### Abstract

In this paper, using the notion of the tight integral closure, we will give a criterion for F-rationality of Rees algebras of m-primary ideals in a Cohen-Macaulay local ring. As its application, we prove the following results: (1) In dimension two, if A is F-rational and I is integrally closed, then the Rees algebra R(I) is F-rational. On the other hand, in higher dimensions, we construct many examples of Cohen-Macaulay, normal Rees algebras which are not F-rational. (2) If both A and R(I) are F-rational, then so is the extended Rees algebra R'(I). (3) If R(I) is F-rational and a(G(I)) ≠ - 1, then A is F-rational. On the other hand, using resolution of singularities, we will prove that a two-dimensional rational singularity always admits F-rational Rees algebras. In particular, this theorem gives another way than that devised by Watanabe (1997, J. Pure Appl. Algebra 122, 323-328) to construct counterexamples to the Boutot-type theorem for F-rational rings.

Original language | English |
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Pages (from-to) | 153-190 |

Number of pages | 38 |

Journal | Journal of Algebra |

Volume | 247 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2002 Jan 1 |

### Keywords

- F-rationality
- Rational singularity
- Rees ring
- Tight integral closure

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Journal of Algebra*,

*247*(1), 153-190. https://doi.org/10.1006/jabr.2001.8998