On the relation type of systems of parameters
type: Item
instance of: scholarly article
title: On the relation type of systems of parameters
zbMATH Open document ID: 0846.13016
author: Yung-Hui Lai
published in: Journal of Algebra
publication date: 1996-09-22T00:00:00Z
review text: Let \(A\) be a commutative ring. A relation on a sequence of elements \(x_1, \dots, x_n\) of \(A\) is a form of positive degree in \(S = A[U_1, \dots, U_n]\) that yields 0 when evaluated at \(x_1, \dots, x_n\). Let \(R_j\) denote the ideal of \(S\) generated by the relations on \(x_1, \dots, x_n\) of degree at most \(j\). The least integer (possibly infinite) \(t\) such that \(R_t=\bigcup_j R_j\) is called the relation type of the ideal \((x_1, \dots, x_n)\). In section two the following theorem is proven: Let \(B\) be a Buchsbaum local ring, \(I\) an ideal and \(A = B/I\). Then \(J^n \cap I = J^{n - k} (J^k \cap I)\) for any ideal \(J\) with \(JA\) being a parameter ideal of \(A\) and for \(n \geq k\) if and only if the relation type of parameter ideals of \(A\) is uniformly bounded by \(k\). In section three the main theorem is given, that is: Let \((A,m)\) be an equidimensional local ring of dimension \(d\) which possesses a dualizing complex. Then there is a certain power \(I\) of \(\prod_{i < d} \text{Ann} H^i_m (A)\) such that if \(x_1, \dots, x_d\) and \(y_1, \dots, y_d\) are systems of parameters with \(x_i - y_i\) in \(I\) for \(i = 1, \dots, d\), the relation type of \((x_1, \dots, x_d)\) and that of \((y_1, \dots, y_d)\) are the same. Using this theorem, the uniform boundedness of the relation type of systems of parameters for certain local rings is shown, e.g., for a local ring with finite local cohomology and with finite residue field.
reviewed by: Yoichi Aoyama
Mathematics Subject Classification ID: 13H99
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 13B25
zbMATH DE Number: 783926
zbMATH Keywords: uniform Artin-Rees property
zbMATH Keywords: relation on a sequence of elements
zbMATH Keywords: local ring
zbMATH Keywords: dualizing complex
zbMATH Keywords: uniform boundedness of the relation type
zbMATH Keywords: systems of parameters
MaRDI profile type: Publication
full work available at URL: jabr.1995.1190
OpenAlex ID: W2072339135
DOI: 10.1006/JABR.1995.1190
description: scientific article; zbMATH DE number 783926