dd-SEQUENCES AND PARTIAL EULER–POINCARÉ CHARACTERISTICS OF KOSZUL COMPLEX

Abstract

Communicated by S. R. López-PermouthThe aim of this paper is to introduce a new notion of sequences called dd-sequences and show that this notion may be convenient for studying the polynomial property of partial Euler-Poincaré characteristics of the Koszul complex with respect to the powers of a system of parameters. Some results about the dd-sequences, the partial Euler-Poincaré characteristics and the lengths of local cohomology modules are also presented in the paper. There are also applications of dd-sequences on the structure of sequentially Cohen-Macaulay modules.

On sequentially Cohen-Macaulay modules

“…Note that the notion of sequentially Cohen-Macaulay module was introduced first by Stanley [12] for graded case (see also Herzog-Sbara [8]). In our study of sequentially Cohen-Macaulay modules, the notion of dd-sequences defined in [4] is used frequently. For convenience, we recall briefly the definition and some basic results of dd-sequences presented in [4].…”

On the structure of sequentially generalized Cohen–Macaulay modules

“…. , x n d d )M) is a polynomial, the authors in [5] have introduced a notion of dd-sequences. For the definition we need the notion of d-sequence of Huneke [12].…”

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.…”

“…([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, .…”

“…. , 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.…”