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depth; local cohomology; Serre subcategory; $ZD$-module
Let $R$ be a Noetherian ring, and $I$ and $J$ be two ideals of $R$. Let $S$ be a Serre subcategory of the category of $R$-modules satisfying the condition $C_I$ and $M$ be a $ZD$-module. As a generalization of the $S$-${\rm depth}(I, M)$ and ${\rm depth}(I, J, M)$, the $S$-${\rm depth}$ of $(I, J)$ on $M$ is defined as $S$-${\rm depth}(I, J, M)=\inf \{S$-${\rm depth}(\frak {a}, M) \colon \frak {a}\in \widetilde {\rm W}(I,J)\}$, and some properties of this concept are investigated. The relations between $S$-${\rm depth}(I, J, M)$ and $H^{i}_{I,J}(M)$ are studied, and it is proved that $S$-${\rm depth}(I, J, M)=\inf \{i \colon H^{i}_{I,J}(M)\notin S\}$, where $S$ is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology modules with respect to a pair of ideals coincide with ordinary local cohomology modules under these conditions. Let ${\rm Supp}_R H^{i}_{I,J}(M)$ be a finite subset of ${\rm Max}(R)$ for all $i<t$, where $M$ is an arbitrary $R$-module and $t$ is an integer. It is shown that there are distinct maximal ideals $\frak m_1, \frak m_2,\ldots ,\frak m_k\in {\rm W}(I, J)$ 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 $i<t$.
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