On ring-theoretical invariants of singular points in positive characteristic

正特征奇点的环理论不变量

基本信息

批准号：
11640021
负责人：
YOSHIDA Kenichi
金额：
$ 2.24万
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：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

We have studied Hilbert-Kunz multiplicity as an invariant of singular points in positive char acteristic for three years. The most important result in our work is to give a characterization of regular local rings in terms of Hilbert-Kunz multiplicity. Actually, many researchers tried to gener alize our theorem. After this research, we have studied Hilbert-Kunz multiplicity of ideals defined by the dual graph of the resolution of singularities. Note that Hilbert-Kunz multiplicity for such an ideal is a ring-theoretical invariant associated to isolated singularity in positive characteristic. As one of our results, for integrally closed ideals in a rational double point, we obtained algorithm for calculating their Hilbert-Kunz multiplicities in terms of the dual graph. On the other hand, we have tried calculation of Hilbert-Kunz multiplicity for blow-up rings, but we could not get complete algorithm. As a partial result, we get some inequalities with respect to blow-up rings and the basering.Also, we introduced the notion of the minimal Hilbert-Kunz multiplicity and gave several method for calculation. This invariant can be described as the difference of the Hilbert-Kunz multiplicities of some pairs of ideals. Furthermore, we found that this invariant is equal to the invariant which is defined by other researchers. We gave a presentation of our results as above at Symposium on Commutative algebra and at Symposium on Algebra on Summer in 2001. Also, we have a project to study blow-up rings in positive characteristic.

我们研究希尔伯特-昆茨重数作为正特征奇点的不变量已经三年了。我们工作中最重要的结果是根据希尔伯特-昆茨重数给出了规则局部环的特征。事实上，许多研究人员试图推广我们的定理。经过这项研究，我们研究了由奇点分辨率的对偶图定义的希尔伯特-昆茨理想重数。请注意，这种理想的希尔伯特-昆茨多重性是与正特性中的孤立奇点相关的环理论不变量。作为我们的结果之一，对于有理双点中的积分封闭理想，我们获得了根据对偶图计算其 Hilbert-Kunz 重数的算法。另一方面，我们也尝试过计算爆炸环的Hilbert-Kunz重数，但未能得到完整的算法。作为部分结果，我们得到了关于爆炸环和基环的一些不等式。此外，我们引入了最小Hilbert-Kunz重数的概念，并给出了几种计算方法。这个不变量可以描述为一些理想对的希尔伯特-昆茨重数之差。此外，我们发现这个不变量等于其他研究人员定义的不变量。我们在交换代数研讨会和2001年夏季代数研讨会上介绍了我们的结果。此外，我们还有一个研究正特性爆炸环的项目。