On length of generalized fractions \(1/(x_1^n,\dots, x_d^n,1)\)
type: Item
instance of: scholarly article
title: On length of generalized fractions \(1/(x_1^n,\dots, x_d^n,1)\)
zbMATH Open document ID: 1072.13005
author: Le Thanh Nhan
author: Marcel Morales
published in: Vietnam Journal of Mathematics
publication date: 2005-02-11T00:00:00Z
review text: Let \(M\) be a finitely generated module over a noetherian local ring \((R,m)\) with \(\dim M= d\). For a system of parameters \((x_{1},x_{2},\dots, x_{d})\) of \(M\) and a positive integer \(n\), consider the generalized fraction \(1/(x_{1}^{n}, x_{2}^{n},\dots, x_{d}^{n},1)\) [concept introduced by \textit{H. Sharp} and \textit{H. Zakeri}, Mathematika 29, 32--41 (1982; Zbl 0497.13006)]. One of the questions on this object is whether its length can be seen as a polynomial in \(n\), for \(n\) large enough. The present paper answers in the negative this question, by indicating a counterexample for any \(d\) \(\geq 3\) . Moreover, the above problem is related with the study of the length of a ring \(R\)/(\(x_{1}^{n}\), \( x_{2}^{n}\),\dots, \(x_{d}^{n}\), \(I\)), where \(I\subset R\) is an arbitrary ideal of \(R\).
reviewed by: Marius Tărnăuceanu
Mathematics Subject Classification ID: 13B30
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13H15
zbMATH DE Number: 2133566
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2133566