ARTINIANNESS OF LOCAL COHOMOLOGY MODULES DEFINED BY A PAIR OF IDEALS

DEFINED BY A PAIR OF IDEALS

Sh. Payrovi and M. Lotfi Parsa

Department of Mathematics, I. K. International University, Postal Code: 34149-1-6818, Qazvin, Iran

[email protected] [email protected]

ABSTRACT. Let R be a commutative Noetherian ring and I, J two ideals of R.

Let M be a finitely generated R-module; it is shown that (1) if dim R/(I+J) =0, then H_I,Jⁱ (M)/JH_I,Jⁱ (M)is I-cofinite Artinian for all i≥0; let dim_RM/JM=d (2) if R is local and S is a non-zero Serre subcategory of the category of R-modules satisfying the condition C_I, then H_I,J^d (M)/JH_I,J^d (M)∈S (3) if M has finite Krull dimension, then H_I,J^d+1(M)/JH_I,J^d+1(M) =0. Furthermore, 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, whenever M is a ZD-module of dimension n such that the(I,J)-relative Goldie dimension of any quotient of M is finite.

1. INTRODUCTION

Throughout this paper, R is a commutative Noetherian ring with non-zero iden- tity, 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 [5], [6] and [12], if necessary.

The local cohomology theory has been an significant tool in commutative Alge- bra 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 integer t}. The set of elements x of M such that Supp_R(Rx)⊆W(I,J) is said to be (I,J)-torsion submodule of M and is denoted by Γ_I,J(M). It is easy to see that Γ_I,J is a covariant, R-linear functor from the category of R-modules to

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

Key words and Phrases. Artinian modules, Goldie dimension, Local cohomology.

1

itself. For an integer i, the local cohomology functor H_I,Jⁱ with respect to(I,J) is defined to be the i-th right derived functor of Γ_I,J. Also H_I,Jⁱ (M)is called the i-th local cohomology module of M with respect to (I,J). If J=0, then H_I,Jⁱ coincides with the ordinary local cohomology functor H_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 comprehen- sively; see, for example [1], [2], [3] and [4]. In this direction, we study the local cohomology modules with respect to a pair of ideals by the notion of Serre subcat- egory. One of the main results of this paper (2.3) is a generalization of Theorem 3.1, in [9], and shows that if M is a ZD-module and S a Serre subcategory of the category of R-modules satisfying the condition C_I, then the following statements are equivalent: (i) Γ_I,J(M/N)/JΓ_I,J(M/N)∈S for any submodule N of M; (ii) H_I,Jⁱ (M/N)/JH_I,Jⁱ (M/N)∈S for any submodule N of M and all i≥0.

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

I+J =m, M is a finitely generated R-module and t is an integer such that H_I,Jⁱ (M)is Artinian for all i>t, then H_I,J^t (M)/JH_I,J^t (M)is Artinian. In 2.5, we improve this theorem by using the above mentioned result and we show that, for any finitely generated R-module M, H_I,Jⁱ (M)/JH_I,Jⁱ (M)is Artinian for all i≥0, when R is an arbitrary (not necessary local) ring and dim R/(I+J) =0.

As a generalization of the concept of I-relative Goldie dimension, that is intro- duced in [9], we say that M has finite(I,J)-relative Goldie dimension if the Goldie dimension of(I,J)-torsion submodule of M is finite. Let M be a ZD-module with finite Krull dimension n. It is shown that H_Iⁿ_,J(M)/JH_I,Jⁿ (M)is Artinian, whenever (I,J)-relative Goldie dimension of any quotient of M is finite.

2. ARTINIANNESS OFH_Iⁱ_,J(M)

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

Let Z_R(M)denote the set of zero-divisors of M.

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

According to Example 2.2, in [9], the class of zero-divisor modules (ZD-modules) contains finitely generated, Laskerian [11], weakly Laskerian [10], linearly compact and Matlis reflexive 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 subcategory, 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 :_M I)∈S implies that M∈S.

Examples 2.4 and 2.5, in [1], show that the class of zero modules, Artinian mod- ules, I-cofinite Artinian modules, modules with finite support and the class of R- modules M with dim_RM ≤t, where t is a non-negative integer are Serre subcate- gories of the category of R-modules satisfy the condition C_I.

