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.

延续了前人的研究，我们将Hilbert-Kunz重数作为正特征中奇点的不变量进行了研究。另一方面，近两年来，我们主要研究了作为爆炸代数环理论性质之一的里斯代数的F-有理性。我们研究中最重要的成果是给出了F-有理性的判据。 Rees 代数关于 Cohen-Macaulay 局部环中 m 初等理想的合理性。 Fedder 和 Watanabe 将 F 理性的概念定义为特征零中理性奇点的类似物（正特征）。但它们之间确实存在不同的方面。例如，布托定理断言有理奇点的任何直接被加也是有理奇点，它是重要的定理之一，因为该定理确保了线性还原群的不变子环的科恩-麦考利性质。然而，对于F-理性来说，类似的结果一般并不成立。实际上，作为我们结果的应用，我们可以为此提供许多反例。我们研究的另一个贡献是找到了紧闭包的推广，并推广了紧闭包理论中的测试理想的概念。事实上，我们与东北大学的 Hara Nobuo 合作证明了广义检验理想是乘数理想的类似物（具有正特性）。此外，我们还证明了二维有理双点中理想的里斯代数的F-有理性给出了理想的乘数理想与相对于理想的广义检验理想重合的充分条件。我们给出了演示我们在交换环理论研讨会上的结果。