Upper bounds, cofiniteness, and artinianness of local cohomology modules defined by a pair of ideals

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 $\lc^{i}_{I,J}(M)\in\mathcal{S}$ for all $i>n$, then $\lc^{i}_{I,J}(M)/\fa^{j}\lc^{i}_{I,J}(M)\in\mathcal{S}$ for any $\fa\in\tilde{W}(I,J)$, all $i\geq n$, and all $j\geq0$. By introducing the concept of Seree cohomological dimension of $M$ with respect to $(I,J)$, for an integer $r\in\mathbb{N}_0$, $\lc^{j}_{I,J}(R)\in\mathcal{S}$ for all $j>r$ iff $\lc^{j}_{I,J}(M)\in\mathcal{S}$ for all $j>r$ and any finite $R$-module $M$.