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

Theorem 2.3. Let M be a ZD-module such that Γ_I,J(M/N)/JΓ_I,J(M/N)∈S for any submodule 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 that I is not zero, this can be done simply because Γ_I,J is identity functor when I =0. We use induction on i. The case i =0 is triv- ial by assumption. So assume, inductively, that i >0 and we have shown that H_I,Jⁱ⁻¹(M⁰/N⁰)/JH_I,Jⁱ⁻¹(M⁰/N⁰)∈S for any ZD-module M⁰ and any submodule N⁰ of M⁰. Now let M be a ZD-module, N a submodule of M and X =M/N. Then H_I,Jⁱ (X/Γ_I,J(X))∼=H_I,Jⁱ (X)by Corollary 1.13(4), in [12]. 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 Lemma 2.4, in [9], to deduce that I contains an element a which is a non zero-divisor on X . The exact sequence 0−→X −→^a X−→X/aX−→0 induces an exact sequence

· · · −→H_Iⁱ⁻¹_,J (X/aX)−→H_I,Jⁱ (X)−→^a H_Iⁱ_,J(X)−→H_I,Jⁱ (X/aX)−→ · · ·

of local cohomology modules. So we have the exact sequence

H_Iⁱ⁻¹_,J (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.

Since X/aX ∼=M/(aM+N)is a ZD-module, it follows from the inductive hypoth- esis that H_Iⁱ⁻¹_,J (X/aX)/JH_I,Jⁱ⁻¹(X/aX)∈S. So the above exact sequence shows that the R-module 0 :

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

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

Corollary 2.4. Let(R,m)be local,√

I+J=m, S non-zero and M a finitely gener- ated 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 2.3, it is enough to show that Γ_I,J(M/N)/JΓ_I,J(M/N)∈S for any submodule N of M. Assume that N is a submodule of M; Proposition 1.4, in [12], shows that Γ_I,J(M/N) =Γ_I+J,J(M/N) = Γ^√_I+J,J(M/N) = Γ_m,J(M/N).

Since Γ_m,J(M/N)/JΓ_m,J(M/N) is a finitely generated R-module and annihilated by a power ofm; henceΓ_m,J(M/N)/JΓ_m,J(M/N)has finite length. So by Lemma 2.11, in [4], we haveΓ_m,J(M/N)/JΓ_m,J(M/N)∈S.

The following corollary improves 2.4, when S is considered the class of I-cofinite Artinian modules.

Corollary 2.5. Let dim R/(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 2.4.

Let R be local, S non-zero and M a finitely generated R-module of dimension n. Then by using the method of proof of Theorem 7.1.6, in [5], one can see that H_Iⁿ(M)∈S by Lemma 2.11, in [4]. Having this in mind, we get the following theorem which is a generalization of Theorem 2.3, in [7].

Theorem 2.6. Let R be local, S non-zero and M a finitely generated R-module with dim_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 then M = 0. We ar- gue by induction on dim_RM. If dim_RM = 0, then M has finite length. Thus Γ_I,J(M)/JΓ_I,J(M) has finite length. So the result follows by Lemma 2.11, in [4].

Now suppose, inductively, that dim_RM =n>0, and the result has been proved for all R-modules of dimensions smaller than n satisfying the hypothesis. The ex- act sequence 0−→Γ_J(M)−→M−→M/Γ_J(M)−→0(1)induces the long exact sequence

· · · →H_I,Jⁱ (Γ_J(M))→H_I,Jⁱ (M)→H_I,Jⁱ (M/Γ_J(M))→H_I,Jⁱ⁺¹(Γ_J(M))→ · · ·.(2) By Corollary 2.5, in [12], H_I,Jⁱ (Γ_J(M))∼=H_Iⁱ(Γ_J(M)), for all i≥0, sinceΓ_J(M)is J-torsion. On the other hand, dim_RΓ_J(M)≤dim_RM/JM=d. Thus H_I^d(Γ_J(M))∈S by the previous paragraph and H_I^d+1(Γ_J(M)) =0. Therefore, H_I,J^d (Γ_J(M))∈S and H_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 that H_I,J^d (M/Γ_J(M))/JH_I,J^d (M/Γ_J(M))∈S. We have

dim_R(M/Γ_J(M))/J(M/Γ_J(M)) =dim_RM/(JM+Γ_J(M))≤dim_RM/JM=d.(3) So, in view of Theorem 4.3, in [12], we may assume thatΓ_J(M) =0. So the ideal J contains an element a which is non zero-divisor on M. The exact sequence 0−→

