Artinianness of Local Cohomology Modules Defined by a Pair of Ideals

MALAYSIANMATHEMATICAL

SCIENCESSOCIETY http://math.usm.my/bulletin

Artinianness of Local Cohomology Modules Defined by a Pair of Ideals

SH. PAYROVI ANDM. LOTFIPARSA

Department of Mathematics, Imam Khomeini International University, Qazvin 34149, Iran [email protected], [email protected]

Abstract. LetRbe a commutative Noetherian ring andI,Jtwo ideals ofR. LetMbe a finitely generatedR-module; it is shown that (1) if dimR/(I+J) =0, thenH_I,Jⁱ (M)/JH_I,Jⁱ (M) isI-cofinite Artinian for alli≥0; let dimRM/JM=d(2) ifRis local andSis a non-zero Serre subcategory of the category ofR-modules satisfying the conditionC_I, thenH_I,J^d (M)/

JH_I,J^d (M)∈S(3) ifMhas finite Krull dimension, thenH_I,J^d+1(M)/JH_I,J^d+1(M) =0. Further- more, notion of(I,J)-relative Goldie dimension of modules is defined and it is shown that H_I,Jⁿ (M)/JH_I,Jⁿ (M)is Artinian, wheneverMis aZD-module of dimensionnsuch that the (I,J)-relative Goldie dimension of any quotient ofMis finite.

2010 Mathematics Subject Classification: 13D45, 14B15, 13E10

Keywords and phrases: Artinian modules, Goldie dimension, local cohomology.

1. Introduction

Throughout this paper,Ris a commutative Noetherian ring with non-zero identity,I,Jare two ideals ofRandM is anR-module. For notations and terminologies not given in this paper, the reader is referred to [5, 6] and [12], if necessary.

The local cohomology theory has been an significant tool in commutative Algebra and Algebraic Geometry. As a generalization of the ordinary local cohomology modules, in [12], the authors introduced the local cohomology modules with respect to a pair of ideals.

To be more precise, let W(I,J) ={p∈Spec(R):I^t⊆J+pfor some positive integert}. The set of elementsxofM such that Supp_RRx⊆W(I,J)is said to be(I,J)-torsion submodule ofM and is denoted byΓI,J(M). It is easy to see thatΓI,J is a covariant,R-linear functor from the category ofR-modules to itself. For an integer i, the local cohomology functor H_I,Jⁱ with respect to(I,J)is defined to be thei-th right derived functor ofΓ_I,J. AlsoH_I,Jⁱ (M) is called thei-th local cohomology module ofM with respect to(I,J). IfJ=0, thenH_I,Jⁱ coincides with the ordinary local cohomology functorH_Iⁱ.

Recently, some authors approached the study of local cohomology modules by means of Serre subcategories and it is noteworthy that their approach enables us to deal with several important problems on local cohomology modules comprehensively; see, for example [1–

Communicated byRosihan M. Ali, Dato’.

Received:July 18, 2010;Revised:October 22, 2010.

4]. In this direction, we study the local cohomology modules with respect to a pair of ideals by the notion of Serre subcategory. One of the main results of this paper (Theorem 2.1) is a generalization of [9, Theorem 3.1], and shows that ifM is aZD-module andS a Serre subcategory of the category ofR-modules satisfying the conditionC_I, then the following statements are equivalent: (i)ΓI,J(M/N)/JΓ_I,J(M/N)∈Sfor any submoduleNofM; (ii) H_I,Jⁱ (M/N)/JH_I,Jⁱ (M/N)∈Sfor any submoduleNofMand alli≥0.

Theorem 2.2 in [7] shows that if(R,m)is local,√

I+J=m,Mis a finitely generatedR- module andtis an integer such thatH_I,Jⁱ (M)is Artinian for alli>t, thenH_I,J^t (M)/JH_I,J^t (M) is Artinian. In Corollary 2.2, we improve this theorem by using the above mentioned result and we show that, for any finitely generatedR-moduleM,H_I,Jⁱ (M)/JH_I,Jⁱ (M)is Artinian for alli≥0, whenRis an arbitrary (not necessary local) ring and dimR/(I+J) =0.

