Noncommutative ampleness from finite endomorphisms

重因子的非交换满度

Noncommutative ampleness for multiple divisors

扭曲齐次坐标环是Artin，Van den Bergh等人的非交换射影几何的基本构造之一。Chan将这种构造推广到多齐次情形，使用了可逆层的有限集合的右满度概念和射影概形的自同构。由此他得出某些多齐次环，例如扭曲齐次坐标环的张量积，是右诺特环。我们表明，右和左的振幅是等价的，有一个简单的标准，这样的振幅。因此，我们发现，在自然的假设下，多齐次坐标环是诺特的，并有整数GK-维。

σ-充裕度的判据

Criteria for σ-ampleness

在Artin，Van den Bergh等人的非交换几何中，扭曲齐次坐标环是基本构造之一。这样的环由$σ$ -样本因子定义，其中$σ$是射影概形X的自同构。关于$σ$ -充足因子的许多未决问题仍然存在。我们得到了X上的因子是$σ$ -样本的一个相对简单充要条件。因此，我们证明了右和左$σ$ -振幅是等价的，任何相关的非交换齐次坐标环必须是Noether的，并有有限的，整数GK-维。我们还刻画了哪些自同构$σ$产生一个$σ$ -样本因子。

正则局部环的纯子环、自同态环与Frobenius态射

Pure subrings of regular local rings, endomorphism rings and Frobenius morphisms

本文研究正则局部环纯子环的自同态环，证明其在极大Cohen-Macaulay条件下为非交换crepant消解，并利用Frobenius态射构造Morita等价的另一消解，同时探讨非交换Frobenius态射与整体维数有限性的关系，并应用于wild商奇点的强F-正则性判定。

Noncommutative (crepant) desingularizations and the global spectrum of commutative rings

In this paper we study endomorphism rings of finite global dimension over not necessarily normal commutative rings. These objects have recently attracted attention as noncommutative (crepant) resolutions, or NC(C)Rs, of singularities. We propose a notion of a NCCR over any commutative ring that appears weaker but subsumes all previous notions. Our results yield strong necessary and sufficient conditions for the existence of such objects in many cases of interest. We also give new examples of NCRs of curve singularities, regular local rings and normal crossing singularities. Moreover, we introd…

Cluster tilting modules and noncommutative projective schemes

In this paper, we study the relationship between equivalences of noncommutative projective schemes and cluster tilting modules. In particular, we prove the following result. Let $A$ be an AS-Gorenstein algebra of dimension $d\geq 2$ and ${\mathsf{tails}\,} A$ the noncommutative projective scheme associated to $A$. If $\operatorname{gldim}({\mathsf{tails}\,} A)< \infty$ and $A$ has a $(d-1)$-cluster tilting module $X$ satisfying that its graded endomorphism algebra is $\mathbb N$-graded, then the graded endomorphism algebra $B$ of a basic $(d-1)$-cluster tilting submodule of $X$ is a two-sided …

Natural endomorphisms of quasi-shuffle Hopf algebras

The Hopf algebra of word-quasi-symmetric functions ($\WQSym$), a noncommutative generalization of the Hopf algebra of quasi-symmetric functions, can be endowed with an internal product that has several compatibility properties with the other operations on $\WQSym$. This extends constructions familiar and central in the theory of free Lie algebras, noncommutative symmetric functions and their various applications fields, and allows to interpret $\WQSym$ as a convolution algebra of linear endomorphisms of quasi-shuffle algebras. We then use this interpretation to study the fine structure of quas…