Applications of Frobenius map to Commutative Ring Theory and Algebraic Geometry

Frobenius映射在交换环理论和代数几何中的应用

• 批准号: 07454010
• 负责人: WATANABE Keiichi
• 金额: $ 3.2万
• 依托单位: NIHON UNIVERSITY (1997)Tokai University (1995-1996)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1995
• 资助国家: 日本
• 起止时间: 1995 至 1997
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07454010/
• 关键词: Frobenius map; Algebraic Geometry; Singularities; rational singularity; F-rational ring; Terminal Singularity; regular ring; Hilbert-Kunz multiplicity; rational singularity; F-rational ring; Frobenius写像; regular local ring; Hilbert-Kung multip

1 Characterization of Singularities in Characteristic 0 via Frobenius endomorphism. We found that log-terminal singularity and F-regular rings are equivalent notions in the case the ring is Q-Gorenstein. The same is true for rational singularities and F-rational rings.2 By definig F-terminal rings, the terminal singularities are characterized in 3-dimensional case. F-terminal and Q-Gorenstein imply terminal singularity in any dimension. But in dimension 4, unfortunately, there is a counterexample to the converse and F-rational ring is characterized by the property that its general hyperplane section is F-rational in Gorenstein case.3 The characterization of regular local rings by Hilbert-Kunz multiplicity=1.Namely, an unmixed local ring of characteristic p is regular if and only if its H-K multiplicity is 1.Also we succeeded to classify the 2-dimensional rings with Hibert-Kunz multiplicity less than 9/4.

1通过Frobenius自同态刻画特征0中的奇点。我们发现当环是Q-Gorenstein环时，对数-终端奇点与F-正则环是等价的。有理奇点和F-有理环也是如此。2通过定义F-终结环，刻画了三维情形下的终结奇点。F-终端和Q-Gorenstein意味着在任何维度上的终端奇异性。但在4维中，不幸的是，有一个反例的匡威和F-有理环的特点是它的一般超平面截面是F-有理的Gorenstein情况下的性质。3正则局部环的Hilbert-Kunz重数= 1的特征。即，特征为p的非混合局部环是正则的当且仅当它的H-K重数为1。Hibert-Kunz重数小于9/4的一维环。

项目成果

桔梗宏孝: "応用論理" 共立出版, (1996)

Hirotaka Kikyo：“应用逻辑”Kyoritsu Shuppan，（1996）

Makoto MORI: "Fredholm Matrix and Zeta Functions for 1-dimensional Mappings" Proceedings of Algorithms, Fractals and Dynamics in Kyoto. 161-168 (1995)

Makoto MORI：“一维映射的 Fredholm 矩阵和 Zeta 函数”京都算法、分形和动力学论文集。

Masahiko SUZUKI: "The blow analytically constant stratum of real analytic singularities" "Real analytic and algebraic singularities" (T.Fukuda, T.Fukui, S.Izumiya and S.Koike (eds.). 64-76 (1997)

Masahiko SUZUKI：“实分析奇点的分析恒定层”“实分析和代数奇点”（T.Fukuda、T.Fukui、S.Izumiya 和 S.Koike（编辑）。64-76 (1997)

N. Hara and K. Watanabe: "The injecturity of Frobenius acting on cohomology and local cohomology modules" Manuscripta Math.90. 301-315 (1996)

N. Hara 和 K. Watanabe：“Frobenius 作用于上同调和局部上同调模的注入性”Manuscripta Math.90。

渡辺敬一: "“悪い"特異点のFrobenius写像による特徴付け" 第18回可換環論シンポジウム報告集. 122-126 (1996)

Keiichi Watanabe：“通过 Frobenius 映射表征‘坏’奇点”第 18 届交换代数理论研讨会报告 122-126 (1996)。