On the structure of sequentially generalized Cohen-Macaulay modules (Q2466942)

The authors study some properties and characterizations of sequentially generalized Cohen-Macaulay modules, which were defined by \textit{L. T. Nhan} and the first author [J. Algebra 267, No. 1, 156--177 (2003; Zbl 1068.13009)] as an analogue of generalized Cohen-Macaulay modules and sequentially Cohen-Macaulay modules. Let \(A\) be a noetherian local ring, \(M\) a finitely generated \(A\)-module and \(x_1\), \dots, \(x_d\) a system of parameters for \(M\). It is characterized by the behavior of the length of the residue module \[ M/(x_1^{n_1}, \dots, x_d^{n_d})M \quad (n_1, \dots, n_d > 0) \] whether \(M\) is generalized Cohen-Macaulay or not. The authors observe the length of \(M/(x_1^{n_1}, \dots, x_d^{n_d})M\) when \(M\) is sequentially generalized Cohen-Macaulay and \(x_1\), \dots, \(x_d\) satisfy certain condition. They gave a characterization of sequentially generalized Cohen-Macaulay property of \(M\) by using \(M/(x_1^{n_1}, \dots, x_d^{n_d})M\). They also compute the Hilbert-Samuel function of sequentially generalized Cohen-Macaulay modules.