On the parametric decomposition of powers of parameter ideals in a Noetherian local ring
type: Item
instance of: scholarly article
title: On the parametric decomposition of powers of parameter ideals in a Noetherian local ring
zbMATH Open document ID: 1059.13009
author: Shiro Goto
author: Yasuhiro Shimoda
published in: Tokyo Journal of Mathematics
publication date: 2005-01-04T00:00:00Z
review text: Let \(A\) be a commutative ring and \(a_1, \dots, a_d \in A\). In the present paper, the authors study the equation \[ (a_1, \dots, a_n)^n = \bigcap_{\substack{ n_1, \dots, n_d > 0 \\ n_1 + \dots + n_d = n + d - 1 }} (a_1^{n_1}, \dots, a_d^{n_d}) \quad \text{for any \(n > 0\)}. \tag{1} \] The origin of (1) is given by \textit{W. Heinzer, L. J. Ratliff jun.} and \textit{K. Shah} [Houston J. Math. 21, No. 1, 29--52 (1995; Zbl 0838.13012)]. They showed that (1) is true whenever \(a_1, \dots, a_d\) is an \(A\)-regular sequence. In particular, an arbitrary system of parameters for a Cohen-Macaulay local ring satisfies (1). The authors prove the converse in a weak sense. The main theorem of this paper asserts that all the systems of parameters for a Noetherian local ring~\(A\) of dimension at least~\(2\) satisfy (1) if and only if \(A/H_{\mathfrak m}^0(A)\) is Cohen-Macaulay and \(A\) itself is Buchsbaum. In particular, if \(\text{depth} A > 0\) and all the systems of parameters for~\(A\) satisfy (1), then \(A\) is Cohen-Macaulay. It is a natural question whether \(A\) is Cohen-Macaulay if \(\text{depth} A > 0\) and {some} system of parameters for~\(A\) satisfies (1). The authors give a counterexample for this question but the example is not quasi-unmixed. They also study \(2\)-dimensional generalized Cohen-Macaulay local rings, which are quasi-unmixed, and show that such a ring is Cohen-Macaulay if it is of positive depth and some standard system of parameters satisfies (1).
reviewed by: Takesi Kawasaki
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13C15
zbMATH DE Number: 2124823
zbMATH Keywords: system of parameters
zbMATH Keywords: formal powers of ideals
zbMATH Keywords: depth
zbMATH Keywords: Cohen-Macaulayness
zbMATH Keywords: complete intersection
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208479
OpenAlex ID: W2009705317
DOI: 10.3836/TJM/1244208479
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description: scientific article; zbMATH DE number 2124823