The structure of Sally modules of rank one
type: Item
instance of: scholarly article
title: The structure of Sally modules of rank one
zbMATH Open document ID: 1167.13012
author: Shiro Goto
author: Kazuho Ozeki
author: Kohji Nishida
published in: Mathematical Research Letters
publication date: 2009-02-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/0710.1178
review text: ``A complete structure theorem of Sally modules of \(\mathfrak m\)-primary ideals \(I\) in a Cohen-Macaulay local ring \((A,\mathfrak m)\) satisfying the equality \(e_1(I)=e_0(I)-\ell_A(A/I)+1\) is given, where \(e_0(I)\) and \(e_1(I)\) denote the first two Hilbert coefficients of \(I\).'' Let \((A,\mathfrak m)\) be a Cohen-Macaulay local ring of dimension \(>0\) and with infinite residue field \(k=A/ \mathfrak m\). Let \(I\) be an \(\mathfrak m\)-primary ideal of \(A\) and let \(Q\) be a minimal reduction of \(I\). The \textit{Sally module} of \(I\) with respect to \(Q\), as defined by W. V. Vasconcelos, is \(S=S_Q(I)=IA[It]/IA[Qt]\), where \(t\) is an indeterminate. The authors prove the equivalence of the following three conditions: (1) \(e_1=e_0-\ell(A/I)+1\); (2) \(\mathfrak m S=(0)\) and \(\text{rank}_B S=1\); (3) \(S\cong (X_1,X_2,\dots,X_c)B\) as graded \(T\)-modules for some integer \(0<c\leq d\), where \(X_1,\dots, X_c\) are linearly independent linear forms of the polynomial ring \(B\). Here \(T=A[Qt]\), and \(B=T/ \mathfrak m T\) (which is a polynomial ring over the residue field \(k\)). Further results are included, and connections with previous works on this topic are described in the paper.
reviewed by: Moshe Roitman
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13H15
zbMATH DE Number: 5505926
zbMATH Keywords: Buchsbaum ring
zbMATH Keywords: Cohen-Macaulay local ring
zbMATH Keywords: graded ring
zbMATH Keywords: Hilbert coefficient
zbMATH Keywords: minimal reduction
zbMATH Keywords: Ratliff-Rush closure
zbMATH Keywords: Rees algebra
zbMATH Keywords: Sally module
MaRDI profile type: Publication
arXiv ID: 0710.1178
OpenAlex ID: W2962724207
DOI: 10.4310/MRL.2008.V15.N5.A5
description: scientific article