PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 97(111) (2015), 233–238 DOI: 10.2298/PIM140620001P

BASS NUMBERS OF GENERALIZED LOCAL COHOMOLOGY MODULES

Sh. Payrovi, S. Babaei, and I. Khalili-Gorji

Abstract. Let R be a Noetherian ring, M a finitely generated R-module and N an arbitrary R-module. We consider the generalized local cohomology modules, with respect to an arbitrary ideal I of R, and prove that, for all nonnegative integers r, t and all 𝔭 ∈ Spec(R) the Bass number μ^{r} (𝔭, H_{I}^{t}(M, N)) is bounded above by ∑ _{j=0}^{t} μ^{r}(𝔭, Ext_{R}^{t−j} (M, H_{I}^{j} (N))). A corollary is that Ass (H_{I}^{t}(M, N)) ⊆ ⋃ _{j=0}^{t} Ass ( Ext_{R}^{t−j} (M, H_{I}^{j} (N))). In a slightly different direction, we also present some well known results about generalized local cohomology modules.

1. Introduction

The local cohomology theory has been a significant tool in commutative algebra and algebraic geometry. As a generalization of the ordinary local cohomology modules, Herzog [8] introduced the generalized local cohomology modules and these had been studied further by Suzuki [15] and Yassemi [16] and some other authors. They studied some basic duality theorems, vanishing and other properties of generalized local cohomology modules which also generalize several known facts about Ext and ordinary local cohomology modules.

An important problem in commutative algebra is to determine when the Bass numbers of the i-th local cohomology module is finite. In [9] Huneke conjectured that if (R, 𝔪, k) is a regular local ring, then for any prime ideal 𝔭 of R the Bass numbers μ^{i}(𝔭, H_{I}^{j} (R)) = dim_{k(𝔭)} Ext_{R_{𝔭}}^{i}(k(𝔭), H_{IR_{𝔭}}^{j} (R_{𝔭})) are finite for all i and j. There is evidence that this conjecture is true. It is shown by Huneke and Sharp [10] and Lyubeznik [11] that the conjecture holds for regular local ring containing a field. This conjecture is also true for unramified regular local rings of mixed characteristic; this is part of the main theorem of [12]. On the other hand there is a negative answer to the conjecture (over non-regular ring) that is due to Hartshorne [7]. In [5] Dibaei and Yassemi studied the relationship between the Bass numbers of a module and its local cohomology modules. We would like to study the relationship between the Bass numbers of generalized local cohomology modules H_{I}^{i}(M, N) and Ext_{R}^{i}(M, H_{I}^{j} (N)) whenever R is a Noetherian ring, M is a finitely generated R-module and N is an arbitrary R-module.

2010 Mathematics Subject Classification: Primary 13D45; Secondary 14B15.

Key words and phrases: generalized local cohomology, Bass numbers.

Communicated by Žarko Mijajlović.

2. Main results

Throughout this section R is a Noetherian ring, I is an ideal of R, M is a finitely generated R-module, N is an arbitrary R-module and r, t are non-negative integers. For a prime ideal 𝔭 of R the r-th Bass number of M is denoted by μ^{r}(𝔭, M).

The following lemma will be used to prove the main result of this paper.

Lemma 2.1.

μ^{r}(𝔭, H_{I}^{1}(M, N)) ⩽ μ^{r}(𝔭, Ext_{R}^{1}(M, Γ_{I} (N))) + μ^{r}(𝔭, Hom_{R}(M, H_{I}^{1}(N))).

Proof. In view of [3, Corollary 11.1.6] and [6, Lemma 2.1] μ^{r}(𝔭, H_{I}^{t}(M, N)) = μ^{r}(𝔭R_{𝔭}, H_{I}^{t}(M_{𝔭}, N_{𝔭})); also in view of [3, Corollary 4.3.3]

μ^{r}(𝔭, Ext_{R}^{t−j} (M, H_{I}^{j} (N))) = μ^{r}(𝔭R_{𝔭}, Ext_{R_{𝔭}}^{t−j} (M_{𝔭}, H_{IR_{𝔭}}^{j} (N_{𝔭}))).

