Regular sequences and local cohomology modules with respect to a pair of ideals

Abstract: Let $R$ be a Noetherian ring, $I$ and $J$ two ideals of $R$ and $t$ an integer. Let $S$ be the class of Artinian $R$-modules, or the class of all $R$-modules $N$ with $\dim_RN\leq k$, where $k$ is an integer. It is proved that $\inf{i: H^{i}_{I,J}(M)\notin S}=\inf{S-\depth_\frak{a}(M): \frak{a}\in \tilde{\rm W}(I,J)}$, where $M$ is a finitely generated $R$-module, or is a $ZD$-module such that $M/\frak{a}M\notin S$ for all $\frak{a}\in \tilde{\rm W}(I,J)$. Let $\Supp_R H^{i}_{I,J}(M)$ be a finite subset of $\Max(R)$ for all $i<t$. It is shown that there are maximal ideals $\frak m_1, \frak m_2,\ldots,\frak m_k$ of $R$ such that $H^{i}_{I,J}(M)\cong H^{i}_{\frak m_1}(M)\oplus H^{i}_{\frak m_2}(M)\oplus\cdots\oplus H^{i}_{\frak m_k}(M)$ for all $i<t$.