A variant of Wang's theorem
type: Item
instance of: scholarly article
title: A variant of Wang's theorem
zbMATH Open document ID: 1267.13009
published in: Journal of Algebra
publication date: 2013-04-17T00:00:00Z
review text: In this nicely written paper, the authors use characteristic \(p\) techniques to give a new variant, as the title suggests, of \textit{H. J. Wang}'s theorem [Math. Z. 256, 749--756 (2007; Zbl 1123.13018)]. Wang's theorem states that for a Cohen-Macaulay local ring \((R, \mathfrak{m})\) of dimension \(d \geq 2\) with \(d \geq 3\) if \(R\) is regular, if \(J \subseteq \mathfrak{m}^s\) for some \(s \geq 2\), then \(L:=J: \mathfrak{m}^s \subseteq \mathfrak{m}^s\) and \(L^2=JL\). The authors' interest in Wang's theorem stems from finding the Goto number of a parameter ideal in a more general setting. The Goto number of a parameter ideal \(J\) in a Noetherian local ring \((R,\mathfrak{m})\) was defined as \(g(J):=\text{max}\{t \in \mathbb{N} \mid J:\mathfrak{m}^t \text{ is integral over } J\}\) by \textit{W. Heinzer} and \textit{I. Swanson} [J. Algebra 321, 152--166 (2009; Zbl 1166.13018)]. The authors' first result is to show that in a quasi-unmixed Noetherian local ring or characteristic \(p\), for any subsystem of parameters \(J=(a_1,\ldots, a_m)\), the integral closure of \(J\) is equal to the \(J^{m-1}\)-tight closure of \(J\). They use this fact along with known results on \(\mathfrak{a}^t\)-test ideals to show that \(J:\overline{J}=J+\tau(J^{d-1})\) for any parameter ideal in a complete or \(F\)-finite normal Gorenstein local ring. Relying on work of \textit{N. Hara} and \textit{K. Yoshida} [Trans. Am. Math. Soc. 355, No. 8, 3143--3174 (2003; Zbl 1028.13003)] they then prove some Skoda-type theorems pertaining to the \(\mathfrak{m}^t\)-test ideals for \(t \geq d-1\) where \(d\) is the dimension of the ring, their strongest result being \(\tau(\mathfrak{m}^t) \subseteq \mathfrak{m}^{t-d+3}\) when \((R, \mathfrak{m})\) is a complete Gorenstein local ring with multiplicity \(e(R) \geq 3\). They then proceed to prove their new variant of Wang's theorem which states that if \(R\) is a Cohen-Macaulay ring of dimension \(d\geq 2\) and \(J \subseteq \mathfrak{m}^s\) is a p…
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 6155227
zbMATH Keywords: Gorenstein ring
zbMATH Keywords: Cohen-Macaulay ring
zbMATH Keywords: regular ring
zbMATH Keywords: test ideal
zbMATH Keywords: multiplier ideal
zbMATH Keywords: integral closure
zbMATH Keywords: Goto number
author: Kei-Ichi Watanabe
author: Ken-Ichi Yoshida
reviewed by: Janet Vassilev
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.07.012
OpenAlex ID: W1997540416
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DOI: 10.1016/J.JALGEBRA.2012.07.012
description: scientific article; zbMATH DE number 6155227