## Abstract

Let (R, m) be a Noetherian local ring and let I be an R-ideal. Inspired by the work of Hübl and Huneke, we look for conditions that guarantee the Cohen-Macaulayness of the special fiber ring ℱ = script R sign/mscript R sign R of I, where script R sign denotes the Rees algebra of I. Our key idea is to require 'good' intersection properties as well as 'few' homogeneous generating relations in low degrees. In particular, if I is a strongly Cohen-Macaulay R-ideal with G_{ℓ} and the expected reduction number, we conclude that ℱ is always Cohen-Macaulay. We also obtain a characterization of the Cohen-Macaulayness of script R sign/Kscript R sign for any m-primary ideal K. This result recovers a well-known criterion of Valabrega and Valla whenever K=I. Furthermore, we study the relationship between the Cohen-Macaulay property of the special fiber ring ℱ and the Cohen-Macaulay property of the Rees algebra script R sign and the associated graded ring script g sign of I. Finally, we focus on the integral closedness of mI. The latter question is motivated by the theory of evolutions.

Original language | English |
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Pages (from-to) | 3713-3734 |

Number of pages | 22 |

Journal | Communications in Algebra |

Volume | 31 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2003 |

### Bibliographical note

Funding Information:Let (R, m) be a Noetherian local ring and let I be an R-ideal. Inspired by the work of Hübl and Huneke, we look for conditions that guarantee the Cohen-Macaulayness of the special fiber ring F = R/mR R of I, where R denotes the Rees algebra of I. Our key idea is to require ‘good’ intersection properties as well as ‘few’ homogeneous #Dedicated to Steven L. Kleiman on the occasion of his 60th birthday. †Partially supported by NSF. *Correspondence: Alberto Corso, Department of Mathematics, University Kentucky, Lexington, KY 40506, USA; E-mail: corso@ms.uky.edu.

Funding Information:

The last two authors gratefully acknowledge partial support from the NSF.

## Keywords

- Associated graded ring
- Cohen-Macaulayness
- Integrally closed ideals
- Reduction number
- Special fiber ring

## ASJC Scopus subject areas

- Algebra and Number Theory