As a generalization of the concept ofI-relative Goldie dimension, that is introduced in [9], we say thatM has finite(I,J)-relative Goldie dimension if the Goldie dimension of (I,J)-torsion submodule ofMis finite. LetMbe aZD-module with finite Krull dimension n. It is shown thatH_I,Jⁿ (M)/JH_I,Jⁿ (M)is Artinian, whenever(I,J)-relative Goldie dimension of any quotient ofMis finite.

2. Artinianness ofH_I,Jⁱ (M)

Recall thatRis a Noetherian ring,I,Jare two ideals ofRandMis anR-module. LetZ_R(M) denote the set of zero-divisors ofM.

Definition 2.1. An R-module M is said to be zero-divisor module if for any submodule N of M, the set Z_R(M/N)is a finite union of prime ideals inAss_R(M/N).

According to [9, Example 2.2], the class of zero-divisor modules (ZD-modules) contains finitely generated, Laskerian [11], weakly Laskerian [10], linearly compact and Matlis re- flexive modules. Also it contains modules whose quotients have finite Goldie dimension and modules with finite support, in particular Artinian modules.

Definition 2.2. A full subcategory of the category of R-modules is said to be Serre subcate- gory, if it is closed under taking submodules, quotients and extensions. A Serre subcategory S is said to be satisfy the condition C_I if for any I-torsion R-module M,0 :_MI∈S implies that M∈S.

Examples 2.4 and 2.5 in [1] show that the class of zero modules, Artinian modules,I- cofinite Artinian modules, modules with finite support and the class ofR-modulesM with dim_RM ≤t, wheret is a non-negative integer are Serre subcategories of the category of R-modules satisfy the conditionCI.

In the rest of the paper, S denotes a Serre subcategory of the category of R-modules satisfying the conditionC_I. The following result is a generalization of [9, Theorem 3.1].

Theorem 2.1. Let M be a ZD-module such thatΓI,J(M/N)/JΓ_I,J(M/N)∈S for any sub- module N of M. Then H_I,Jⁱ (M/N)/JH_I,Jⁱ (M/N)∈S for any submodule N of M and all i≥0.

Proof. We may assume thatIis not zero, this can be done simply becauseΓ_I,J is identity functor whenI=0. We use induction oni. The casei=0 is trivial by assumption. So assume, inductively, thati>0 and we have shown thatH_I,Jⁱ⁻¹(M⁰/N⁰)/JH_I,Jⁱ⁻¹(M⁰/N⁰)∈S for anyZD-moduleM⁰and any submoduleN⁰ofM⁰. Now letMbe a ZD-module,Na sub- module ofMandX =M/N. ThenH_I,Jⁱ (X/Γ_I,J(X))∼=H_I,Jⁱ (X)by [12, Corollary 1.13(4)].

Also X/Γ_I,J(X)is a (ZD-module) (I,J)-torsion free R-module. We therefore assume in addition that X is an (I,J)-torsion free R-module. We now use [9, Lemma 2.4] to de- duce thatIcontains an element awhich is a non zero-divisor onX. The exact sequence 0−→X−→^a X−→X/aX−→0 induces an exact sequence

· · · −→H_I,Jⁱ⁻¹(X/aX)−→H_I,Jⁱ (X)−→^a H_I,Jⁱ (X)−→H_I,Jⁱ (X/aX)−→ · · · of local cohomology modules. So we have the exact sequence

H_I,Jⁱ⁻¹(X/aX)/JH_I,Jⁱ⁻¹(X/aX)→H_I,Jⁱ (X)/JH_I,Jⁱ (X)→^a aH_I,Jⁱ (X)/aJH_I,Jⁱ (X)→0.

