On the unmixedness and universal catenaricity of local rings and local cohomology modules

Journal of Algebra 321 (2009) 303–311

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Journal of Algebra

www.elsevier.com/locate/jalgebra

On the unmixedness and universal catenaricity of local rings

and local cohomology modules ✩

Le Thanh Nhan a,∗, Tran Nguyen An b

a College of Natural Sciences, Thai Nguyen University, Thai Nguyen, Viet Nam

b College of Education, Thai Nguyen University, Thai Nguyen, Viet Nam

article info abstract

Article history:

Received 18 June 2008

Available online 23 September 2008

Communicated by Kazuhiko Kurano

Keywords:

Local cohomology modules

Unmixedness

Universal catenaricity

Multiplicity

Let (R,m) be a Noetherian local ring and M a finitely generated

R-module. Let i 0 be an integer. Consider the following property

for the Artinian local cohomology module Hi

m(M)

AnnR (0 :

Hi

m(M) p) = p for all p ∈ Var

AnnR

Hi

m(M)

. (∗)

In this paper, we study the property (∗) of Hi

m(M) in order

to investigate the universal catenaricity of the ring R/AnnR M

and the unmixedness of the ring R/p for certain p in Supp M.

We also characterize the property (∗) for Hi

m(M) and obtain the

associativity formulae for multiplicity of Hi

m(M) in case where

Hi

m(M) satisfies the property (∗).

© 2008 Elsevier Inc. All rights reserved.

1. Introduction

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

finitely generated R-module. For each ideal I of R, we denote by Var(I) the set of all prime ideals

containing I. For a subset T of Spec(R), we denote by min(T ) the set of all minimal elements of T

under the inclusion.

It is clear that AnnR (M/pM) = p for all p ∈ Var(AnnR M). Therefore it is natural to ask the dual

property for Artinian modules:

AnnR (0 :A p) = p for all p ∈ Var(AnnR A). (∗)

✩ This paper is the result of the Scientific Research Project at Ministrial level in Mathematics.

* Corresponding author.

E-mail addresses: [email protected] (L.T. Nhan), [email protected] (T.N. An).

0021-8693/$ – see front matter © 2008 Elsevier Inc. All rights reserved.

doi:10.1016/j.jalgebra.2008.09.005