A local ring is CM if and only if its residue field has a CM syzygy
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.
scientific article; zbMATH DE number 60856
scientific article