So, we may assume that R is a local ring with maximal ideal 𝔭. We denote μ^{r}(𝔭, M) by μ^{r}(M) and we have to show that

μ^{r}(H_{I}^{1}(M, N)) ⩽ μ^{r}(Ext_{R}^{1}(M, Γ_{I} (N))) + μ^{r}(Hom_{R}(M, H_{I}^{1}(N))).

By Theorem 11.38 of [14] there is a Grothendieck spectral sequence

E_{2}^{p,q} := Ext_{R}^{p}(M, H_{I}^{q}(N)) ⇒_{p} H_{I}^{p+q} (M, N) = E^{p+q}

and there exists a finite filtration 0 = ϕ^{2}E^{1} ⊆ ϕ^{1}E^{1} ⊆ ϕ^{0}E^{1} = H_{I}^{1}(M, N) such that E_{∞}^{0,1} = E^{1}/ϕ^{1}E^{1} and E_{∞}^{1,0} ≅ ϕ^{1}E^{1}. Thus μ^{r}(E^{1}) ⩽ μ^{r}(E_{∞}^{0,1}) + μ^{r}(E_{∞}^{1,0}).

Now, by the sequence 0 −→ E_{2}^{0,1} −→^{d_{2}^{0,1}} E_{2}^{2,0} we have E_{∞}^{0,1} ≅ E_{3}^{0,1} ≅ kerd_{2}^{0,1}. Also

E_{∞}^{1,0} ≅ E_{2}^{1,0}. Hence, μ^{r}(E^{1}) ⩽ μ^{r}(kerd_{2}^{0,1}) + μ^{r}(E_{2}^{1,0}) ⩽ μ^{r}(E_{2}^{0,1}) + μ^{r}(E_{2}^{1,0}) from which the result follows. □

Theorem 2.1. We have μ^{r}(𝔭, H_{I}^{t}(M, N)) ⩽ ∑ _{j=0}^{t} μ^{r}(𝔭, Ext_{R}^{t−j} (M, H_{I}^{j} (N))).

Proof. The proof which we include for the reader’s convenience, is based on [5, Theorem 2.1]. By the same argument as Lemma 2.1 we may assume that R is a local ring with maximal ideal 𝔭. We have to show that

μ^{r}(H_{I}^{t}(M, N)) ⩽ ∑_{j=0}^{t} μ^{r}(Ext_{R}^{t−j} (M, H_{I}^{j} (N))).

We use induction on t. In the case t = 0, we have H_{I}^{0}(M, N) = Hom_{R}(M, Γ_{I} (N)) so that there is nothing to prove. In the case when t = 1, the claim follows from Lemma 2.1. We therefore assume, inductively, that t > 1 and the result has been proved for smaller values of t. Then the exact sequence 0 −→ Γ_{I} (N) −→ N −→ N/Γ_{I} (N) −→ 0 induces a long exact sequence

⋯−→ H_{I}^{t}(M, Γ_{I} (N)) −→ H_{I}^{t}(M, N) −→ H_{I}^{t}(M, N/Γ_{I} (N)) −→ ⋯

which implies that

μ^{r}(H_{I}^{t}(M, N)) ⩽ μ^{r}(H_{I}^{t}(M, Γ_{I} (N))) + μ^{r}(H_{I}^{t}(M, N/Γ_{I} (N))).

