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Some examples of the class of co-cohen-macaulay modules in dimension >s
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Some examples of the class of co-cohen-macaulay modules in dimension >s

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Some examples of the class

of co-Cohen-Macaulay modules in dimension > s

Nguyen Thi Dung - Agriculture and Forestry University

Abstract. In this paper, using the concept of A-cosequences in dimension > s introduced

by [NH], we give some examples of the class of co-Cohen-Macaulay modules in dimension

> s defined by [D3] in connection with some of the previous classes of modules.

1 Introduction

Throughout this paper, let (R, m) be Noetherian local ring and A an Artinian R-module.

The class of co-Cohen-Macaulay modules (co-CM for short) introduced by Tang and Zakeri

[TZ] plays an important role in the catergory of Artinian modules. There are some classes

of modules that contain the class co-CM modules, among which are co-filter modules and

co-CM modules in dimension > s introduced by [D1], [D3] satisfying the condition that

every system of parameters is a filter coregular sequence and a A-cosequence in dimension

> s, respectively. Moreover, some properties and characterizations of these modules via

systems of parameters, the dimension of the local homology modules Hm

i

(A) introduced

by Cuong-Nam [CN], the polynomial type ld(A) of A given by Minh [MIN] and the

multiplicity e(x; A) of A with respect to a s.o.p. x defined by [CNh1] were given.

The purpose of this paper is to give some examples of the class of co-CM modules

in dimension > s defined by [D3] in the connection with some of the previous classes of

modules.

2 Some examples

Let s ≥ −1 be an integer. An A-cosequence in dimension > s was introduced in [NH] as

an expansion of the notion of coregular sequence in [O].

Definition 2.1. A sequence (x1, . . . , xk) of elements in m is called an A-cosequence in

dimension > s if xi ∈/ p for all attached primes p ∈ AttR(0 :A (x1, . . . , xi−1)R) satisfying

dim(R/p) > s for all i = 1, . . . , k.

Note that an A-cosequence in dimension > −1, 0 are exactly an A-cosequence in sense

of A. Ooishi [O] and f-coregular sequence in sense of [D1], respectively.

The A-cosequence in dimension > s still has many nice properties that are used to

proved a finiteness result for attached primes of certain local cohomology modules and to

give the new class of modules (see [ND], [D3]).

156Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn

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