(1)BASS NUMBERS OF GENERALIZED LOCAL COHOMOLOGY MODULES Sh

BASS NUMBERS OF GENERALIZED LOCAL COHOMOLOGY MODULES Sh. Payrovi, S. Babaei, and I. Khalili-Gorji

Abstract. Let Rbe a Noetherian ring, M a finitely generated R-module andN an arbitraryR-module. We consider the generalized local cohomology modules, with respect to an arbitrary idealIofR, and prove that, for all non- negative integersr, tand allp∈ Spec(R) the Bass numberµ^r(p, H_I^t(M, N)) is bounded above by Pt

j=0µ^r p,Ext^t−j_R (M, H_I^j(N)). A corollary is that Ass H_I^t(M, N)

⊆St

j=0Ass Ext^t_R^−j(M, H_I^j(N))

.In a slightly different di- rection, we also present some well known results about generalized local coho- mology modules.

1. Introduction

The local cohomology theory has been a significant tool in commutative alge- bra 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 gen- eralized 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 idealp of R the Bass numbers µⁱ(p, H_I^j(R)) = dimk(p)Extⁱ_Rp(k(p), H_IR^j

p(Rp)) 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 Harts- horne [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

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

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

Communicated by Žarko Mijajlović.

233

the relationship between the Bass numbers of generalized local cohomology modules H_Iⁱ(M, N) and Extⁱ_R(M, H_I^j(N)) wheneverR is a Noetherian ring,M is a finitely generated R-module andN is an arbitraryR-module.

2. Main results

Throughout this sectionRis a Noetherian ring,Iis an ideal ofR,M is a finitely generatedR-module,N is an arbitraryR-module andr, tare non-negative integers.

For a prime idealpofR ther-th Bass number ofM is denoted byµ^r(p, M).

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

Lemma 2.1.

µ^r(p, H_I¹(M, N))6µ^r(p,Ext¹_R(M,ΓI(N))) +µ^r(p,HomR(M, H_I¹(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(pRp, H_I^t(Mp, Np)); also in view of [3, Corollary 4.3.3]

µ^r(p,Ext^t_R^−j(M, H_I^j(N))) =µ^r(pRp,Ext^t_R^−j

p(Mp, H_IR^j

p(Np))).

So, we may assume thatRis a local ring with maximal idealp. We denoteµ^r(p, M) byµ^r(M) and we have to show that

µ^r(H_I¹(M, N))6µ^r(Ext¹_R(M,ΓI(N))) +µ^r(HomR(M, H_I¹(N))).

By Theorem 11.38 of [14] there is a Grothendieck spectral sequence E₂^p,q := Ext^p_R(M, H_I^q(N))⇒pH_I^p+q(M, N) =E^p+q

and there exists a finite filtration 0 = φ²E¹ ⊆ φ¹E¹ ⊆ φ⁰E¹ =H_I¹(M, N) such that E∞^0,1 = E¹/φ¹E¹ and E^1,0∞ ∼= φ¹E¹. Thus µ^r(E¹) 6 µ^r(E∞^0,1) +µ^r(E^1,0∞).

Now, by the sequence 0 −→ E₂^{0,1 d}

0,1

−→2 E₂^2,0 we haveE∞^0,1 ∼=E₃^0,1 ∼= kerd^0,1₂ . Also E∞^1,0∼=E₂^1,0. Hence,µ^r(E¹)6µ^r(kerd^0,1₂ ) +µ^r(E₂^1,0)6µ^r(E₂^0,1) +µ^r(E₂^1,0) from

which the result follows.

Theorem 2.1. We haveµ^r(p, H_I^t(M, N))6Pt

j=0µ^r(p,Ext^t_R^−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 thatR is a local ring with maximal ideal p. We have to show that

µ^r(HI^t(M, N))6

t

X

j=0

µ^r(Ext^t−j_R (M, H_I^j(N))).