SinceX/aX∼=M/(aM+N)is aZD-module, it follows from the inductive hypothesis that H_I,Jⁱ⁻¹(X/aX)/JH_I,Jⁱ⁻¹(X/aX)∈S. So the above exact sequence shows that theR-module 0 :_Hi

I,J(X)/JH_I,Jⁱ (X)a∈S. Hence,H_I,Jⁱ (X)/JH_I,Jⁱ (X)∈Sby [1, Lemma 2.3]. This completes the inductive step. The result follows by induction.

The following result is an improvement of [7, Theorem 2.2].

Corollary 2.1. Let(R,m) be local, √

I+J=m, S non-zero and M a finitely generated R-module. Then H_I,Jⁱ (M/N)/JH_I,Jⁱ (M/N)∈S for any submodule N of M and all i≥0.

Proof. In view of Theorem 2.1, it is enough to show thatΓI,J(M/N)/JΓ_I,J(M/N)∈Sfor any submoduleNofM. Assume thatNis a submodule ofM; [12, Proposition 1.4] shows that

ΓI,J(M/N) =ΓI+J,J(M/N) =Γ^√_I+J,J(M/N) =Γm,J(M/N).

SinceΓm,J(M/N)/JΓ_m,J(M/N)is a finitely generatedR-module and annihilated by a power ofm; henceΓm,J(M/N)/JΓ_m,J(M/N)has finite length. So by [4, Lemma 2.11], we have Γm,J(M/N)/JΓ_m,J(M/N)∈S.

The following corollary improves Corollary 2.1, when S is considered the class of I- cofinite Artinian modules.

Corollary 2.2. Let dimR/(I+J) =0 and M be a finitely generated R-module. Then H_I,Jⁱ (M)/JH_I,Jⁱ (M)is I-cofinite Artinian for all i≥0.

Proof. The proof is similar to that of Corollary 2.1.

LetRbe local,Snon-zero andMa finitely generatedR-module of dimensionn. Then by using the method of proof of [5, Theorem 7.1.6], one can see thatH_Iⁿ(M)∈Sby [4, Lemma 2.11]. Having this in mind, we get the following theorem which is a generalization of [7, Theorem 2.3].

Theorem 2.2. Let R be local, S non-zero and M a finitely generated R-module withdim_RM/

JM=d. Then H_I,J^d (M)/JH_I,J^d (M)∈S.

Proof. When dim_RM=−1, there is nothing to prove, as thenM =0. We argue by in- duction on dim_RM. If dim_RM=0, thenMhas finite length. ThusΓI,J(M)/JΓ_I,J(M)has finite length. So the result follows by [4, Lemma 2.11]. Now suppose, inductively, that dim_RM=n>0, and the result has been proved for allR-modules of dimensions smaller thannsatisfying the hypothesis. The exact sequence

(2.1) 0−→ΓJ(M)−→M−→M/Γ_J(M)−→0

induces the long exact sequence

(2.2) · · · →H_I,Jⁱ (Γ_J(M))→H_I,Jⁱ (M)→H_I,Jⁱ (M/Γ_J(M))→H_I,Jⁱ⁺¹(Γ_J(M))→ · · ·. By [12, Corollary 2.5],H_I,Jⁱ (Γ_J(M))∼=H_Iⁱ(Γ_J(M)), for alli≥0, sinceΓ_J(M)isJ-torsion.

On the other hand, dim_RΓ_J(M)≤dim_RM/JM=d. ThusH_I^d(Γ_J(M))∈Sby the previous paragraph andH_I^d+1(Γ_J(M)) =0. Therefore,H_I,J^d (Γ_J(M))∈SandH_I,J^d+1(Γ_J(M)) =0. Now by the exact sequence

H_I,J^d (Γ_J(M))/JH_I,J^d (Γ_J(M))−→H_I,J^d (M)/JH_I,J^d (M)

−→H_I,J^d (M/Γ_J(M))/JH_I,J^d (M/Γ_J(M))−→0 we only have to show thatH_I,J^d (M/Γ_J(M))/JH_I,J^d (M/Γ_J(M))∈S. We have

