6_137723_Parsa.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 5, No. 1, 2012, 55-58

ISSN 1307-5543 – www.ejpam.com

SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA

29 JUNE - 02 JULY 2011, ISTANBUL TURKEY

On the Vanishing Properties of Local Cohomology Modules

Defined by a Pair of Ideals

M. Lotfi Parsa,∗, Sh. Payrovi

Department of Mathematics, I. K. International University, Qazvin, Iran

Abstract. As a generalization of the ordinary local cohomology modules, recently some authors intro-

duced the local cohomology modules with respect to a pair of ideals. In this paper, we get some results

on Artinianness, vanishing, finiteness and other properties of these modules. Let R be a commutative

Noetherian ring, I , J two ideals of R and M a finitely generated R-module such that dimR M = n. We

prove that Hn
I,J
(M)/JHn

I,J
(M) is I -cofinite Artinian and Hn

I,J
(M)/IHn

I,J
(M) has finite length. Also we

show that, if R is local with dim R/I + J = 0 and dimR M/J M = d > 0, then Hd
I,J
(M) is not finitely

generated.

2000 Mathematics Subject Classifications: 13D45, 13E05, 13E10.

Key Words and Phrases: Artinian module, Cofinite module, Local cohomology, Noetherian module.

1. Introduction

Throughout this paper, R is a commutative Noetherian ring with non-zero identity, I , J

are two ideals of R and M is an R-module. For notations and terminologies not given in this

paper, the reader is referred to [1] and [6], if necessary.

As a generalization of the ordinary local cohomology modules, Takahashi, Yoshino and

Yoshizawa, in [6], introduced the local cohomology modules with respect to a pair of ideals

(I , J). To be more precise, let W(I , J) = {p ∈ Spec(R) : I t ⊆ p+ J for some positive integer t}.
The set of elements x of M such that SuppRRx ⊆W(I , J), is said to be (I , J)-torsion submodule

of M and is denoted by ΓI ,J(M). It is easy to see that ΓI ,J is a covariant, R-linear functor from

the category of R-modules to itself. For an integer i, the local cohomology functor H i
I ,J with

respect to (I , J), is defined to be the i-th right derived functor of ΓI ,J . Also H i
I ,J(M) is called

∗Corresponding author.

Email addresses: lotfi.parsa�ikiu.a
.ir (M. Parsa), shpayrovi�ikiu.a
.ir (Sh. Payrovi)

http://www.ejpam.com 55 c© 2012 EJPAM All rights reserved.



M. Parsa, Sh. Payrovi / Eur. J. Pure Appl. Math, 5 (2012), 55-58 56

the i-th local cohomology module of M with respect to (I , J). If J = 0, then H i
I ,J coincides

with the ordinary local cohomology functor H i
I .

Some authors studied the properties of these extended modules; see, for example, [2, 3, 5,

7]. In this direction, we study Artinianness, vanishing and finiteness of the local cohomology

modules defined by a pair of ideals. Suppose that M is finitely generated with dimR M = n. It

is well known that Hn
I (M) is I-cofinite Artinian; [see 4, Proposition 5.1]. We generalize this

result and prove that Hn
I ,J(M)/JHn

I ,J(M) is I-cofinite Artinian.

Let R be local and M finitely generated with dimR M = n> 0. It follows by Grothendieck’s

Non-vanishing Theorem that Hn
I (M) is not finitely generated, whenever dim R/I = 0. As a

generalization of this result, we show that if dim R/I + J = 0 and dimR M/J M = d > 0, then

Hd
I ,J(M) is not finitely generated.

2. Main Results

Recall that R is a Noetherian ring, I , J are two ideals of R and M is an R-module. The

following result improves [5, Corollary 3.5].

Theorem 1. Let M be finitely generated with dimR M = n. Then Hn
I ,J(M)/JHn

I ,J(M) is I-cofinite

Artinian.

Proof. We use induction on n. If n= 0, then M has finite length. Therefore ΓI ,J(M)/JΓI ,J(M)

has finite length and so ΓI ,J(M)/JΓI ,J(M) is I-cofinite Artinian. Now suppose, inductively,

that n > 0, and the result has been proved for all R-modules of dimensions smaller than

n satisfying the hypothesis. Since Hn
I ,J(M/ΓI ,J(M))

∼= Hn
I ,J(M) by [6, Corollary 1.13(4)],

we may assume in addition that M is an (I , J)-torsion free R-module. Thus I contains an

element a which is non zero-divisor on M . Since dim M/aM ≤ n− 1, it follows by the induc-

tive hypothesis that Hn−1
I ,J (M/aM)/JHn−1

I ,J (M/aM) is I-cofinite Artinian. The exact sequence

0→ M
a
→ M → M/aM → 0 induces an exact sequence

· · · → Hn−1
I ,J (M/aM)→ Hn

I ,J(M)
a
→ Hn

I ,J(M)→ 0

of local cohomology modules. Now the exact sequence

Hn−1
I ,J (M/aM)/JHn−1

I ,J (M/aM)→ Hn
I ,J(M)/JHn

I ,J(M)
a
→ Hn

I ,J(M)/JHn
I ,J(M)→ 0

implies that 0 :Hn
I ,J (M)/JHn

I ,J (M)
a is I-cofinite Artinian. Therefore Hn

I ,J(M)/JHn
I ,J(M) is I-cofinite

Artinian, by [4, Proposition 4.1]. This completes the inductive step. The result follows by

induction.

