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.

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.

A finitely generated module M over a local ring is called a sequentially generalized Cohen-Macaulay module if there is a filtration of submodules of M : M 0 ⊂ M 1 ⊂ · · · ⊂ M t = M such that dim M 0 < dim M 1 < · · · < dim M t and each M i /M i−1 is generalized Cohen-Macaulay. The aim of this paper is to study the structure of this class of modules. Many basic properties of these modules are presented and various characterizations of sequentially generalized Cohen-Macaulay property by using local cohomology modules, theory of multiplicity and in terms of systems of parameters are given. We also show that the notion of dd-sequences defined in [N.T. Cuong, D.T. Cuong, dd-Sequences and partial Euler-Poincaré characteristics of Koszul complex, J. Algebra Appl. 6 (2) (2007) 207-231] is an important tool for studying this class of modules.

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