(2.3) dim_R(M/Γ_J(M))/J(M/Γ_J(M)) =dim_RM/(JM+ΓJ(M))≤dim_RM/JM=d. So, in view of [12, Theorem 4.3], we may assume thatΓ_J(M) =0. So the idealJcontains an elementawhich is a non zero-divisor onM. The exact sequence 0−→M−→^a M−→

M/aM−→0 induces the exact sequence

· · · −→H_I,J^d (M)−→^a H_I,J^d (M)−→H_I,J^d (M/aM)−→0 of local cohomology modules, see [12, Theorem 4.3]. Now the exact sequence

H_I,J^d (M)/JH_I,J^d (M)−→^a H_I,J^d (M)/JH_I,J^d (M)−→H_I,J^d (M/aM)/JH_I,J^d (M/aM)−→0 shows that

H_I,J^d (M/aM)/JH_I,J^d (M/aM)∼=H_I,J^d (M)/(J+Ra)H_I,J^d (M) =H_I,J^d (M)/JH_I,J^d (M).

We have dim_RM/aM=n−1 and

dim_R(M/aM)/J(M/aM) =dim_RM/(J+Ra)M=dim_RM/JM=d.

Thus, by the inductive hypothesisH_I,J^d (M/aM)/JH_I,J^d (M/aM)∈S. This completes the in- ductive step.

Letkbe a field andR=k[x]the polynomials ring in an indeterminatex, with coefficients ink. LetI= (x−1)andJ=I∩(x) = (x²−x). Then one has dim_RR/J=0 andH_I,J¹ (R)6=0;

see [12, Remark 4.6 (2)]. Nevertheless, we have the following result.

Theorem 2.3. Let M be a finitely generated R-module of finite Krull dimension. Ifdim_RM/

JM=d, then H_I,J^d+1(M)/JH_I,J^d+1(M) =0.

Proof. IfJM=M, then(1+a)M=0 for somea∈Jby Nakayama’s Lemma. ThusJx=Rx for all x∈M and soM is (I,J)-torsion. Hence, ΓI,J(M)/JΓ_I,J(M) =0. Now suppose that d ≥0. We use induction on dim_RM. If dim_RM=0, then H_I,J¹ (M)/JH_I,J¹ (M) =0 by [12, Theorem 4.7(1)]. So assume, inductively, that dim_RM=n>0 and we established the result for R-modules of dimension smaller thann satisfying the hypothesis. By an- other using of [12, Theorem 4.7(1)], we have H_I,J^d+1(Γ_J(M)) =H_I,J^d+2(Γ_J(M)) =0 since dim_RΓJ(M)≤dim_RM/JM=d. Therefore, the exact sequence (2.2) shows thatH_I,J^d+1(M)∼= H_I,J^d+1(M/Γ_J(M)). Hence, it is enough to show thatH_I,J^d+1(M/Γ_J(M))/JH_I,J^d+1(M/Γ_J(M)) = 0. Also by using the exact sequence (2.3) and [12, Theorem 4.7(2)], we may assume ΓJ(M) =0. The argument now proceeds like that used in the proof of Theorem 2.2.

Now we get some results on the finiteness of the support of the local cohomology mod- ules.

Corollary 2.3. Let M be a ZD-module such thatΓI,J(M/N)/JΓ_I,J(M/N)has finite sup- port for any submodule N of M. Then H_I,Jⁱ (M/N)/JH_I,Jⁱ (M/N)has finite support for any submodule N of M and all i≥0.

Proof. Apply Theorem 2.1 and the fact that the class of modules with finite support is a Serre subcategory of the category ofR-modules satisfying the conditionC_I.

Corollary 2.4. Let R be local and M a finitely generated R-module such that for any sub- module N of M and for all p∈Supp_RΓI(M/N), dim_RR/p≤1. Then H_Iⁱ(M) has finite support for all i≥0.

