Abstract
Abstract. The notion of weakly Laskerian modules was introduced recently by the authors. Let R be a commutative Noetherian ring with identity, a an ideal of R, and M a weakly Laskerian module. It is shown that if a is principal, then the set of associated primes of the local cohomology module H i a (M ) is finite for all i ≥ 0. We also prove that when R is local, then Ass R (H 
Introduction
Throughout, let R be a commutative Noetherian ring with identity. Let a be an ideal of R and M an R-module. For each i ≥ 0, the i-th local cohomology module of M with respect to a is defined as The reader can refer to [3] for basic properties of local cohomology. In [7] , Hartshorne defined an R-module M to be a-cofinite if Supp R M ⊆ V(a) and Ext i R (R/a, M) is finitely generated for all i ≥ 0. He then asked when the local cohomology modules of a finitely generated module are a-cofinite. In this regard, the best known result is that for a finitely generated R-module M if either a is principal or R is local and dim R/a = 1, then the H It is easy to see that an a-cofinite module has only finitely many associated primes. Huneke [8] raised the following question: If M is a finitely generated R-module, then the set of associated primes of H i a (M) is finite for all ideals a of R and all i ≥ 0.
Introduction
Singh [16] gives a counter-example to this question (also see [17] for some other counter-examples). On the other hand, Brodmann and Lashgari [2, Theorem 2.2] showed that if for a finitely generated R-module M and an integer t, the local cohomology module H i a (M) is finitely generated for all i < t, then Ass R (H t a (M)) is finite. For another proof of this result, see [10] . As in [6] , an R-module M is called weakly Laskerian if Ass R (M/N) is finite for all submodules N. Let M be a weakly Laskerian module and t ∈ N. In [6] , we proved the set of associated primes of the first non a-cofinite local cohomology module of M is finite. That clearly implies the result mentioned above due to Brodmann and Lashgari. Also regarding the Artinianness of local cohomology, it is shown in [5] , that in many cases weakly Laskerian modules behave similar to finitely generated modules.
Introduction
In this article, we continue studying the set of associated primes of local cohomology of weakly Laskerian modules. Example 2.3 shows that the class of weakly Laskerian modules is much larger than that of Noetherian modules. Our main aim in this paper is to show that in conjunction with finiteness properties of local cohomology, in many cases weakly Laskerian modules behave similar to finitely generated modules. By applying the known techniques we can get the analogues of some finiteness results for finitely generated modules for weakly Laskerian modules .
Introduction
In section 2, we first review weakly Laskerian modules. Then we introduce the notion of weak cofiniteness and show that many results concerning cofiniteness of finitely generated modules are valid for weak cofiniteness of weakly Laskerian modules. In particular, it is proved that the Change of Ring Principle for weak cofiniteness holds.
Introduction
The main results of this article appear in section 3. We prove that the local cohomology modules of a weakly Laskerian module are weakly cofinite in several cases. As a result, we deduce that the sets of associated primes of the local cohomology modules of a weakly Laskerian module are finite in these cases. For instance, it is shown that if a is a principal ideal of R and M a weakly Laskerian R-module, then Ass R (H i a (M)) is finite for all i ≥ 0. Also, we prove that if R is a local ring and M a weakly Laskerian R-module, then Ass R (H 
Weakly cofinite modules
In this section, we first recall the definition of weakly Laskerian modules. Then we bring a lemma which is needed in the sequel. Definition 2.1. i) An R-module M is said to be Laskerian if any submodule of M is an intersection of a finite number of primary submodules. ii) (See [6] ) An R-module M is said to be weakly Laskerian if the set of associated primes of any quotient module of M is finite.
Weakly cofinite modules
Obviously, any Noetherian module is Laskerian and it is clear that any Laskerian module is weakly Laskerian. So, any Noetherian module is weakly Laskerian. We need to the following lemma in the sequel. iii) Let N be a submodule of M. We have
Weakly cofinite modules
Thus the set of associated primes of any quotient module of M is finite. Now, we provide several examples for this class of modules.
Weakly cofinite modules
Example 2.3. i) An R-module M is said to be minimax if M has a finitely generated submodule S such that M/S is Artinian (see [18] ). By Lemma 2.2, it follows that any minimax R-module is weakly Laskerian.
Weakly cofinite modules
ii) Let E be the minimal injective cogenerator of R. If for an R-module M the natural map from M to Hom R (Hom R (M, E), E) is an isomorphism, then M is said to be Matlis reflexive. Also, an R-module M is said to be linearly compact if each system of congruences
Weakly cofinite modules
indexed by a set I and where the M i are submodules of M, has a solution x whenever it has a solution for every finite subsystem. It is known that if either M is reflexive or linearly compact, then M has a finitely generated submodule S such that M/S is Artinian (see e.g. [6, Example 2.2]). Hence any reflexive R-module and also any linearly compact R-module is weakly Laskerian.
