On ring-theoretical properties of blow-up rings over singular points in positive characteristic

正特性奇点上爆炸环的环理论性质

基本信息

批准号：
14540020
负责人：
YOSHIDA Ken-ichi
金额：
$ 1.73万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540020/
关键词：
F-regular F-rational tight closure Frobenius map blow-up Rees algebra Hilbert-Kunz multiplicity rational singularity Rees環重複度 Ress環

项目摘要

It continued for the previous research, and we have studied Hilbert-Kunz multiplicity as an invariant of singular points in positive characteristic. On the other hand, for last two years, we have studied mainly the F-rationality of Rees algebras as one of ring-theoretical properties of blow-up algebras.The most important result in our research is to give a criterion for the F-rationality of Rees algebras with respect to m-primary ideals in Cohen-Macaulay local rings. The notion of F-rationality was defined by Fedder and Watanabe as an analogue (in positive characteristic) of that of rational singularity in characteristic zero. But there are certainly different aspects between them. For instance, Boutot's theorem, which asserts that any direct summand of a rational singularity is also a rational singularity, is one of important theorems, because this theorem ensures the Cohen-Macaulay property of invariant subrings of linearly reductive groups. However, as for F-rationality, the similar result does not hold in general. Actually, as an application of our result, we can provide many counterexamples for such this.Another contribution of our research is to find a generalization of tight closure, and to generalize the notion of test ideal in the theory of tight closures. In fact, we showed that the generalized test ideal is an analogue (in positive characteristic) of a multiplier ideal in collaboration with Hara Nobuo at Tohoku University. Furthermore, we showed that the F-rationality of Rees algebra of an ideal in a rational double point in dimension two gives a sufficient condition for the multiplier ideal of the ideal and the generalized test ideal with respect to the ideal coincides.We gave a presentation of our results as above at Symposium on Commutative ring theory.

它继续进行了先前的研究，我们研究了希尔伯特·昆兹（Hilbert-Kunz）的多重性，是积极特征中奇异点的不变性。另一方面，在过去的两年中，我们主要研究了REES代数的F理性是爆炸代数的环理论特性之一。我们的研究最重要的结果是为REES代数的F理性提供标准，以相对于Cohen-Macaulay本地RING的M-Primary Ideals的F理性。 Fedder和Watanabe将F理性的概念定义为特征零中理性概念的类似物（以积极的特征）。但是它们之间肯定有不同的方面。例如，Boutot的定理断言，任何有理概念性的直接求和也是一个理性的概念，这是重要的定理之一，因为该定理可确保线性还原群体不变子环的Cohen-Macaulay属性。但是，至于f理性，相似的结果一般不产生。实际上，作为我们结果的应用，我们可以为此提供许多反示例。我们的研究的另一个贡献是找到紧密封闭的概括，并在紧密封闭理论中概括了测试理想的概念。实际上，我们表明，在与Tohoku University的Hara Nobuo合作中，广义测试理想是乘数理想的类似物（以积极特征）。此外，我们表明，在二维二维中，理想双点的REES代数的F理性为理想的乘数和普遍测试的乘数提供了足够的条件，相对于理想的重合，我们给出了我们的结果的介绍，如通勤环理论上的上述成果。