On sequentially Cohen-Macaulay modules

Abstract

In this paper we present characterizations of sequentially Cohen-Macaulay modules in terms of systems of parameters, which are generalizations of well-known results on Cohen-Macaulay and generalized Cohen-Macaulay modules. The sequentially Cohen-Macaulayness of Stanley-Reisner rings of small embedding dimension are also examined.

Cohen-Macaulayness versus the vanishing of the first Hilbert coefficient of parameter ideals

“…We refer the reader to [6] for the definition and details. See also [2,14,15]. We use the following characterization.…”

On the structure of sequentially generalized Cohen–Macaulay modules

“…Lemma 2.6. (See [6,Lemma 3.5]. ) Every system of parameters of M, which is also a dd-sequence on M, is a good system of parameters, and therefore it is a good system of parameters with respect to any filtration F satisfying the dimension condition of M.…”

Hilbert coefficients and sequentially Cohen–Macaulay modules

“…This paper is divided into four sections. In the next section we recall the notions of dimension filtration, good parameter ideals and distinguished parameter ideals following [14,4,2,3], and prove some preliminary results on the dimension filtration. We discuss in Section 3 the relationship between Hilbert coefficients and arithmetic degrees (see [1,16]) of an m-primary ideal.…”

“…. , x d such that q = (x).Now let us briefly give some facts on the dimension filtration and good systems of parameters (see[2][3][4]). Let N be the set of all positive integers.…”

“…([2][3][4]). A filtration D :M = D 0 ⊃ D 1 ⊃ · · · ⊃ D s = H 0 m (M) of submodules of M is said to be a dimension filtration if D i is the largest submodule of D i−1 with dim D i < dim D i−1 for all i = 1, .…”