We use induction ont. In the caset= 0, we haveH_I⁰(M, N) = HomR(M,ΓI(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 −→ΓI(N)−→ N −→

N/ΓI(N)−→0 induces a long exact sequence

· · · −→H_I^t(M,ΓI(N))−→H_I^t(M, N)−→H_I^t(M, N/ΓI(N))−→ · · ·

which implies that

µ^r(H_I^t(M, N))6µ^r(H_I^t(M,ΓI(N))) +µ^r(H_I^t(M, N/ΓI(N))).

LetE be an injective hull of ΓI(N) and letL=E/ΓI(N). Then by the sequence 0−→ΓI(N)−→E−→L−→0 it follows thatH_Iⁱ⁻¹(M, L))∼=H_Iⁱ(M,ΓI(N)), for alli>2. Thus by induction hypothesis fort−1, we have

µ^r(H_I^t−1(M, L))6

t−1

X

j=0

µ^r(Ext^t_R^−1−j(M, H_I^j(L))) =µ^r(Ext^t_R(M,ΓI(N)))

since H_Iⁱ(L) ∼=H_Iⁱ⁺¹(ΓI(N)) = 0, for all i > 1. Also, the exact sequence 0 −→

ΓI(N)−→ ΓI(E) −→ΓI(L) −→0 shows Ext^t−_R¹(M, H_I⁰(L))∼= Ext^tR(M,ΓI(N))) because ΓI(E) is an injectiveR-module. LetE^′be an injective hull ofN/ΓI(N) and letK=E^′/N/ΓI(N). Then by the sequence 0−→N/ΓI(N)−→E^′−→K−→0 it follows thatH_Iⁱ⁻¹(M, K))∼=H_Iⁱ(M, N/ΓI(N)), for alli>2. Thus by induction hypothesis fort−1, we have

µ^r(H_I^t−1(M, K))6

t−1

X

j=0

µ^r(Ext^t_R^−1−j(M, H_I^j(K)))

=

t−1

X

j=1

µ^r(Ext^t_R^−1−j(M, H_I^j(K))) +µ^r(Ext^t_R⁻¹(M, H_I⁰(K)))

=

t−1

X

j=1

µ^r(Ext^t_R^−1−j(M, H_I^j+1(N/ΓI(N))))+µ^r(Ext^t_R⁻¹(M, H_I⁰(K)))

=

t−1

X

j=1

µ^r(Ext^t−_R^1−j(M, H_I^j+1(N))) +µ^r(Ext^t_R⁻¹(M, H_I⁰(K)))

=

t

X

j=2

µ^r(Ext^t_R^−j(M, H_I^j(N))) +µ^r(Ext^t−_R¹(M, H_I⁰(K))).

Now, the exact sequence 0−→ΓI(E^′)−→ΓI(K)−→H_I¹(N/ΓI(N))−→0 shows that Ext^t_R⁻¹(M, H_I⁰(K))∼= Ext^t_R⁻¹(M, H_I¹(N/ΓI(N)))∼= Ext^t_R⁻¹(M, H_I¹(N)). Thus

µ^r(H_I^t−1(M, K))6

t

X

j=2

µ^r(Ext^t−j_R (M, H_I^j(N))) +µ^r(Ext^t_R⁻¹(M, HI¹(N)))

=

t

X

j=1

µ^r(Ext^t_R^−j(M, H_I^j(N))).

Hence,

µ^r(HI^t(M, N))6µ^r(HI^t(M,ΓI(N))) +µ^r(HI^t(M, N/ΓI(N)))

=µ^r(H_I^t−1(M, L)) +µ^r(H_I^t−1(M, K)) 6µ^r(Ext^t_R(M,ΓI(N))) +

t

X

j=1

µ^r(Ext^t_R^−j(M, H_I^j(N)))

which is claimed.

Corollary 2.1. [13, Theorem 1.1]We have Ass(H_I^t(M, N))⊆

t

[

j=0

Ass(Ext^t_R^−j(M, H_I^j(N))).

Proof. Letp∈Ass(H_I^t(M, N)). Thenµ⁰(p, H_I^t(M, N))6= 0. So, by Theorem 2.1 we havePt

j=1µ⁰(p,Ext^t_R^−j(M, H_I^j(N)))6= 0. Hence, there exists 06j6tsuch that µ⁰(p,Ext^t−j_R (M, H_I^j(N))) 6= 0. Therefore,p ∈St

j=0Ass(Ext^t−j_R (M, H_I^j(N))).

