Some criteria for the Cohen-Macaulay property and local cohomology
Let \(R\) be a commutative noetherian ring and \(A\) be a proper ideal of \(R\). The author obtains a characterization of Cohen-Macaulay rings using relative homological algebra. It is shown that \(R\) is Cohen-Macaulay if and only if for any module \(M\), height of \(P\) is less or equal to the Gorenstein flat dimension of \(M\) for all prime idea \(P\) with the 0-th Bass number of \(M\) in \(P\), \(\mu_0(P,M)\), not zero. Moreover, if \((R,{\mathcal M})\) is a complete local ring, the filter depth of \(M\) in \(A\), i.e. the length of a maximal filter regular sequence of \(M\) in \(A\), is greater of equal to \(d\), the depth of \(D(H_{\mathcal M}^i(M))\geq i-1\) for all \(2\leq i<d\) and \(H_{\mathcal M}^i(M)\not=0\), then \(D(H_{\mathcal M}^d(M))\) is a Cohen-Macaulay \(R\)-module of dimension \(d\).
scientific article; zbMATH DE number 5629547