On the equality of ordinary and symbolic powers of ideals
type: Item
instance of: scholarly article
title: On the equality of ordinary and symbolic powers of ideals
zbMATH Open document ID: 1282.13007
DOI: 10.1216/JCA-2012-4-2-281
author: Aline Hosry
author: Youngsu Kim
author: Javid Validashti
published in: Journal of Commutative Algebra
publication date: 2013-03-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1304.5833
full work available at URL: https://projecteuclid.org/euclid.jca/1343846895
review text: Let \(R\) be a Noetherian regular local ring of dimension \(d\) and let \(P\) be a prime ideal of height \(d-1\) in \(R\). For all \(n \in \mathbb N\) let \(P^{(n)} = P^nR_P \cap R\) denote the \(n\)-th symbolic power of \(P\). Craig Huneke asked the following question: ``If \(P^n = P^{(n)}\) for all \(n \leq d-1\), then is \(P^n = P^{(n)}\) for all \(n\)?''. He gave positive answers in embedding dimension \(d=3\), and embedding dimension \(d= 4\) provided \(R/P\) is Gorenstein. Furthermore, there are supporting examples showing that the Gorenstein property of \(R/P\) may already be forced by the equality \(P^2 = P^{(2)}\). Thus it is natural to ask: ``If \(P^2 = P^{(2)}\), then is \(R/P\) Gorenstein?'' In this article, the authors study these questions in the case of a monomial curve \(k[[t^{a_1},\ldots,t^{a_d}]]\). They provide validity evidence in some subcases of \(d=4\), and for any embedding dimension \(d\) either assuming that the numerical sequence \(a_1,\ldots,a_n\) contains an arithmetical subsequence of length \(5\), or assuming some upper bounds on the Hilbert-Samuel multiplicity of \(R/P\). Finally, using a deformation argument due to \textit{M. Morales} [J. Algebra 140, No. 1, 12--25 (1991; Zbl 0738.13012)], they extend their results to successive modifications of the originally considered monomial curve.
reviewed by: Alessandro De Stefani
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 6148282
zbMATH Keywords: symbolic powers
zbMATH Keywords: monomial curves
zbMATH Keywords: Gorenstein
MaRDI profile type: Publication
arXiv ID: 1304.5833
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