Let E be an injective hull of Γ_{I} (N) and let L = E/Γ_{I} (N). Then by the sequence 0 −→ Γ_{I} (N) −→ E −→ L −→ 0 it follows that H_{I}^{i−1} (M, L)) ≅ H_{I}^{i}(M, Γ_{I} (N)), for all i ⩾ 2. Thus by induction hypothesis for t − 1, we have

μ^{r}(H_{I}^{t−1} (M, L)) ⩽ ∑_{j=0}^{t−1} μ^{r}(Ext_{R}^{t−1−j} (M, H_{I}^{j} (L))) = μ^{r}(Ext_{R}^{t}(M, Γ_{I} (N)))

since H_{I}^{i}(L) ≅ H_{I}^{i+1} (Γ_{I} (N)) = 0, for all i ⩾ 1. Also, the exact sequence 0 −→ Γ_{I} (N) −→ Γ_{I} (E) −→ Γ_{I} (L) −→ 0 shows Ext_{R}^{t−1} (M, H_{I}^{0}(L)) ≅ Ext_{R}^{t}(M, Γ_{I} (N))) because Γ_{I} (E) is an injective R-module. Let E′be an injective hull of N/Γ_{I} (N) and let K = E′/N/Γ_{I} (N). Then by the sequence 0 −→ N/Γ_{I} (N) −→ E′ −→ K −→ 0 it follows that H_{I}^{i−1} (M, K)) ≅ H_{I}^{i}(M, N/Γ_{I} (N)), for all i ⩾ 2. Thus by induction hypothesis for t − 1, we have

μ^{r}(H_{I}^{t−1} (M, K)) ⩽ ∑_{j=0}^{t−1} μ^{r}(Ext_{R}^{t−1−j} (M, H_{I}^{j} (K)))

= ∑_{j=1}^{t−1} μ^{r}(Ext_{R}^{t−1−j} (M, H_{I}^{j} (K))) + μ^{r}(Ext_{R}^{t−1} (M, H_{I}^{0}(K)))

= ∑_{j=1}^{t−1} μ^{r}(Ext_{R}^{t−1−j} (M, H_{I}^{j+1} (N/Γ_{I} (N))))+μ^{r}(Ext_{R}^{t−1} (M, H_{I}^{0}(K)))

= ∑_{j=1}^{t−1} μ^{r}(Ext_{R}^{t−1−j} (M, H_{I}^{j+1} (N))) + μ^{r}(Ext_{R}^{t−1} (M, H_{I}^{0}(K)))

= ∑_{j=2}^{t} μ^{r}(Ext_{R}^{t−j} (M, H_{I}^{j} (N))) + μ^{r}(Ext_{R}^{t−1} (M, H_{I}^{0}(K))).

Now, the exact sequence 0 −→ Γ_{I} (E′) −→ Γ_{I} (K) −→ H_{I}^{1}(N/Γ_{I} (N)) −→ 0 shows that Ext_{R}^{t−1} (M, H_{I}^{0}(K)) ≅ Ext_{R}^{t−1} (M, H_{I}^{1}(N/Γ_{I} (N))) ≅ Ext_{R}^{t−1} (M, H_{I}^{1}(N)). Thus

μ^{r}(H_{I}^{t−1} (M, K)) ⩽ ∑_{j=2}^{t} μ^{r}(Ext_{R}^{t−j} (M, H_{I}^{j} (N))) + μ^{r}(Ext_{R}^{t−1} (M, H_{I}^{1}(N)))

= ∑_{j=1}^{t} μ^{r}(Ext_{R}^{t−j} (M, H_{I}^{j} (N))).

Hence,

μ^{r}(H_{I}^{t}(M, N)) ⩽ μ^{r}(H_{I}^{t}(M, Γ_{I} (N))) + μ^{r}(H_{I}^{t}(M, N/Γ_{I} (N)))

= μ^{r}(H_{I}^{t−1} (M, L)) + μ^{r}(H_{I}^{t−1} (M, K))

⩽ μ^{r}(Ext_{R}^{t}(M, Γ_{I} (N))) + ∑_{j=1}^{t} μ^{r}(Ext_{R}^{t−j} (M, H_{I}^{j} (N)))

which is claimed. □

Corollary 2.1. [13, Theorem 1.1] We have

Ass(H_{I}^{t}(M, N)) ⊆ ⋃_{j=0}^{t} Ass(Ext_{R}^{t−j} (M, H_{I}^{j} (N))).

