ON THE ASSOCIATED PRIMES OF THE d-LOCAL COHOMOLOGY MODULES

This paper is concerned to relationship between the sets of associated primes of the d-local cohomology modules and the ordinary local cohomology modules. Let R be a commutative Noetherian local ring, M an R-module and d, t two integers. We prove that Ass(Ht d(M)) = ⋃ I∈Φ Ass(H t I(M)) whenever Hi d(M) = 0 for all i < t and Φ = {I : I is an ideal of R with dimR/I ≤ d}. We give some information about the non-vanishing of the d-local cohomology modules. To be more precise, we prove that Hi d(R) 6= 0 if and only if i = n − d whenever R is a Gorenstein ring of dimension n. This result leads to an example which shows that Ass(Hn−d d (R)) is not necessarily a finite set. Mathematics Subject Classification (2010): 13D45, 14B15


Introduction
Throughout this paper, R denotes a commutative Noetherian ring with non-zero identity.For an ideal I of R and an R-module M , the ith local cohomology module of M with respect to I is defined as The reader can refer to [6] for the basic properties of local cohomology modules.
An important problem in commutative algebra is determining when the set of associated primes of the ith local cohomology module H i I (M ) is finite.In [10], Huneke raised the following question: If M is a finitely generated R-module, then the set of associated primes of H i I (M ) is finite for all ideals I of R and all i ≥ 0. This problem has been studied by many authors, it was shown that it is true in many situations, for examples see [4,5,8].In particular, it is shown in [5] that if for a finitely generated R-module M and integer t, the local cohomology modules H i I (M ) are finitely generated for all i < t, then the set Ass(H t I (M )) is finite.There are several papers devoted to the extension of the above results to more general situations, for examples see [1,9].However there are counterexamples which show that it is not true in general, for examples see [11,16].The purpose of this paper is to make a counterexample to above question in the context of general local cohomology modules.The theory of general local cohomology modules over commutative Noetherian rings introduced by Bijan-Zadeh in [3].General local cohomology theory described as follows.
Let Φ be a non-empty set of ideals of R. We call Φ a system of ideals of R if, whenever I, I ∈ Φ, then there exists J ∈ Φ such that J ⊆ II .Such a system of ideals gives rise to an additive, left exact functor Γ Φ (M ) = {x ∈ M : Ix = 0 for some ideal I ∈ Φ} from the category of R-modules and R-homomorphisms to itself.Γ Φ (−) is called the Φ-torsion functor.For each i ≥ 0, the ith right derived functor of Γ Φ (−) is denoted by H i Φ (−).For an ideal I of R, if Φ = {I n : n ∈ N}, then H i Φ (−) coincides with the ordinary local cohomology functor H i I (−).Let d ≥ 0 be an integer.We denote Γ Φ (−) and H i Φ (−) by Γ d (−) and H i d (−) respectively, for the system of ideals Φ = {I : I is an ideal of R with dim R/I ≤ d}.
The functor Γ d (−) was originally defined in [2] and the modules H i d (M ) were called d-local cohomology modules associated to M were studied in [18,19].After some preliminary results in Section 2, for an R-module M and an integer t we prove that where Φ = {I : I is an ideal of R with dim R/I ≤ d} and H i d (M ) = 0 for all i < t.In Section 3, we shall provide some results concerning the vanishing and non-vanishing of d-local cohomology modules: we shall prove that, over a local ring R, if the non-zero finitely generated R-module M has (Krull) dimension n, then there exists an integer i with 0 This result leads to an example which shows that the Huneke question is not true in the context of d-local cohomology modules.