Proof. In view of [8, Corollary 4.3],ΓI(M/N)has finite support, for any submodulesNof M. Now the result follows by Corollary 2.3.

3. Goldie dimension and Artinianness ofH_I,Jⁱ (M)

For an R-moduleM, the Goldie dimension of M is defined as the cardinal of the set of indecomposable submodules ofER(M), which appear in a decomposition ofER(M)into direct sum of indecomposable submodules. We shall use GdimM to denote the Goldie dimension ofM. Letµ⁰(p,M)denote the 0-th Bass number ofMwith respect to prime ideal p. It is clear that GdimM=∑p∈spec(R)µ⁰(p,M). In [9], the authors, offered a generalization of the notion of Goldie dimension and introduced the concept ofI-relative Goldie dimension ofMas Gdim_IM=∑p∈V(I)µ⁰(p,M), where V(I)denotes the set of prime ideals ofRwhich are containingI. We first generalize this concept as follows.

Definition 3.1. Let I, J be two ideals of R. For an R-module M, we define (I,J)-relative Goldie dimension of M asGdim_I,JM=∑p∈W(I,J)µ⁰(p,M). HereW(I,J)denotes the set of prime idealspof R such that I^t⊆p+J for some positive integer t.

It is easy to see that finitely generated modules, Artinian modules, quotients of the Matlis reflexive modules and quotients of the linearly compact modules have finite(I,J)-relative Goldie dimension, see [9, Example 2.2]. Also it is clear that ifJ=0, then W(I,J) =V(I) and so Gdim_I,JM=Gdim_IM. Moreover

Gdim_IM≤Gdim_I,JM≤GdimM.

But the following example shows that these inequalities may be strict. LetI=2Z,J=3Z andM=Z/2Z⊕Z/3Z⊕Z/5Z. Then V(I) ={2Z}, W(I,J)∩Ass_ZM={2Z,5Z} and E_Z(M) =E_Z(Z/2Z)⊕E_Z(Z/3Z)⊕E_Z(Z/5Z). Therefore Gdim_IM =1, Gdim_I,JM =2 and GdimM=3.

Theorem 3.1. Let M be a ZD-module such thatGdim_IRq,JRqM_qis finite, for any prime ideal qwhich is maximal inAss_RM. ThenGdim_I,JM is finite.

Proof. Let{q₁,q₂, . . . ,q_t}be the set of all prime ideals with the property being maximal in Ass_RM; note that this set is finite by [9, Lemma 2.3]. It is easy to see that ifp∈W(I,J)and p⊆q, thenpRq∈W(IRq,JRq), whereqis an arbitrary prime ideal ofR. Thus

Gdim_I,JM=

∑

p∈W(I,J)

µ⁰(p,M)≤

t

i=1

∑ ∑

p∈W(I,J),p⊆q_i

µ⁰(p,M)

≤

t

i=1

∑ ∑

pRqi∈W(IR_q

i,JRqi)

µ⁰(pR_qi,M_qi)

=

t

∑

i=1

Gdim_IRq

i,JRqiM_qi so the claim follows.

In the following we show that, for anyR-moduleM,(I,J)-relative Goldie dimension of M is equal to Goldie dimension of its(I,J)-torsion submodule. Precisely, we shall show that:

Lemma 3.1. If M is an R-module, thenGdim_I,JM=GdimΓ_I,J(M).

Proof. LetE_R(M)∼=⊕_p∈spec(R)µ⁰(p,M)E_R(R/p)be a decomposition ofE_R(M)as the direct sum of indecomposable injectiveR-modules, whereE_R(R/p)denotes the injective hull of R/pandµ⁰(p,M)denotes the 0-th Bass number ofMwith respect to prime idealp. Then by using [12, Proposition 1.11], we haveΓI,J(E_R(M))∼=⊕_p∈W(I,J)µ⁰(p,M)E_R(R/p)and so it is an injectiveR-module. We have to show thatΓI,J(E_R(M))is an essential extension of ΓI,J(M). Suppose xbe a non-zero element of ΓI,J(E_R(M)). Thus there existsr∈R and a positive integert such that I^tx⊆Jxand 06=rx∈M∩Rx. So that I^t(rx)⊆J(rx) and 06=rx∈ΓI,J(M)∩Rx. Hence,ΓI,J(E_R(M))is an injective essential extension ofΓI,J(M).

