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

在前人研究的基础上，我们研究了希尔伯特-昆兹重数作为正特征奇点的不变量。另一方面，近两年来，作为爆破代数的环论性质之一，我们主要研究了Rees代数的F-有理性，其中最重要的结果是给出了Rees代数关于Cohen-Macaulay局部环的m-素数理想的F-有理性的一个判据。F-理性的概念由Fedder和Watanabe定义为特征为零的有理奇点的类比(正特征)。但他们之间肯定有不同的方面。例如，布托特定理是重要的定理之一，它断言有理奇点的任何直和也是有理奇点，因为这个定理保证了线性约化群的不变子环的Cohen-Macaulay性质。然而，对于F-理性，类似的结果并不普遍成立。实际上，作为我们结果的一个应用，我们可以为这种情况提供许多反例。我们的研究的另一个贡献是找到了紧闭包的一个推广，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，证明了广义测试理想是乘子理想的类比(正特征)。此外，我们还证明了理想的Rees代数在二维有理偶点上的F-有理性给出了理想的乘子理想与广义检验理想关于理想重合的充分条件.我们在交换环论研讨会上给出了上述结果.

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.