On the annihilators and attached primes of top local cohomology modules

Let \frak a be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. It is shown that {\rm Ann}_R(H_{\frak a}^{{\dim M}({\frak a}, M)}(M))= {\rm Ann}_R(M/T_R({\frak a}, M)), where T_R({\frak a}, M) is the largest submodule of M such that {\rm cd}({\frak a}, T_R({\frak a}, M))<{\rm cd}({\frak a}, M). Several applications of this result are given. Among other things, it is shown that there exists an ideal \frak b of R such that {\rm Ann}_R(H_{\frak a}^{\dim M}(M))={\rm Ann}_R(M/H_{\frak b}^{0}(M)). Using this, we show that if H_{\frak a}^{\dim R}(R)=0, then {\rm Att}_RH^{{\dim R}-1}_{\frak a}(R)=\{{\frak p}\in {\rm Spec}\,R|\,{\rm cd}({\frak a}, R/{\frak p})={\dim R}-1\}. These generalize the main results of \cite[Theorem 2.6]{BAG}, \cite[Theorem 2.3]{He} and \cite[Theorem 2.4]{Lyn}.


Introduction
Let R be an arbitrary commutative Noetherian ring (with identity), a an ideal of R and let M be a finitely generated R-module. An important problem concerning local cohomology is determining the annihilators of the i th local cohomology module H i a (M). This problem has been studied by several authors; see for example [9], [10], [11], [14], [15], [16], and has led to some interesting results. More recently, in [2] Bahmanpour et al., proved an interesting result about the annihilator Ann R (H d m (M)), in the case (R, m) is a complete local ring of dimension d.
The purpose of the present paper is to establish some new results concerning of the annihilators of local cohomology modules H i a (M) (i ∈ N 0 ), where a is an ideal in a Noetherian ring R and M a finitely generated module over R.
As a main result in the second section, we determine the annihilators of the top local cohomology module H dim M a (M). More precisely, we shall prove the following theorem which is a generalization of the main result of [2,Theorem 2.6] for an arbitrary ideal a of an arbitrary Noetherian ring R. Here 0 = ∩ n j=1 N j denotes a reduced primary decomposition of the zero submodule 0 in M and N j is a p j -primary submodule of M, for all j = 1, . . . , n and b := Π cd(a,R/p j ) =c p j .
One of the basic problems concerning local cohomology is finding the set of attached primes of H i a (M). In Section 3, we obtain some results about the attached primes of local cohomology modules. In this section among other things, we derive the following consequence of Theorem 1.1 and Corollary 1.2, which provides an upper bound for the attached primes of Att R H cd(a,M ) a (M). This will generalize the main results of [5] and [4].
For an R-module A, a prime ideal p of R is said to be attached prime to A if p = Ann R (A/B) for some submodule B of A. We denote the set of attached primes of A by Att R A. This definition agrees with the usual definition of attached prime if A has a secondary representation (cf. [12,Theorem 2.5]).
Another main result in Section 3 is to give a complete characterization of the attached primes of the local cohomology module H dim R−1 a (R). More precisely, we shall show the following result, which is an extension, as well as, a correction of the main theorem of [7].
One of our tools for proving Theorem 1.4 is the following: Throughout this paper, R will always be a commutative Noetherian ring with non-zero identity and a will be an ideal of R. For any R-module M, the i th local cohomology module of M with support in V (a) is defined by Ext i R (R/a n , M).
The cohomological dimension of M with respect to a is defined as cd(a, M) := sup{i ∈ Z| H i a (M) = 0}. For each R-module L, we denote by Assh R L (resp. mAss R L) the set {p ∈ Ass R L : dim R/p = dim L} (resp. the set of minimal primes of Ass R L). Also, we shall use Att R L to denote the set of attached prime ideals of L. For any ideal a of R, we denote {p ∈ Spec R : p ⊇ a} by V (a). Finally, for any ideal b of R, the radical of b, denoted by Rad(b), is defined to be the set {x ∈ R : x n ∈ b for some n ∈ N}. For any unexplained notation and terminology we refer the reader to [3] and [13].

