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 Gℓ 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 |
|---|---|
| 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: [email protected].
Funding Information:
The last two authors gratefully acknowledge partial support from the NSF.
Funding
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: [email protected]. The last two authors gratefully acknowledge partial support from the NSF.
| Funders | Funder number |
|---|---|
| U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China | |
| U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China |
Keywords
- Associated graded ring
- Cohen-Macaulayness
- Integrally closed ideals
- Reduction number
- Special fiber ring
ASJC Scopus subject areas
- Algebra and Number Theory