Asymptotic behaviour of good systems of parameters of sequentially generalized Cohen-Macaulay modules
Let \((R, \mathfrak{m})\) denote a local ring. A finitely generated \(R\)-module \(M\) is called generalized Cohen-Macaulay provided the local cohomology modules \(H^i_{\mathfrak{m}}(M)\) are finitely generated for all \(i \not= d = \dim M\). Let \(\underline{x} = x_1,\ldots,x_d\) denote a system of parameters of \(M\). The index of reducibility \(N_R(\underline{x},M)\) is defined as the socle dimension of \(M/\underline{x}M\). It is an invariant if \(M\) is a Cohen-Macaulay module. \textit{N. T. Cuong} and \textit{H. Le Truong} [J. Algebra 320, No. 1, 158--168 (2008; Zbl 1155.13009)] proved that \(N_R(\underline{x},M)\) is an invariant in a generalized Cohen-Macaulay module for all \(\underline{x} \subset \mathfrak{m}^n\) with \(n \gg 0\). This generalizes a corresponding result for Buchsbaum modules shown by \textit{S. Gôto} and \textit{N. Suzuki} [J. Algebra 87, 53--88 (1984; Zbl 0538.13003)]. Another generalization of the Cohen-Macaulayness is sequentially Cohen-Macaulay saying that \(M\) admits a Cohen-Macaulay dimension filtration (see \textit{P. Schenzel} in [Lect. Notes Pure Appl. Math. 206, 245--264 (1999; Zbl 0942.13015)]). In their paper [Kodai Math. J. 30, No. 3, 409--428 (2007; Zbl 1139.13011)], \textit{N. T. Cuong} and \textit{D. T. Cuong} generalized the results on the behaviour of socle dimensions of parameter ideals to sequentially Cohen-Macaulay modules. A sequentially generalized Cohen-Macaulay module is one where its dimension filtration has generalized Cohen-Macaulay modules as quotients. In the paper under review there is an extension of the results of Cuong and Cuong [loc. cit.] to sequentially generalized Cohen-Macaulay modules.
sequentially generalized Cohen-Macaulay module
scientific article; zbMATH DE number 6126934