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

Abstract. Let a be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. Let t be a natural integer. It is shown that there is a finite subset X of Spec R, such that Ass R (H t a (M )) is contained in X union with the union of the sets Ass R (Ext j R (R/a, H i a (M ))), where 0 ≤ i < t and 0 ≤ j ≤ t 2 + 1. As an immediate consequence, we deduce that the first non-a-cofinite local cohomology module of M with respect to a has only finitely many associated prime ideals.

Let R be a commutative Noetherian ring, a an ideal of R, and let M be a finitely generated R-module. For a non-negative integer t, we prove that H t a (M) is a-cofinite whenever H t a (M) is Artinian and H i a (M) is a-cofinite for all i < t. This result, in particular, characterizes the a-cofiniteness property of local cohomology modules of certain regular local rings. Also, we show that for a local ring. This result, in conjunction with the first one, yields some interesting consequences. Finally, we extend Grothendieck's non-vanishing Theorem to a-cofinite modules.1. Introduction. Throughout this paper, we assume that R is a commutative Noetherian ring, a an ideal of R, and that M is an R-module. Let t be a non-negative integer. Grothendieck [4] introduced the local cohomology modules H t a (M) of M with respect to a. He proved their basic properties. For example, for a finitely generated module M, he proved that H t m (M) is Artinian for all t whenever R is local with maximal ideal m. In particular, it is shown that Hom R (R/m, H t m (M)) is finitely generated. Later Grothendieck asked in [5] whether a similar statement is valid if m is replaced by an arbitrary ideal. Hartshorne gave a counterexample in [6], where he also defined that an R-module M (not necessarily finitely generated) is a-cofinite, if Supp R (M) V (a) and Ext t R (R/a, M) is a finitely generated R-module for all t. He also asked when the local cohomology modules are a-cofinite. In this regard, the best known result is that when either a is principal or R is local and dim R/a = 1, then the modules H t a (M) are a-cofinite. These results are proved in [8] and [3], respectively. Melkersson [15] characterized those Artinian modules which are a-cofinite. For a survey of recent developments on cofiniteness properties of local cohomology, see Melkersson's interesting article [16]. One of the aim of this note is to show that, for a finitely generated module M, the module H t a (M) is a-cofinite whenever the modules H i a (M) are a-cofinite for all i < t and H t a (M) is Artinian. This result, in particular, characterizes the a-cofiniteness property of local cohomology modules of certain regular local rings (see Remark 2.3(ii)). Next, we assume that R is local with maximal ideal m.

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