Cohen-Macaulayness of special fiber rings

Alberto Corso, Laura Ghezzi, Claudia Polini, Bernd Ulrich

Research output: Contribution to journalArticlepeer-review

20 Scopus citations

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 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 languageEnglish
Pages (from-to)3713-3734
Number of pages22
JournalCommunications in Algebra
Volume31
Issue number8
DOIs
StatePublished - 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

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