Cohen-Macaulayness of special fiber rings

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 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 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 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 among the Cohen-Macaulay property of the special fiber ring F and the one 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. ∗The last two authors gratefully acknowledge partial support from the NSF. AMS 2000 Mathematics Subject Classification. Primary 13A30; Secondary 13B22, 13C40, 13H10.