Singurality Theory and Frobenius Morphism

奇点理论和弗罗贝尼乌斯态射

• 批准号: 17540043
• 负责人: WATANABE Keiichi
• 金额: $ 2.47万
• 依托单位: Nihon University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540043/
• 关键词: Frobenius endomorphism; rational singularity; F-rational ring; tight closure; integral closure; totally reflexive module; F-threshold; multiplicity; F-rational ring; F-regular ring; 重複度; 乗数イデアル; totally reflexive module; 特異点; 標数p>0の手法; mult; Frobenius endomorphism; rational singularity; F-rational ring; tight closure; integral closure; totally reflexive module; F-threshold; multiplicity; F-rational ring; F-regular ring; 重複度; 乗数イデアル; totally reflexive module; 特異点; 標数p>0の手法; mult

Frobenius endomorphism hi characteristic p > 0 is a very powerful tool in commutative ring theory as well as singularity theory or algebraic geometry iover a field of characteristic 0 via reduction mod p.We applied the Frobenius endomorphism to various problems in commutative algebra and singularity theory. Inparticular, we showed the followings ;1. F-thresholds ; F-threshold is defined in a Noetherian ring of characteristic p>0 to a pair (I,J) of two ideals of A.This notion was originally introduced to describe multiplier ideals and jumping numbers in a regular local ring. But in our research, it turned out that this notion is closely related to tight closures and integral closures and also we have a nice conjecture concerning F-threshold and multiplicity of a parameter ideal.2. Multi-graded rings, rational singularity and F-rational rings ;The notion of multi-graded rings and their diagonal algebras is a very interesting object and very useful in making many interesting examples. In a joint paper with A Singh and E. Sato, K. Kurano and K. Watanabe made a new example with discrete divisor class group whose local cohomology modules shows very interesting feature. Also we showed a criterion for diagonal subalgebras of multi-graded hypersurfaces to be f-rational or F-regular in terms of the degree.3. Totally reflexive modules ;In the theory of totally reflexive modules, examples of non-trivial totally reflexive modules are very few. Watanabe and R. Takahashi constructed a family of non-trivial totally reflexive modules using geometry of curves of genus greater than 1. This is the first case that algebraic geometry is used in this theory.

Frobenius内态性HI特征p> 0是通勤环理论以及奇异理论或代数几何的非常强大的工具。通过还原mod p.我们将frobenius内态性应用于交换性代数和奇异理论的各种问题。内部，我们显示了以下; 1。 F阈值； F-阈值在一个特征性p> 0的noetherian环上定义为两个理想的对（i，j）。该概念最初是为了描述常规本地环中的乘数理想和跳跃数字。但是在我们的研究中，事实证明，这个概念与紧密的封闭和整体封闭密切相关，并且我们对参数理想的f-阈值和多重性有一个很好的猜想。2。多级别的环，合理的概念和F理性的环；多级环及其对角线代数的概念是一个非常有趣的对象，对于制作许多有趣的例子非常有用。在与Singh和E. Sato的联合论文中，K。Kurano和K. Watanabe与离散的Divisor class组做了一个新的示例，其本地共同体学模块显示出非常有趣的功能。同样，我们还展示了多层超曲面的对角线亚代galgebras的标准，其程度是F理性的或f-groumar的标准。3。完全反身模块；在完全反身模块的理论中，非平凡的完全反身模块的示例很少。渡边和高桥河（R.