A local ring is CM if and only if its residue field has a CM syzygy
type: Item
instance of: scholarly article
title: A local ring is CM if and only if its residue field has a CM syzygy
zbMATH Open document ID: 0778.13020
author: Satoru Okiyama
published in: Tokyo Journal of Mathematics
publication date: 1992-09-27T00:00:00Z
review text: Let \((R,{\mathfrak m},k)\) be a noetherian local ring and \(\Omega_ i\) the \(i\)-th syzygy of \(k\). It is well known that (1) \(R\) is regular \(\Leftrightarrow\) there exists an integer \(i\geq n=\dim R\) such that \(\Omega_ i\) is \(R\)-free \(\Leftrightarrow\) for all \(i\geq n\), \(\Omega_ i\) is \(R\)-free. (2) \(R\) is Gorenstein \(\Leftrightarrow\) there exists an integer \(i\geq n\) such that \(\text{Ext}^ 1_ R(\Omega_ i,R)=0\Leftrightarrow\) for all \(i\geq n\), \(\text{Ext}^ 1_ R(\Omega_ i,R)=0\). Here it is shown that \(R\) is Cohen-Macaulay \(\Leftrightarrow\) there exists an integer \(i\geq n\) such that \(\Omega_ i\) is CM\(\Leftrightarrow\) for all \(i\geq n\), \(\Omega_ i\) is CM or 0. There are also sufficient conditions for \(\Omega_ i\), \(i>0\), to be Buchsbaum modules.
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13D02
zbMATH DE Number: 60856
zbMATH Keywords: Cohen-Macaulay modules
zbMATH Keywords: noetherian local ring
zbMATH Keywords: syzygy
zbMATH Keywords: Buchsbaum modules
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270130388
OpenAlex ID: W1980106270
DOI: 10.3836/TJM/1270130388
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description: scientific article; zbMATH DE number 60856