$有倾斜对象.特别地，范畴${\underline {\operatorname {CM}}}^{\Bbb Z}(S^G)$是三角形等价于有限维代数的导出范畴。

AS-正则代数上的充分群作用与非交换分次孤立奇点

Ample Group Action on AS-regular Algebras and Noncommutative Graded Isolated Singularities

本文引入右Noether分次代数$A$上的群作用$G$的满度概念，并证明了它与第二作者提出的分次孤立奇点$A^G$的概念有很强的联系.此外，如果$S$是Noether AS-正则代数，$G$是作用在$S$上的有限样本群，则证明了${\mathcal D}^b(\operatorname{tails} S^G)\cong {\cal D}^b(\operatorname{mod} \nabla S*G)$，其中$\nabla S$是$S$的Beilinson代数.我们还将显式地计算一个$Q_{S, G}$，使得当$S$为2维时，${\mathcal D}^b(\operatorname{tails} S^G)\cong {\mathcal D}^b(\operatorname{mod} kQ_{S, G})$。

The integral monodromy of isolated quasihomogeneous singularities

The integral monodromy on the Milnor lattice of an isolated quasihomogeneous singularity is subject of an almost untouched conjecture of Orlik from 1972. We prove this conjecture for all iterated Thom-Sebastiani sums of chain type singularities and cycle type singularities. The main part of the paper is purely algebraic. It provides tools for dealing with sums and tensor products of ${\mathbb Z}$-lattices with automorphisms of finite order and with cyclic generators. The calculations are involved. They use fine properties of unit roots, cyclotomic polynomials, their resultants and discriminant…

具有孤立奇点的完全交域上的Poisson结构

Poisson structures over a complete intersection with isolated singularities

研究了奇异簇上的Poisson结构。在这一目的中，我们考虑与完全交的方程相联系的Koszul复形。这个复形形成一个微分分次代数，它等价于簇的代数。我们证明了泊松结构等价于Koszul复形上的多重导子序列。如果我们的簇有孤立奇点，那么我们可以构造一个约化形式的多重导子序列。

非对易奇点解的几何

The Geometry of Noncommutative Singularity Resolutions

我们介绍了一个几何实现的非交换奇点决议。为此，我们首先提出了一种新的几何方法获得传统的决议使用坐标环的矩阵值函数。利用适当的代数证明了这个猜想对所有的循环商曲面奇点、Kleinian D_n和E_6曲面奇点、锥状奇点和非孤立奇点都成立.这个猜想提供了一个可能的新的推广的经典麦凯对应。然后，使用辛减少这些环内，我们得到新的，非传统的决议是隐藏的，如果只考虑交换函数。几何上，这些非传统的决议，从收缩例外轨迹分歧（非Azumaya）点状球体的结果。