Toward a theory of generalized Cohen-Macaulay modules

Commutative Algebra and Combinatorics

§ l. Introdudion The. purpose of this paper is to give, apply:ing a decomposition theorem of maximal Buchsbaum modules over regular local rings, a structure theorem for generalized Cohen,-Macaulay modules relative to so.-called standard systems of parameters. Before stating the results more precisely, let us recall the definition of Buchsbaum modules-and generalized Cohen-Macaulay modules respectively (see (2 .. 8) and (4.l}for a further detail). Throughout let A den-ete a N oetherian local ring. with maximal ideal m and M a :finitely generated A-module of dim..1.M=s. Then Mis said to be Buchsbaum (resp. generalized Cohen-Macaulay), if there is given a numerical invariant l..i.(M) of M so that the equality

arXiv: Commutative Algebra, 2020

Let a be a proper ideal of a commutative Noetherian ring R with identity. Using the notion of a-relative system of parameters, we introduce a relative variant of the notion of generalized Cohen-Macaulay modules. In particular, we examine relative analogues of quasi-Buchsbaum, Buchsbaum and surjective Buchsbaum modules. We reveal several interactions between these types of modules that extend some of the existing results in the classical theory to the relative one.

Proceedings of the American Mathematical Society, 1990

Let R R be a Cohen-Macaulay local ring with maximal ideal m m and suppose that dim ⁡ R ≥ 2 \dim R \geq 2 . Then R R is regular if (and only if) for any Buchsbaum R R -module M M and for any integer i , i ≠ dim R M i,i \ne {\dim _R}M , the canonical map Ext R i ( R / m , M ) → H m i ( M ) := lim → n E x t R i ( R / m n , M ) {\text {Ext}}_R^i\left ( {R/m,M} \right ) \to H_m^i\left ( M \right ): = \lim _{\substack {\to \\n}} \mathrm {Ext}_R^i \left (R/m^n, M \right ) is surjective. The hypothesis that R R is Cohen-Macaulay is not superfluous. Two examples are given.

Journal of Algebra, 2007

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 [5] is an important tool for studying this class of modules.

Nagoya Mathematical Journal

Let A be a Noetherian local ring of dimensiondand with maximal idealm. ThenAis called Buchsbaum if every system of parameters is a weak sequence. This is equivalent to the condition that, for every parameter idealq, the differenceis an invariantI(A) ofAnot depending on the choice ofq. (See Section 2 for the detail.) The concept of Buchsbaum rings was introduced by Stückrad and Vogel [8], and the theory of Buchsbaum singularities is now developing (cf. [6], [7], [9], [10], and [12]).

We study two classes of Artinian modules over commutative Noetherian rings called coBuchsbaum modules and generalized co-Cohen-Macaulay modules. Some properties on q−weak co-sequences, co-standard sequences, multiplicity, local homology modules, localization, . . . , of these modules are presented. MIRAMARE – TRIESTE July 2004 ntcuong@math.ac.vn xsdung0507@yahoo.com Junior Associate of the Abdus Salam ICTP. trtrnhan@yahoo.com

Bulletin of The Belgian Mathematical Society-simon Stevin, 1994

We consider the possibility of characterizing Buchsbaum and some special generalized Cohen-Macaulay rings by systems of parameters having certain properties of regular sequences. As an application, we give a bound on Castelnuovo-Mumford regularity of so-called (k, d)-Buchsbaum graded Kalgebras.

arXiv: Commutative Algebra, 2016

Let $(R, \frak m)$ be a homomorphic image of a Cohen-Macaulay local ring and $M$ a finitely generated $R$-module. We use the splitting of local cohomology to shed a new light on the structure of non-Cohen-Macaulay modules. Namely, we show that every finitely generated $R$-module $M$ is associated by a sequence of invariant modules. This modules sequence expresses the deviation of $M$ with the Cohen-Macaulay property. This result generalizes the unmixed theorem of Cohen-Macaulayness for any finitely generated $R$-module. As an application we construct a new extended degree in sense of W. Vasconcelos.

Journal of Algebra, 2016

Let M be a finitely generated module over a Noetherian local ring R. In this paper we introduce the notion of sequential polynomial type of M , which is denoted by sp(M), in order to measure how far M is different from the sequential Cohen-Macaulayness. We study the sequential polynomial type under localization and m-adic completion. We investigate an ascentdescent property of sequential polynomial type between M and M/xM for certain parameter x of M. When R is a quotient of a Gorenstein local ring, we describe sp(M) in term of the deficiency modules of M .