Some remarks on generalized Cohen-Macaulay rings
Let \(A\) be a Noetherian local ring (resp., graded \(K\)-algebra) of dimension \(d\) and \({\mathfrak m}\) its maximal (resp., maximal homogeneous) ideal. A is called a generalized Cohen-Macaulay (C-M for short) ring if all the local cohomology modules \(H^i_{\mathfrak m} (A)\), \(i < d\), are of finite length. For \(k \geq 1\), we say (in this review only) that \(A\) is a generalized C-M ring of order \(k\) if \({\mathfrak m}^k\) annihilates \(H^i_{\mathfrak m} (A)\) for every \(i < d\). A is a \((k,d)\)-Buchsbaum ring if, for every system of parameters \(x_1, \ldots, x_d\) of \(A\), \(A/(x_1, \ldots, x_j)\) is a generalized C-M ring of order \(k\) for every \(j < d\). In the paper under review, the authors give a characterization of \((k,d)\)-Buchsbaum rings with \(H^i_{\mathfrak m} (A) = 0\) for \(i \neq \text{depth} (A)\), \(d\), in terms of the existence of a system of parameters with certain regularity properties. They also prove a bound for the Castelnuovo-Mumford regularity of a graded \((k,d)\)-Buchsbaum \(K\)- algebra, generalizing a result of \textit{Ngô Viêt Trung} [Proc. Am. Math. Soc. 101, 229-236 (1987; Zbl 0641.13016)].
scientific article; zbMATH DE number 722052