M−→^a M−→M/aM−→0 induces the exact sequence

· · · −→H_I,J^d (M)−→^a H_I^d_,J(M)−→H_I,J^d (M/aM)−→0

of local cohomology modules, see Theorem 4.3, in [12]. Now the exact sequence H_I^d_,J(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^d_,J(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 hypothesis H_I^d_,J(M/aM)/JH_I,J^d (M/aM)∈

S. This completes the inductive step.

Let k be a field and R =k[x] the polynomials ring in an indeterminate x, with coefficients in k. Let I= (x−1)and J=I∩(x) = (x²−x). Then one has dim_RR/J= 0 and H_I,J¹ (R)6=0; see Remark 4.6 (2), in [12]. Nevertheless, we have the following result.

Theorem 2.7. Let M be a finitely generated R-module of finite Krull dimension. If dim_RM/JM=d, then H_I,J^d+1(M)/JH_I,J^d+1(M) =0.

Proof. If JM =M, then (1+a)M =0 for some a∈ J by Nakayama’s Lemma.

Thus Jx=Rx for all x∈M and so M is(I,J)-torsion. Hence,Γ_I,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 Theorem 4.7(1), in [12]. So assume, inductively, that dim_RM=n>0 and we established the result for R-modules of dimension smaller than n satisfying the hypothesis. By another using of Theorem 4.7(1), in [12], we have H_I,J^d+1(Γ_J(M)) =H_I^d+2_,J (Γ_J(M)) =0 since dim_RΓ_J(M)≤dim_RM/JM=d.

Therefore, the exact sequence (2) shows that H_I,J^d+1(M)∼=H_I,J^d+1(M/Γ_J(M)). Hence, it is enough to show that H_I,J^d+1(M/Γ_J(M))/JH_I,J^d+1(M/Γ_J(M)) =0. Also by using (3) and Theorem 4.7(2), in [12], we may assume Γ_J(M) =0. The argument now

proceeds like that used in the proof of 2.6.

Now we get some results on the finiteness of the support of the local cohomology modules.

Corollary 2.8. Let M be a ZD-module such that Γ_I,J(M/N)/JΓ_I,J(M/N) has fi- nite support for any submodule N of M. Then H_Iⁱ_,J(M/N)/JH_Iⁱ_,J(M/N) has finite support for any submodule N of M and all i≥0.

Proof. Apply 2.3 and the fact that the class of modules with finite support is a Serre subcategory of the category of R-modules satisfying the condition C_I. Corollary 2.9. Let R be local and M a finitely generated R-module such that for any submodule N of M and for allp∈Supp_RΓ_I(M/N), dim_RR/p≤1. Then H_Iⁱ(M) has finite support for all i≥0.

Proof. In view of Corollary 4.3, in [8], Γ_I(M/N) has finite support, for any sub-

modules N of M. Now the result follows by 2.8.

3. GOLDIE DIMENSION ANDARTINIANNESS OFH_I,Jⁱ (M)

For an R-module M, the Goldie dimension of M is defined as the cardinal of the set of indecomposable submodules of E_R(M), which appear in a decomposition of E_R(M) into direct sum of indecomposable submodules. We shall use GdimM to denote the Goldie dimension of M. Let µ⁰(p,M) denote the 0-th Bass number of M with 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 of I-relative Goldie dimension of M as Gdim_IM =

