Integrally closed and componentwise linear ideals

Abstract

In two dimensional regular local rings integrally closed ideals have a unique factorization property and have a Cohen-Macaulay associated graded ring. In higher dimension these properties do not hold for general integrally closed ideals and the goal of the paper is to identify a subclass of integrally closed ideals for which they do. We restrict our attention to 0-dimensional homogeneous ideals in polynomial rings $R$ of arbitrary dimension and identify a class of integrally closed ideals, the Goto-class $\G^*$, that is closed under product and that has a suitable unique factorization property. Ideals in $\G^*$ have a Cohen-Macaulay associated graded ring if either they are monomial or $\dim R\le 3$. Our approach is based on the study of the relationship between the notions of integrally closed, contracted, full and componentwise linear ideals.

revised version, references added, to appear in Math. Z

References in corpus (2)

• Castelnuovo-Mumford regularity of products of ideals
• Evolutions, Symbolic Squares, and Fitting Ideals