Proof. Let 𝔭 ∈ Ass(H_{I}^{t}(M, N)). Then μ^{0}(𝔭, H_{I}^{t}(M, N)) ≠ 0. So, by Theorem 2.1 we have ∑ _{j=1}^{t} μ^{0}(𝔭, Ext_{R}^{t−j} (M, H_{I}^{j} (N))) ≠ 0. Hence, there exists 0 ⩽ j ⩽ t such that μ^{0}(𝔭, Ext_{R}^{t−j} (M, H_{I}^{j} (N))) ≠ 0. Therefore, 𝔭 ∈ ⋃ _{j=0}^{t} Ass(Ext_{R}^{t−j} (M, H_{I}^{j} (N))).

□

Definition 2.1. An R-module X is said to be I-cofinite, whenever Supp(X) ⊆ V (I) and Ext_{R}^{i}(R/I, X) is finitely generated R-module, for all i ⩾ 0.

Corollary 2.2. If Supp(M) ⊆ V (I) and H_{I}^{j} (N) is I-cofinite, for all 0 ⩽ j ⩽ t, then μ^{r}(𝔭, H_{I}^{i}(M, N)) < ∞, for all 0 ⩽ i ⩽ t. In particular, if I is an ideal of R with Spec(R) = V (I) and H_{I}^{j} (N) is I-cofinite, for all 0 ⩽ j ⩽ t, then μ^{r}(𝔭, H_{I}^{i}(N)) < ∞, for all 0 ⩽ i ⩽ t.

Proof. In view of [4, Proposition 1] Ext_{R}^{t−j} (M, H_{I}^{j} (N)) is a finitely generated R-module, for all 0 ⩽ j ⩽ t. Now, the claim is obvious by Theorem 2.1. □

Corollary 2.3. If Supp(M) ⊆ V (I) and N is a finitely generated R-module for which H_{I}^{j} (N) is finitely generated, for all 0 ⩽ j < t, then μ^{r}(𝔭, H_{I}^{i}(M, N)) < ∞, for all 0 ⩽ i ⩽ t. In particular, if I is an ideal of R with Spec(R) = V (I) and N is a finitely generated R-module for which H_{I}^{j} (N) is finitely generated, for all 0 ⩽ j < t, then μ^{r}(𝔭, H_{I}^{i}(N)) < ∞, for all 0 ⩽ i ⩽ t.

Proof. In view of [1, Theorem 2.5] and [4, Proposition 1] Ext_{R}^{t−j} (M, H_{I}^{j} (N)) is a finitely generated R-module, for all 0 ⩽ j ⩽ t. Now, the claim follows by Theorem 2.1. □

Corollary 2.4. [2, Proposition 5.2] Let pdM < ∞ and let dimN < ∞, then H_{I}^{t}(M, N) = 0, for all t > pdM + dimN. In particular,

μ^{r}(𝔭, H_{I}^{pd M+dim N} (M, N)) ⩽ μ^{r}(𝔭, Ext_{R}^{pd M} (M, H_{I}^{dim N} (N))).

Proof. By Theorem 2.1 it follows that μ^{0}(𝔭, H_{I}^{t}(M, N)) = 0, for any prime ideal 𝔭 of R and for all t > pdM + dimN. Thus H_{I}^{t}(M, N)) = 0. The second assertion also follows by Theorem 2.1. □

Corollary 2.5. [16, Theorem 2.5] Let pdM < ∞ and let ara(I) < ∞; then H_{I}^{t}(M, N) = 0, for all t > pdM + ara(I), where ara(I), the arithmetic rank of the ideal I, is the least number of elements of R required to generate an ideal which has the same radical as I. In particular,

μ^{r}(𝔭, H_{I}^{pd M+ara(I)} (M, N)) ⩽ μ^{r}(𝔭, Ext_{R}^{pd M} (M, H_{I}^{ara(I)} (N))).