Let W̃(I , J) denote the set of ideals a of R such that I t ⊆ a+ J for some positive integer t.

It is easy to see that, for any a ∈ W̃(I , J), Γa(M) is a subset of ΓI ,J(M).

Theorem 2. Let M be finitely generated with dimR M = n and t a positive integer. If

H i
I ,J(M) = 0, for all i > t, then H t

I ,J(M)/aH t
I ,J(M) = 0, for any a ∈ W̃(I , J).



M. Parsa, Sh. Payrovi / Eur. J. Pure Appl. Math, 5 (2012), 55-58 57

Proof. Let a ∈ W̃(I , J) be fixed. We prove the claim by using induction on n. If n= 0, then

the claim is clear. Assume, inductively, that n > 0 and the result has been proved for any R-

module of dimension less than n satisfying the hypothesis. Since H i
I ,J(M/ΓI ,J(M))

∼= H i
I ,J(M)

for all i > 0, by [6, Corollary 1.13(4)], we may assume in addition that ΓI ,J(M) = 0. We

have Γa(M) ⊆ ΓI ,J(M), thus Γa(M) = 0, and therefore a contains an element a which is non

zero-divisor on M . The exact sequence 0 → M
a
→ M → M/aM → 0 induces the following

exact sequence

· · · → H i
I ,J(M)

a
→ H i

I ,J(M)→ H i
I ,J(M/aM)→ H i+1

I ,J (M)→ · · ·

of local cohomology modules. In view of the hypothesis and the above exact sequence,

H i
I ,J(M/aM) = 0 for all i > t. Since a is non zero-divisor on M , we have dim M/aM ≤ n− 1,

and therefore the inductive hypothesis implies that H t
I ,J(M/aM)/aH t

I ,J(M/aM) = 0. The

above exact sequence implies that H t
I ,J(M)/aH t

I ,J(M)
∼= H t

I ,J(M/aM). Since a ∈ a, therefore

H t
I ,J(M)/aH t

I ,J(M)
∼= H t

I ,J(M/aM)/aH t
I ,J (M/aM).

The inductive step is complete. The result follows by induction.

Corollary 1. Let M be a finitely generated module such that dimR M = n. Then Hn
I ,J(M)/aHn

I ,J(M)

has finite length, for any a ∈ W̃(I , J). Specially, Hn
I ,J(M)/IHn

I ,J (M) has finite length.

Proof. Let a ∈ W̃(I , J) be fixed. If n= 0, then M has finite length and so ΓI ,J(M)/aΓI ,J(M)

has finite length. Now assume that n > 0. It follows by [6, Theorem 4.7(1)] and Theorem 2,

that Hn
I ,J(M)/aHn

I ,J(M) = 0.

Corollary 2. Let M be finitely generated of finite dimension such that dimR M/J M = d. Then

Hd+1
I ,J (M)/aHd+1

I ,J (M) is finitely generated, for any a ∈ W̃(I , J). Specially, Hd+1
I ,J (M)/IHd+1

I ,J (M)

is finitely generated.

Proof. Let a ∈ W̃(I , J) be fixed. If d = −1, then the claim is trivial. Now assume that d ≥ 0.

It follows by [6, Theorem 4.7(2)] and Theorem 2, that Hd+1
I ,J (M)/aHd+1

I ,J (M) = 0.

Corollary 3. Let R be local and M a finitely generated module such that dimR M/J M = d. Then

Hd
I ,J(M)/aHd

I ,J(M) is finitely generated, for any a ∈ W̃(I , J). In particular, Hd
I ,J(M)/IHd

I ,J (M) is

finitely generated.

Proof. Let a ∈ W̃(I , J) be fixed. If d = 0, then the claim is trivial. Now assume that d > 0.

It follows by [6, Theorem 4.3] and Theorem 2, that Hd
I ,J(M)/aHd

I ,J(M) = 0.

Proposition 1. Let R be local, M finitely generated and t a non-negative integer. If H i
I ,J(M) is

finitely generated, for all i > t, then H i
I ,J(M) = 0, for all i > t.



REFERENCES 58

Proof. We may assume that I 6= R, otherwise ΓI ,J is identity functor. Proposition 4.10, in

[6], says that H i
I ,J(M) = 0, for all i > ara(IR), where R = R/

p

J +AnnR(M). Let s = ara(IR).

When t ≥ s, there is nothing to prove. Now, assume that t < s. In view of Theorem 2, we

have Hs
I ,J(M)/IHs

I ,J (M) = 0, so Nakayama’s Lemma shows that Hs
I ,J(M) = 0. By keeping this

process, we deduce that H i
I ,J(M) = 0, for all i > t.

Corollary 4. Let R be local with dim R/I + J = 0 and M finitely generated. Then Hd
I ,J(M) is not

finitely generated, where dimR M/J M = d > 0.

Proof. Note that sup{i : H i
I ,J(M) 6= 0}= d , by [6, Theorem 4.5]. Now the claim follows by

Proposition 1.

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