From associated graded modules to blowing-ups of generalized Cohen-Macaulay modules
type: Item
instance of: scholarly article
title: From associated graded modules to blowing-ups of generalized Cohen-Macaulay modules
zbMATH Open document ID: 0628.13017
published in: Journal of Mathematics of Kyoto University
publication date: 1984-01-01T00:00:00Z
review text: A denotes a local ring with maximal ideal \(m\) and M a generalized Cohen- Macaulay module over A, i.e. \(d:=\dim (M)>1\) and \(\ell (H^ i_{{\mathfrak m}}(M))<\infty\) for \(i=0,...,d-1\), where \(H^ i_{{\mathfrak m}}(M)\) denotes the i-th local cohomology module of M with respect to \({\mathfrak m}.\) Let I(M) denote the maximum of the differences I(\({\mathfrak q}:M):=\ell (M/{\mathfrak q}M)-e({\mathfrak q};M)\), where \({\mathfrak q}\) runs through all parameter ideals of M and e(\({\mathfrak q};M)\) is the multiplicity of M with respect to \({\mathfrak q}\). Then M being a generalized Cohen-Macaulay module just means that \(I(M)<\infty.\) If \({\mathfrak q}=(a_ 1,...,a_ d)\) and I(\({\mathfrak q};M)=I(M)\), we call \(a_ 1,...,a_ d\) a standard system of parameters of M. It is known that there exist ideals \({\mathfrak a}\) of A such that every system of parameters of M contained in \({\mathfrak a}\) is standard. Such ideals are called M-standard ideals. In particular, M is a Buchsbaum module if and only if \({\mathfrak m}\) is a M-standard ideal. We compute the local cohomology modules of the associated graded module \(G_{{\mathfrak a}}(M):=\oplus^{\infty}_{n=1}{\mathfrak a}^ nM/{\mathfrak a}^{n+1}M\quad and\) of the Rees module (blowing-up) \(R_{{\mathfrak a}}(M):=\oplus^{\infty}_{n=1}{\mathfrak a}^ nM\quad with\) respect to the maximal graded ideal of \(G_{{\mathfrak a}}(A)\) and \(R_{{\mathfrak a}}(A)\). Sometime, we will imagine of \(R_{{\mathfrak a}}(A)\) as the subring A[\({\mathfrak a}T]\) of A[T], where T is some indeterminate. In particular, since \(G_{{\mathfrak a}}(A)\cong A[{\mathfrak a}T]/{\mathfrak a}A[{\mathfrak a}T]\), we may consider \(G_{{\mathfrak a}}(A)\) and hence \(G_{{\mathfrak a}}(M)\) as a graded \(R_{{\mathfrak a}}(A)\)-module. Let P denote the maximal graded ideal of \(R_{{\mathfrak a}}(M)\) and Q the ideal (\({\mathfrak a},{\mathfrak a}T)\) of \(R_{{\mathfrak a}}(M)\). Main result: Theorem 1.1. Suppose that there exists an integer \(r…
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 4021305
zbMATH Keywords: generalized Cohen-Macaulay module
zbMATH Keywords: standard ideals
zbMATH Keywords: Rees module
zbMATH Keywords: blowing- up
zbMATH Keywords: standard system of parameters
zbMATH Keywords: Buchsbaum ring
author: Trung Ngo Viet
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250521221
OpenAlex ID: W1603962918
DOI: 10.1215/KJM/1250521221
description: scientific article; zbMATH DE number 4021305