Cofiniteness and Artinianness of certain local cohomology modules

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 HI,Ji(M)∈S for all i> n, then HI,Ji(M)/ajHI,Ji(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∈ N0, HI,Jj(R)∈S for all j> r iff HI,Jj(M)∈S for all j> r and any finite R-module M. © 2015, Università degli Studi di Napoli "Federico II".