Annihilators of top local cohomology modules
Let us, firstly, recall the important concept of the cohomological dimension of an Rmodule L with respect to an ideal a of a commutative Noetherian ring R. Denoted by cd(a, L), is the largest integer i such that H i a (L) = 0. The main result of this section is Theorem 2.3. The following lemma plays a key role in the proof of that theorem.
Consequently, we can (and do) assume that T R (a, M) = 0, and with this assumption our aim is to show that Ann R (H c a (M)) = Ann R (M). To this end, as , it is enough for us to prove that We can, and do, assume henceforth in this proof that cd(a, R) = c = dim R and that M is a faithful R-module. Hence it is sufficient for us to show that  Remark 2.6. Let R be a Noetherian ring, a an ideal of R and M a non-zero finitely generated R-module with finite cohomological dimension c := cd(a, M). Proof. The assertion follows from Theorem 2.3 and Remark 2.6.
Corollary 2.8. Let R be a Noetherian ring with finite dimension c and a an ideal of R such that cd(a, R) = c. Then j=1 q j is a reduced primary decomposition of the zero ideal of R, q j is a p j -primary ideal of R, for all 1 ≤ j ≤ n and b = Π cd(a,R/p j ) =c p j .
Proof. The result follows from Corollary 2.7.   Hence T R (a, R) ⊆ Ann R (H c a (R)), and so T R (a, R) = 0. Now, the assertion follows from Remark 2.6.

Attached primes of local cohomology modules
It will be shown in this section that the subjects of the previous section can be used to investigate the attached prime ideals of local cohomology modules. In fact, we will generalize and improve the main result of Hellus (cf. [7, Theorem 2.3]). The main result is Theorem 3.7. The following proposition will serve to shorten the proof of the main theorem. We begin with Definition 3.1. Let L be an R-module. We say that a prime ideal p of R is an attached prime of L, if there exists a submodule K of L such that p = Ann R (L/K). We denote by Att R L the set of attached primes of L.
When M is representable in the sense of [12] (e.g. Artinian or injective), our definition of Att R L coincides with that of Macdonald, Sharp [12], [17].
It follows easily from the definition that, if p ∈ Att R L, then L ⊗ R R/p = 0. This is used in the proof of Theorem 3.3.
Lemma 3.2. Let R be a Noetherian ring and L an R-module. Then, the set of minimal elements of V (Ann R (L)) coincides with that of Att R L. In particular, Proof. The assertion follows from the Definition 3.1 and the fact that p = Ann R (L/pL) for each minimal prime p over Ann R (L).
We are now ready to state and prove the first main result of this section, that gives us an upper bound for the attached primes of Att R H cd(a,M ) a (M). Before this, we note that as Ann R (M) ⊆ Ann R (H i a (M)), it follows that Att R (H i a (M)) ⊆ Supp M, for every finitely generated R-module M.   R (a, R)). Proof. In order to prove the first containment, in view of Theorem 3.3, it is enough to show that {p ∈ Spec R| cd(a, R/p) = c = dim R/p} ⊆ Att R H c a (R). To this end, let p ∈ Spec R be such that cd(a, R/p) = c = dim R/p. Then in view of Corollary 3.4, we have p ∈ Att R H c a (R/p). Now, from the exact sequence 0 −→ p −→ R −→ R/p −→ 0, and the right exactness of H c a (·), we deduce that p ∈ Att R H c a (R), as required. In addition, in order to show the last inclusion, use Corollary 2.5 and the fact that (R/J t )) ⊆ q t , for all integers t. Consequently, we obtain that and so the Krull's Intersection Theorem implies that JR p = 0. As R is an integral domain, it follows that J = 0, which is a contradiction.  The following lemma, which is a consequence of the Lichtenbaum-Hartshorne vanishing theorem, is assistant in the proof of Theorem 3.9.  If, in addition, R is complete, then Proof. For the proof of Assh R R ⊆ Att R H d−1 a (R), let p ∈ Assh R R. Then, in view of Theorem 3.7, it is enough to show that cd(a, R/p) = d − 1. To do this, as Assh R R = {q ∩ R| q ∈ AsshRR}, dimR/aR = 1 and cd(aR,R/pR) = cd(a, R/p), by using Lemma 2.1 without loss of generality, we may assume that R is complete. Now, since H d a (R) = 0, it follows that H d a (R/p) = 0. Thus a + p is not m-primary, and so dim R/(a + p) = 1 (note that dim R/a = 1). Let q be a minimal prime over a + p such that dim R/q = 1. Then, it is easy to see that q is also minimal over a. Next, as R is catenary, it yields that dim R q /pR q = d − 1, and so we have We end the paper with the following question: Question.