∑_p∈V(I)µ⁰(p,M), where V(I)denotes the set of prime ideals of R which are con- taining I. 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 as Gdim_I,JM=∑_p∈W(I,J)µ⁰(p,M). Here W(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 Example 2.2, in [9]. Also it is clear that if J=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. Let I=2Z, J =3Zand M =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 3.2. Let M be a ZD-module such that Gdim_IR

q,JRqM_q is finite, for any prime idealqwhich is maximal in Ass_RM. Then Gdim_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 Lemma 2.3, in [9]. It is easy to see that ifp∈W(I,J)andp⊆q, thenpR_q∈W(IR_q,JR_q), whereqis an arbitrary prime

ideal of R. Thus

Gdim_I,JM =

∑

p∈W(I,J)

µ⁰(p,M)≤

∑

i=1

∑

p∈W(I,J),p⊆q_i

µ⁰(p,M)

≤

∑

i=1

∑

pR_q

i∈W(IR_q

i,JR_q

i)

µ⁰(pR_q

i,M_q

i)

=

∑

Gdim_IR

qi,JR_q

i

M_q

i

so the claim follows.

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

Lemma 3.3. If M is an R-module, then Gdim_I,JM=GdimΓ_I,J(M).

Proof. Let E_R(M)∼=⊕_p∈spec(R)µ⁰(p,M)E_R(R/p)be a decomposition of E_R(M)as the direct sum of indecomposable injective R-modules, where E_R(R/p)denotes the injective hull of R/pandµ⁰(p,M)denotes the 0-th Bass number of M with respect to prime idealp. Then by using Proposition 1.11, in [12], we haveΓ_I,J(E_R(M))∼=

⊕_p∈W(I,J)µ⁰(p,M)E_R(R/p) and so it is an injective R-module. We have to show that Γ_I,J(E_R(M)) is an essential extension of Γ_I,J(M). Suppose x be a non-zero element of Γ_I,J(E_R(M)). Thus there exists r∈R and a positive integer t such that I^tx⊆Jx and 06=rx∈M∩Rx. So that I^t(rx)⊆J(rx)and 0 6=rx∈Γ_I,J(M)∩Rx.

Hence, Γ_I,J(E_R(M)) is an injective essential extension of Γ_I,J(M). Therefore we have E_R(Γ_I,J(M))∼=Γ_I,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 Corollary 3.3(ii), in [9].

Theorem 3.4. Let M be a ZD-module of dimension n such that(I,J)-relative Goldie dimension 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 2.6. We use induction on n. If n=0, then Ass_RΓ_I,J(M)⊆

Ass_RM⊆Max(R). Hence E_R(Γ_I,J(M)) is a finite direct sum of E_R(R/m), where mis a maximal ideal of R. Therefore E_R(Γ_I,J(M))and soΓ_I,J(M)/JΓ_I,J(M)is Ar- tinian. We therefore assume, inductively, that n>0 and the result has been proved for any R-module of dimension less than n satisfying the hypothesis. The exact sequence (1) induces the long exact sequence

· · · →H_I,Jⁿ (Γ_J(M))→H_I,Jⁿ (M)→H_Iⁿ_,J(M/Γ_J(M))→H_Iⁿ⁺¹_,J (Γ_J(M))→ · · ·. By Corollary 2.5, in [12], H_I,Jⁱ (Γ_J(M))∼=H_Iⁱ(Γ_J(M)), for all i≥0, sinceΓ_J(M)is J- torsion. On the other hand, we have dim_RΓ_J(M)≤dim_RM=n, thus H_Iⁿ(Γ_J(M))is Artinian and H_Iⁿ⁺¹(Γ_J(M)) =0. Hence H_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 thatΓ_J(M) =0. Thus J contains an element a which is a non zero- divisor on M, by Lemma 2.4, in [9]. Since dim_RM/aM ≤n−1, thus it follows either from inductive hypothesis or from Theorem 3.2, in [12], and Grothendieck’s Vanishing Theorem, Theorem 6.1.2, in [5], 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_Iⁿ_,J(M)→0.

Now we have 0 :_Hn

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

Corollary 3.5. Let M be a finitely generated R-module of dimension n. Then H_I,Jⁿ (M)/JH_I,Jⁿ (M)is Artinian.

4. ACKNOWLEDGMENT

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

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