Weakly cofinite modules
Next, we present a generalization of the notion of cofiniteness.
Weakly cofinite modules
Definition 2.4. Let a be an ideal of R and M an R-module. We say that M is a-weakly cofinite if Supp R M ⊆ V(a) and Ext
Weakly cofinite modules
If M is weakly Laskerian, then by Lemma 2.2 ii), it turns out that M is a-weakly cofinite. In particular, if either M is finitely generated or Artinian, then M is a-weakly cofinite.
Weakly cofinite modules
ii) Every a-cofinite module is a-weakly cofinite.
Weakly cofinite modules
We collect some important known properties of cofinite modules in a lemma. Then we study the corresponding properties for weakly cofinite modules.
Weakly cofinite modules
is an exact sequence of R-modules and two of the modules in the exact sequence are a-cofinite, then so is the third one. iii) If M is a a-cofinite R-module, then Ass R M is finite. iv) For every a-cofinite R-module M, the R-module M/aM is finitely generated. Remark 2.7. i) Example 2.5 i) shows that the analogue of Lemma 2.6 i) holds for weak cofiniteness. Also, in view of Lemma 2.2 i), it follows that the analogue of Lemma 2.6 ii) holds for weak cofiniteness. ii) Because the set of associated primes of a weakly Laskerian module is finite, by using the same argument as in the proof of Lemma 2.6 iii), we can deduce that the set of associated primes of a a-weakly cofinite module is finite. iii) Recall that a module M is said to have finite Goldie dimension if M does not contain an infinite direct sum of nonzero submodules, or equivalently, the injective envelope E(M) of M decomposes as a finite direct sum of indecomposable injective submodules. By [13, Proposition 1.3], the Goldie dimension of an a-cofinite module is finite. This is not the case for a-weakly cofinite modules. To this end, take M = ⊕ i∈N R/m, where m is a fixed maximal ideal of R. Since Supp R M = {m} by Lemma 2.2 iii), it turns out that M is weakly Laskerian. Therefore M is m-weakly cofinite, although its Goldie dimension is not finite.
Weakly cofinite modules
Next, we are going to prove the analogue of Lemma 2.6 iv) for weak cofiniteness, but we first need some preliminary results. In view of Lemma 2.2, by using the same proof as that used in [4, Proposition 1], we can deduce the following. for all p, q ≥ 0. Also, for each n ∈ N 0 there is a bounded filtration
Weakly cofinite modules
. . , n. Now, by using Lemma 2.2 i) successively, we deduce that Ext n R (R/a, M) is a weakly Laskerian R-module for all n ≥ 0. That is M is a-weakly cofinite.
Weakly cofinite modules
Conversely, assume that M is a-weakly cofinite. By induction on n, we show that E n,0 2 = Ext n T (T /aT, M) is a weakly Laskerian T -module for all n ≥ 0. Since the module
Weakly cofinite modules
is weakly Laskerian, the claim holds for n = 0. Now, assume that n > 0 and that E p,0 2 is weakly Laskerian for all p < n. By Lemma 2.8, it turns out that E p,q 2 is weakly Laskerian for all p < n and q ≥ 0. We have E Proof. Let a 1 , a 2 , . . . , a n be generators of a. Let Θ : R[X 1 , X 2 , . . . X n ] −→ R be the natural ring homomorphism defined by Θ(f ) = f (a 1 , a 2 , . . . , a n ). Then Θ is surjective and (X 1 , X 2 , . . . X n )R = a. Denote the ideal (X 1 , X 2 , . . . X n ) by b. By Theorem 2.9, Ext
Weakly cofinite modules
. . X n ) denote the Koszul complex of the ring R[X 1 , X 2 , . . . X n ] with respect to X 1 , X 2 , . . . X n . Since X 1 , X 2 , . . . X n is a regular sequence in the ring R[X 1 , X 2 , . . . X n ], it follows that
Weakly cofinite modules
Therefore M/aM is a weakly Laskerian R-module.
Associated primes of local cohomology modules
In this section, we prove that the local cohomology modules of a weakly Laskerian module are weakly cofinite in several cases.
Associated primes of local cohomology modules
Let M be a finitely generated R-module and t a nonnegative integer. In [12,
Associated primes of local cohomology modules
is also a-cofinite. In the following we show that the same result holds for weak cofiniteness. Then, as a result we deduce several results concerning finiteness of associated primes of local cohomology modules. 
is weakly
Laskerian for all p ≥ 0. This completes the proof. By using the above result, we can deduce the following corollary. The following extends the main result of [9] . Proof. Since dim R/a ≤ 1, it follows that V(a) is contained in the set of minimal prime ideals of a union with {m}. Thus Proof. Let a be an ideal of R. As in the proof of Theorem 3.7, we may assume that t + 1 ≥ grade(a, M). Also, we can and do assume that H 

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