Definition2.1. AnR-moduleXis said to beI-cofinite, whenever Supp(X)⊆ V(I) and ExtⁱR(R/I, X) is finitely generatedR-module, for alli>0.

Corollary 2.2. If Supp(M) ⊆ V(I) and H_I^j(N) is I-cofinite, for all 0 6 j 6 t, then µ^r(p, H_Iⁱ(M, N)) < ∞, for all 0 6 i 6 t. In particular, if I is an ideal of R with Spec(R) =V(I) andH_I^j(N) isI-cofinite, for all 0 6j 6t, then µ^r(p, H_Iⁱ(N))<∞, for all 06i6t.

Proof. In view of [4, Proposition 1] Ext^t−j_R (M, H_I^j(N)) is a finitely generated R-module, for all 06j6t. Now, the claim is obvious by Theorem 2.1.

Corollary 2.3. If Supp(M)⊆V(I) andN is a finitely generatedR-module for whichH_I^j(N)is finitely generated, for all06j < t, thenµ^r(p, H_Iⁱ(M, N))<∞, for all 0 6i 6t. 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 06j < t, thenµ^r(p, H_Iⁱ(N))<∞, for all 06i6t.

Proof. In view of [1, Theorem 2.5] and [4, Proposition 1] Ext^t_R^−j(M, H_I^j(N)) is a finitely generated R-module, for all 0 6 j 6 t. Now, the claim follows by

Theorem 2.1.

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

µ^r(p, H_I^pdM+dimN(M, N))6µ^r(p,Ext^pd_R ^M(M, H_I^dimN(N))).

Proof. By Theorem 2.1 it follows that µ⁰(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 letara(I)<∞; then H_I^t(M, N) = 0, for all t >pdM + ara(I), where ara(I), the arithmetic rank of the idealI, is the least number of elements ofRrequired to generate an ideal which has the same radical as I. In particular,

µ^r(p, H_I^pdM+ara(I)(M, N))6µ^r(p,Ext^pd_R ^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^tR(M, HI⁰(N)))6

t

X

j=2

µ^r(p,Ext^t_R^−j(M, H_I^j−1(N))) +µ^r(p, HI^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^t_R(M, H_I⁰(N)))6

t

X

j=2

µ^r(Ext^t_R^−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 E₂^p,q:= Ext^p_R(M, H_I^q(N))⇒pH_I^p+q(M, N) =E^p+q.

Now, the exact sequence

· · · −→E₂^t−2,1d

t−2,1

−→2 E₂^t,0−→0

and E₃^t,0 =E₂^t,0/Imd^t₂^−2,0 show that µ^r(E^t,0₂ ) 6µ^r(E₂^t−2,1) +µ^r(E₃^t,0). Also, the exact sequence

· · · −→E₃^t−3,2d

t−3,2

−→3 E₃^t,0−→0

andE₄^t,0=E₃^t,0/Imd^t₃^−3,2 show thatµ^r(E₃^t,0)6µ^r(E₄^t,0) +µ^r(E₃^t−3,2). Hence, µ^r(E₂^t,0)6µ^r(E₂^t−2,1) +µ^r(E₃^t−3,2) +µ^r(E₄^t,0)

6µ^r(E₂^t−2,1) +µ^r(E₂^t−3,2) +µ^r(E₄^t,0) 6· · ·=

t

X

j=2

µ^r(E₂^t−j,j−1) +µ^r(E_t+1^t,0)

=

t

X

j=2

µ^r(E₂^t−j,j−1) +µ^r(E^t)

since E_k^t−j,j−1 is a subquotient of E₂^t−j,j−1, for all 3 6 k 6 t, and E_t+1^t,0 is a subquotient ofE^t, whereE∞^t,0∼=φ^tE^t/φ^t+1E^t∼=φ^tE^t⊂E^tand

0 =φ^tE^t⊆ · · · ⊆φ¹E^t⊆φ⁰E^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

[email protected] [email protected] [email protected]