Integrally closed and componentwise linear ideals

There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

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.

Related collections

Most cited references 14

Record: found
Abstract: found
Article: not found

Componentwise linear ideals

Jürgen Herzog, Takayuki Hibi (1999)

A componentwise linear ideal is a graded ideal I of a polynomial ring such that, for each degree q, the ideal generated by all homogeneous polynomials of degree q belonging to I has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal I Δ arising from a simplicial complex Δ is componentwise linear if and only if the Alexander dual of Δ is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.