Proof. It follows by the same argument as that of Corollary 2.4. □

Theorem 2.2. We have

μ^{r}(𝔭, Ext_{R}^{t}(M, H_{I}^{0}(N))) ⩽ ∑_{j=2}^{t} μ^{r}(𝔭, Ext_{R}^{t−j} (M, H_{I}^{j−1} (N))) + μ^{r}(𝔭, H_{I}^{t}(M, N)).

Proof. We may assume that R is a local ring with maximal ideal 𝔭. So we have to show that

μ^{r}(Ext_{R}^{t}(M, H_{I}^{0}(N))) ⩽ ∑_{j=2}^{t} μ^{r}(Ext_{R}^{t−j} (M, H_{I}^{j−1} (N))) + μ^{r}(H_{I}^{t}(M, N)).

Theorem 11.38 of [14] shows that there is a Grothendieck spectral sequence

E_{2}^{p,q} := Ext_{R}^{p}(M, H_{I}^{q}(N)) ⇒_{p} H_{I}^{p+q} (M, N) = E^{p+q} .

Now, the exact sequence

⋯−→ E_{2}^{t−2,1} −→^{d_{2}^{t−2,1}} E_{2}^{t,0} −→ 0

and E_{3}^{t,0} = E_{2}^{t,0}/Imd_{2}^{t−2,0} show that μ^{r}(E_{2}^{t,0}) ⩽ μ^{r}(E_{2}^{t−2,1} ) + μ^{r}(E_{3}^{t,0}). Also, the exact sequence

⋯−→ E_{3}^{t−3,2} −→^{d_{3}^{t−3,2}} E_{3}^{t,0} −→ 0

and E_{4}^{t,0} = E_{3}^{t,0}/ Imd_{3}^{t−3,2} show that μ^{r}(E_{3}^{t,0}) ⩽ μ^{r}(E_{4}^{t,0}) + μ^{r}(E_{3}^{t−3,2} ). Hence,

μ^{r}(E_{2}^{t,0}) ⩽ μ^{r}(E_{2}^{t−2,1} ) + μ^{r}(E_{3}^{t−3,2} ) + μ^{r}(E_{4}^{t,0})

⩽ μ^{r}(E_{2}^{t−2,1} ) + μ^{r}(E_{2}^{t−3,2} ) + μ^{r}(E_{4}^{t,0})

⩽ ⋯= ∑_{j=2}^{t} μ^{r}(E_{2}^{t−j,j−1} ) + μ^{r}(E_{t+1}^{t,0} )

= ∑_{j=2}^{t} μ^{r}(E_{2}^{t−j,j−1} ) + μ^{r}(E^{t})

since E_{k}^{t−j,j−1} is a subquotient of E_{2}^{t−j,j−1} , for all 3 ⩽ k ⩽ t, and E_{t+1}^{t,0} is a subquotient of E^{t}, where E_{∞}^{t,0} ≅ ϕ^{t}E^{t}/ϕ^{t+1} E^{t} ≅ ϕ^{t}E^{t} ⊂ E^{t} and

0 = ϕ^{t}E^{t} ⊆ ⋯⊆ ϕ^{1}E^{t} ⊆ ϕ^{0}E^{t} = E^{t}

is a finite filtration. This completes the proof. □

Acknowledgment. The authors are deeply grateful to the referee for his/her careful reading of the manuscript and very helpful suggestions.

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Department of Mathematics (Received 06 11 2013) Imam Khomeini International University Qazvin, Iran

 shpayrovi@sci.ikiu.ac.ir
 sbabaei@educ.ikiu.ac.ir
 i.khalili@edu.ikiu.ac.ir