On Cohen-Macaulay local rings with embedding dimension \(e+d-2\)
type: Item
instance of: scholarly article
title: On Cohen-Macaulay local rings with embedding dimension \(e+d-2\)
zbMATH Open document ID: 0873.13010
author: Hsin-Ju Wang
published in: Journal of Algebra
publication date: 1997-11-02T00:00:00Z
review text: The main result of this paper is the following theorem 3.1. If \((R, {\mathfrak m})\) is a \(d\)-dimensional local Cohen-Macaulay ring of embedding dimension \(e+d-2\) with infinite residue field, then \(\text{depth} G({\mathfrak m}) \geq d-1\), where \(G({\mathfrak m}) =R/{\mathfrak m} \oplus {\mathfrak m}/{\mathfrak m}^2 \oplus \dots\) is the associated graded ring of \(R\) with respect to \({\mathfrak m}\). -- This answers a question of \textit{J. D. Sally} [J. Algebra 83, 393-408 (1983; Zbl 0517.13013)]. The author uses properties of Hilbert coefficients and of the Sally module \[ S_J(I)= \bigoplus^\infty_{n=0} I^{n+1}/J^nI, \] where \(I\) is an \({\mathfrak m}\)-primary ideal with a minimal reduction \(J\); see also: \textit{H.-J. Wang}, Commun. Algebra 25, No. 1, 303-309 (1997; see the preceding review).
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13D40
zbMATH DE Number: 1013973
zbMATH Keywords: depth
zbMATH Keywords: Cohen-Macaulay ring
zbMATH Keywords: associated graded ring
zbMATH Keywords: Hilbert coefficients
zbMATH Keywords: Sally module
reviewed by: Moshe Roitman
MaRDI profile type: Publication
full work available at URL: jabr.1996.6894
OpenAlex ID: W2010153679
cites work: A 𝑑-dimensional extension of a lemma of Huneke’s and formulas for the Hilbert coefficients
cites work: Hilbert Coefficients and the Depths of Associated Graded Rings
cites work: On the associated graded ring of a local Cohen-Macaulay ring
cites work: Super-regular sequences
cites work: Cohen-Macaulay local rings of embedding dimension e+d-2
cites work: Q4295735
DOI: 10.1006/JABR.1996.6894
description: scientific article; zbMATH DE number 1013973