Normality of Ideals and Modules

TL;DR

This paper studies conditions under which the Rees algebra of integrally closed ideals in regular local rings is Cohen-Macaulay and normal, extending known results to higher dimensions and modules.

Contribution

It provides new sufficient conditions for the Rees algebra to be Cohen-Macaulay and normal, including cases involving the number of generators and modules.

Findings

Rees algebra is Cohen-Macaulay and normal if the ideal contains a part of a regular system of parameters.

Results extend to ideals with a bounded number of generators, up to d+2.

Generalization to integrally closed torsionfree modules of finite colength.

Abstract

We investigate when the Rees algebra of an integrally closed $\mathfrak{m}$-primary ideal in a regular local ring is a Cohen-Macaulay normal domain. While this property always holds in dimension two, it fails in general in higher dimensions, prompting a search for sufficient conditions on the ideal. We show that if an integrally closed ideal contains a part of regular system of parameters of length $d-2$, where $d$ is the dimension of the regular local ring, then its Rees algebra is Cohen-Macaulay and normal. We also extend results of Goto and Ciuperc\u{a} by proving the same conclusion when the minimal number of generators of an ideal is at most $d+2$. Furthermore, we treat the case of integrally closed zero-dimensional ideals generated by $d+3$ homogeneous polynomials. Finally, using generic Bourbaki ideals, we generalize these results to integrally closed torsionfree modules of finite colength.