Therefore we haveER(Γ_I,J(M))∼=ΓI,J(E_R(M))and so Gdim_I,JM=

∑

p∈W(I,J)

µ⁰(p,M) =GdimΓ_I,J(M).

The following result is a generalization of [9, Corollary 3.3(ii)].

Theorem 3.2. Let M be a ZD-module of dimension n such that (I,J)-relative Goldie di- mension of any quotient of M is finite. Then H_I,Jⁿ (M)/JH_I,Jⁿ (M)is Artinian.

Proof. The proof, which we include for the reader’s convenience, proceeds like that used in the proof of Theorem 2.2. We use induction onn. Ifn=0, then Ass_RΓ_I,J(M)⊆Ass_RM⊆ Max(R). HenceE_R(Γ_I,J(M))is a finite direct sum ofE_R(R/m), wheremis a maximal ideal ofR. ThereforeE_R(ΓI,J(M))and soΓI,J(M)/JΓI,J(M)is Artinian. We therefore assume, inductively, thatn>0 and the result has been proved for anyR-module of dimension less thannsatisfying the hypothesis. The exact sequence (2.1) induces the long exact sequence

· · · →H_I,Jⁿ (Γ_J(M))→H_I,Jⁿ (M)→H_I,Jⁿ (M/Γ_J(M))→H_I,Jⁿ⁺¹(Γ_J(M))→ · · ·. By [12, Corollary 2.5],H_I,Jⁱ (Γ_J(M))∼=H_Iⁱ(Γ_J(M)), for alli≥0, sinceΓJ(M)isJ-torsion.

On the other hand, we have dim_RΓJ(M)≤dim_RM =n, thusH_Iⁿ(Γ_J(M))is Artinian and H_Iⁿ⁺¹(Γ_J(M)) =0. HenceH_I,Jⁿ (Γ_J(M))is Artinian and H_I,Jⁿ⁺¹(Γ_J(M)) =0. Now by the exact sequence

H_I,Jⁿ (Γ_J(M))/JH_I,Jⁿ (Γ_J(M))−→H_I,Jⁿ (M)/JH_I,Jⁿ (M)

−→H_I,Jⁿ (M/Γ_J(M))/JH_I,Jⁿ (M/Γ_J(M))−→0

we can assume thatΓJ(M) =0. ThusJcontains an elementawhich is a non zero-divisor onM, by [9, Lemma 2.4]. Since dim_RM/aM≤n−1, thus it follows either from inductive hypothesis or from [12, Theorem 3.2], and Grothendieck’s Vanishing Theorem [5, Theorem

6.1.2], that H_I,Jⁿ⁻¹(M/aM)/JH_I,Jⁿ⁻¹(M/aM)is Artinian. The exact sequence 0−→M−→^a M−→M/aM−→0 induces the exact sequence

H_I,Jⁿ⁻¹(M/aM)/JH_I,Jⁿ⁻¹(M/aM)→H_I,Jⁿ (M)/JH_I,Jⁿ (M)→^a H_I,Jⁿ (M)/JH_I,Jⁿ (M)→0.

Now we have 0 :_Hⁿ

I,J(M)/JH_I,Jⁿ (M)a is Artinian and soH_I,Jⁿ (M)/JH_I,Jⁿ (M)is Artinian by [1, Lemma 2.3]. This completes the inductive step.

Corollary 3.1. Let M be a finitely generated R-module of dimension n. Then H_I,Jⁿ (M)/

JH_I,Jⁿ (M)is Artinian.

Acknowledgement.The authors are deeply grateful to the referees for their careful reading and many helpful suggestions.

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