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

继续前人的研究，我们研究了Hilbert-Kunz多重性作为正特征上奇异点的不变量。另一方面，在过去的两年中，我们主要研究了Rees代数的f -合理性作为爆破代数的环理论性质之一。我们的研究中最重要的结果是给出了Cohen-Macaulay局部环中关于m-原初理想的Rees代数的f -合理性判据。f -合理性的概念被Fedder和Watanabe定义为特征零点的理性奇点的类比（在正特征上）。但它们之间肯定有不同的方面。例如Boutot定理，它断言任何有理奇点的直接和也是一个有理奇点，是一个重要的定理，因为这个定理保证了线性约化群的不变子的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，作为我们的结果的应用，我们可以提供许多反例。本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法器理想的类似物（在正特性上）。进一步证明了二维有理双点上理想的Rees代数的f -合理性，给出了该理想的乘子理想与广义检验理想重合的充分条件。我们在交换环理论研讨会上发表了上述结果。

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.