Integrally closed and componentwise linear ideals

TL;DR

This paper identifies a special class of integrally closed ideals in polynomial rings, called Goto-class G*, which have desirable properties like Cohen-Macaulay associated graded rings under certain conditions, extending known results to higher dimensions.

Contribution

It introduces the Goto-class G* of integrally closed ideals in polynomial rings, demonstrating their closure under product and their Cohen-Macaulay properties in specific cases.

Findings

Goto-class G* is closed under product

Ideals in G* have Cohen-Macaulay associated graded rings if monomial or in dimension ≤ 3

The study links integrally closed, contracted, full, 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 $\dim R\leq 3$. Our approach is based on the study of the relationship between the notions of integrally closed, contracted, full and componentwise linear ideals.