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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.

在前人研究的基础上，我们研究了Hilbert-Kunz重数作为奇点正特征不变量的性质。另一方面，近两年来，我们主要研究了作为Blow-up代数环论性质之一的Rees代数的F-合理性，其中最重要的结果是给出了Cohen-Macaulay局部环中Rees代数关于m-准素理想的F-合理性的一个判别准则。Fedder和Watanabe定义了F-合理性的概念，作为特征零中的合理奇点的类似物（在正特征中）。但它们之间肯定有不同的方面。例如，Boutot定理，它断言一个有理奇点的任何直和项也是一个有理奇点，是一个重要的定理，因为这个定理确保了线性约化群的不变子环的Cohen-Macaulay性质。然而，对于F-合理性，类似的结果一般不成立。实际上，作为我们结果的应用，我们可以提供许多反例。我们研究的另一个贡献是找到了紧闭包的一个推广，推广了紧闭包理论中的测试理想的概念。事实上，我们证明了广义测试理想是一个类似物（在积极的特征）的乘数理想在东北大学原伸夫合作。此外，我们还证明了二维有理二重点上理想的Rees代数的F-有理性给出了理想的乘子理想与广义测试理想关于理想重合的充分条件，并在交换环理论研讨会上作了介绍.