In a two dimensional regular local ring integrally closed ideals have a unique factorization property and their associated graded ring is Cohen–Macaulay. In higher dimension these properties do not hold 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. We identify a class of integrally closed ideals, the Goto-class G∗, which is closed under product and it has a suitable unique factorization property. Ideals in G∗ have a Cohen–Macaulay associated graded ring if either they are monomial or dim R ≤ 3. Our approach is based on the study of the relationship between the notions of integrally closed, contracted, full and componentwise linear ideals.
INTEGRALLY CLOSED AND COMPONENTWISE LINEAR IDEALS
CONCA, ALDO;DE NEGRI, EMANUELA;ROSSI, MARIA EVELINA
2010-01-01
Abstract
In a two dimensional regular local ring integrally closed ideals have a unique factorization property and their associated graded ring is Cohen–Macaulay. In higher dimension these properties do not hold 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. We identify a class of integrally closed ideals, the Goto-class G∗, which is closed under product and it has a suitable unique factorization property. Ideals in G∗ have a Cohen–Macaulay associated graded ring if either they are monomial or dim R ≤ 3. Our approach is based on the study of the relationship between the notions of integrally closed, contracted, full and componentwise linear ideals.
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