On pseudo supports and non-Cohen–Macaulay locus of finitely generated modules

Journal of Algebra 323 (2010) 3029–3038

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Journal of Algebra

www.elsevier.com/locate/jalgebra

On pseudo supports and non-Cohen–Macaulay locus of

finitely generated modules ✩

Nguyen Tu Cuong a,∗, Le Thanh Nhan b, Nguyen Thi Kieu Nga c

a Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Viet Nam

b Thai Nguyen College of Sciences, Thai Nguyen, Viet Nam

c Hanoi Pedagogical University N02, Vinh Phuc, Viet Nam

article info abstract

Article history:

Received 15 December 2009

Available online 20 March 2010

Communicated by Kazuhiko Kurano

MSC:

13D45

13E05

Keywords:

Pseudo supports

Non-Cohen–Macaulay locus

Catenarity

Serre conditions

Unmixedness

Let (R,m) be a Noetherian local ring and M a finitely generated R￾module with dim M = d. Let i 0 be an integer. Following M. Brod￾mann and R.Y. Sharp (2002) [BS1], the i-th pseudo support of M is

the set of all prime ideals p of R such that Hi−dim(R/p)

pRp (Mp) = 0.

In this paper, we study the pseudo supports and the non-Cohen–

Macaulay locus of M in connections with the catenarity of the ring

R/AnnR M, the Serre conditions on M, and the unmixedness of the

local rings R/p for certain prime ideals p in SuppR (M).

© 2010 Elsevier Inc. All rights reserved.

1. Introduction

Throughout this paper, let (R,m) be a Noetherian local ring and M a finitely generated R-module

with dim M = d. For each ideal I of R, denote by Var(I) the set of all prime ideals containing I. Let

i 0 be an integer. Following M. Brodmann and R.Y. Sharp [BS1], the i-th pseudo support of M, denoted

by Psuppi

R M, is defined by

Psuppi

R (M) =

p ∈ Spec(R)

Hi−dim(R/p)

pRp (Mp) = 0



.

✩ The authors are supported by the Vietnam National Foundation for Science and Technology Development (Nafosted).

* Corresponding author.

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

0021-8693/$ – see front matter © 2010 Elsevier Inc. All rights reserved.

doi:10.1016/j.jalgebra.2010.03.006