Cohen-Macaulayness of special fiber rings

Abstract

Let $(R, {\mathfrak m})$ be a Noetherian local ring and let $I$ be an $R$-ideal. Inspired by the work of H\"ubl and Huneke, we look for conditions that guarantee the Cohen-Macaulayness of the special fiber ring ${\mathcal F}={\mathcal R}/{\mathfrak m}{\mathcal R}$ of $I$, where ${\mathcal R}$ 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 $G_{\ell}$ and the expected reduction number, we conclude that ${\mathcal F}$ is always Cohen-Macaulay. We also obtain a characterization of the Cohen-Macaulayness of ${\mathcal R}/K{\mathcal R}$ for any ${\mathfrak m}$-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 ${\mathcal F}$ and the one of the Rees algebra ${\mathcal R}$ and the associated graded ring ${\mathcal G}$ of $I$. Finally, we focus on the integral closedness of ${\mathfrak m}I$. The latter question is motivated by the theory of evolutions.