Project/Area Number  14540020 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Nagoya University 
Principal Investigator 
YOSHIDA Kenichi Nagoya University, Graduate School of Mathematics, Assistant, 大学院・多元数理科学研究科, 助手 (80240802)

CoInvestigator(Kenkyūbuntansha) 
WATANABE Keiichi Nihon University, Department of Mathematics College of Humanities and Sciences, Professor, 文理学部, 教授 (10087083)
HASHIMOTO Mitsuyasu Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (10208465)
OKADA Soichi Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (20224016)

Project Period (FY) 
2002 – 2003

Project Status 
Completed(Fiscal Year 2003)

Budget Amount *help 
¥2,700,000 (Direct Cost : ¥2,700,000)
Fiscal Year 2003 : ¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 2002 : ¥1,600,000 (Direct Cost : ¥1,600,000)

Keywords  Fregular / Frational / tight closure / Frobenius map / blowup / Rees algebra / HilbertKunz multiplicity / rational singularity / Rees環 / 重複度 / Ress環 
Research Abstract 
It continued for the previous research, and we have studied HilbertKunz multiplicity as an invariant of singular points in positive characteristic. On the other hand, for last two years, we have studied mainly the Frationality of Rees algebras as one of ringtheoretical properties of blowup algebras. The most important result in our research is to give a criterion for the Frationality of Rees algebras with respect to mprimary ideals in CohenMacaulay local rings. The notion of Frationality 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 CohenMacaulay property of invariant subrings of linearly reductive groups. However, as for Frationality, 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 Frationality 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.