The associated primes
It is our intention in this section to present the relationship between the sets of associated primes of the d-local cohomology modules and the ordinary local ON THE ASSOCIATED PRIMES OF THE d-LOCAL COHOMOLOGY MODULES 57 cohomology modules.So throughout this section, R will denote a ring and I is an ideal of R, d is an integer and Φ = {I : I is an ideal of R with dim R/I ≤ d}.
Lemma 2.1.Let M be an R-module and t be an integer such that H i d (M ) = 0 for all i < t.Then the following statements are true: ) for all I ∈ Φ so, by [14,Theorem 11.38], the Grothendieck spectral sequence E p,q for all q ≤ t.On the other hand, for all q ≤ t, by the sequence 0 for all i ≤ t.The proof is therefore complete.
(ii) It is similar to that of (i).
Corollary 2.2.Let M be an R-module and t be an integer such that H i d (M ) = 0, for all i < t.Then the following statements are true: ) for all I, J ∈ Φ with I ⊆ J and i ≤ t.
Proof.(i) By a similar argument to that of Lemma 2.1 (i), one can shows that , for all I ∈ Φ and all i ≤ t.Therefore, Ass(Hom R (R/I, We are now in a position to prove that the main result of this section. Theorem 2.3.Let M be an R-module and t be an integer such that H i d (M ) = 0, for all i < t.Then Ass(H t p (M )), where Proof.First of all, we show that For an ideal I of R, it is clear that 0 : Hence, H t c (M ) = 0.The converse is true by Lemma 2.1 (ii).

The non-vanishing theorems
In this section, we shall provide some results concerning the vanishing and nonvanishing of d-local cohomology modules.Throughout R is a local ring with maximal ideal m and d is a non negative integer.
We are now in a position to prove that the non-vanishing theorems in the d-local cohomology modules.Theorem 3.1.Let (R, m) be a local ring and let M be a non-zero finitely generated R-module of dimension n.Then there is at least one j with 0 ≤ j ≤ d for which Then by [12,Theorems 18.1 and 18.8] we have On the other hand, by Theorem 3.1, there is one j with 0 ≤ j ≤ d for which (ii) For all I ∈ Φ and all i ≥ 0, Thus there is a Grothendieck spectral sequence Hence, , for all i > 0.
Theorem 3.9.Let R be Gorenstein of dimension n and let be an injective resolution of R. Then the following statements are true: (i) Ext j R (H i d (M ), R) = 0, for all R-module M , 0 ≤ j < n − d and i ≥ 0. (ii) Ext j R (E i , R) = 0 for all i ≥ 1 and j < i.
Proof.(i) It follows by Lemma 3.8.
(ii) For i ≥ 1 we have is an exact sequence that induces the long exact sequence ), for all j < i. Hence Ext j R (E i , R) = 0 for all j < i every Gorenstein local ring is catenary and biequidimensional, see [12, Theorem 17.3].Moreover, H n−d pRp (R p ) = 0, by [6, Theorem 7.3.2].Thus p is a minimal element of Supp(H n−d d (R)) and then p ∈ Ass(H n−d d (R)).(iii) By (ii), Ass(H n−d d (R)) ⊆ Max(R) so H n−d d (R) is not Artinian.(iv) It is obvious by the proof of (i).(v) See [17, Theorem 2.1].Example 3.5.Let K be a field and let

1 ( 1 (
is a local Goenstein ring of dimension n.So by Theorem 3.4 we haveAss(H n−1 R m )) = {pR m ∈ Spec(R m ) : dim(R/p) m = 1}.Now, Ass(H n−1 R m ))has infinite members by [15, Exercise 15.3].Theorem 3.6.Let R be Gorenstein of dimension n, I an ideal of R and let 0 ≤ d ≤ n.Then the following statements are true:

Corollary 3 . 7 .Lemma 3 . 8 .
Let R be Gorenstein of dimension n and let 0 ≤ d ≤ n.Then dim H n−d d (R) = depthH n−d d (R) = d and H n−d d (R) is not a finitely generated Rmodule.Proof.It follows by the proof of Theorems 3.2(ii), 3.4, 3.6 and [7, Exercise 9.1.12(c)].Let {p λ : p λ ∈ Φ} λ∈Λ be a family of prime ideals of R. Then for any d-torsion R-module M , Hom R (M, ⊕ λ∈Λ E(R/p λ )) = 0.Proof.Suppose that Hom R (M, E(R/p)) = 0, for some p ∈ Φ.Thus there is a nonzero element f ∈ Hom R (M, E(R/p)) so f (x) = 0 for some x ∈ M .By assumption there is an I ∈ Φ such that Ix = 0 and so If (x) = 0. Hence, I ⊆ p. Otherwise for each a ∈ I \p we have the automorphism E(R/p) a → E(R/p), contrary to If (x) = 0.