Let R be a commutative Noetherian ring, I, J be two ideals of R, M be an R-module and S be a Serre class of R-modules. A positive answer to the Huneke's conjecture is given for a Noetherian ring R and minimax R-module M of Krull dimension less than 3, with respect to S. There are some results on cofiniteness and Artinianness of local cohomology modules with respect to a pair of ideals. For a ZD-module M of finite Krull dimension and an integer n ∈ N, if H i I,J (M) ∈ S for all i > n, then H i I,J (M)/a j H i I,J (M) ∈ S for any a ∈W (I, J ), all i ≥ n, and all j ≥ 0. By introducing the concept of Serre cohomological dimension of M with respect to (I, J ), for an integer r ∈ N 0 , H j I,J (R) ∈ S for all j > r iff H j I,J (M) ∈ S for all j > r and any finite R-module M.
Abstract. Let (R, m) be a local ring and I an ideal. The aim of the present paper is twofold. At first we continue the investigation to compare fgrade(I, R) with depth R/I and further we derive some results on the vanishing of Lyubeznik numbers.
This paper is concerned about the relation between local cohomology modules defined by a pair of ideals and Serre classes of R-modules, as a generalization of results of J. Azami, R. Naghipour and B. Vakili (2009) and M. Asgharzadeh and M.Tousi (2010). Let R be a commutative Noetherian ring, I , J be two ideals of R and M be an R-module. Let a ∈ W (I, J) and t ∈ N 0 be such that Ext t R (R/a, M ) ∈ S and Ext j R (R/a, H i I,J (M )) ∈ S for all i < t and all j ≥ 0. Then for any submodule N of H t I,J (M ) such that Ext 1 R (R/a, N ) ∈ S, we obtain Hom R (R/a, H t I,J (M )/N ) ∈ S.
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