Cohen-Macaulayness of Special Fiber Rings#

Abstract

Let (R, 𝔪) be a Noetherian local ring and let Ibe 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 of I, where ℛ 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 Iis a strongly Cohen-Macaulay R-ideal with G ℓand the expected reduction number, we conclude that ℱ is always Cohen-Macaulay. We also obtain a characterization of the Cohen-Macaulayness of ℛ/Kℛ 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 ℛ and the associated graded ring 𝒢 of I. Finally, we focus on the integral closedness of 𝔪I. The latter question is motivated by the theory of evolutions. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.