Cofiniteness and Artinianness of certain local cohomology modules

Abstract

Let R be a commutative Noetherian ring, I, J be two ideals of R, M be an R-module and ${\mathcal {S}}$ 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 ${\mathcal {S}}$. 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 $n\in {\mathbb {N}}$, if ${{\mathrm{\mathrm{H}}}}^{i}_{I,J}(M)\in {\mathcal {S}}$ for all $i>n$, then ${{\mathrm{\mathrm{H}}}}^{i}_{I,J}(M)/{\mathfrak {a}}^{j}{{\mathrm{\mathrm{H}}}}^{i}_{I,J}(M)\in {\mathcal {S}}$ for any ${\mathfrak {a}}\in \tilde{W}(I,J)$, all $i\ge n$, and all $j\ge 0$. By introducing the concept of Serre cohomological dimension of M with respect to (I, J), for an integer $r\in {\mathbb {N}}_0$, ${{\mathrm{\mathrm{H}}}}^{j}_{I,J}(R)\in {\mathcal {S}}$ for all $j>r$ iff ${{\mathrm{\mathrm{H}}}}^{j}_{I,J}(M)\in {\mathcal {S}}$ for all $j>r$ and any finite R-module M.