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 CI and M be a ZD-module. As a generalization of the S-depth(I, M) and depth(I, J, M), the S-depth of (I, J) on M is defined as $$S{\rm{ - depth}}\left( {I,J,M} \right) = \inf \left\{ {S{\rm{ - depth}}\left( {\mathfrak{a},M} \right): \mathfrak{a}\in \widetilde{W}\left( {I,J} \right)} \right\}$$ , and some properties of this concept are investigated. The relations between S-depth(I, J, M) and H (M) are studied, and it is proved that S-depth(I, J, M) = inf{i: H (M) ∉ 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 SuppRH (M) be a finite subset of 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 $$\mathfrak{m}_1,\mathfrak{m}_2,...\mathfrak{m}_k\in{W(I,J)}$$ such that $$H_{I,J}^i\left( M \right) \cong H_{\mathfrak{m}{_1}}^i\left( M \right) \oplus H_{\mathfrak{m}{_2}}^i\left( M \right) \oplus \ldots \oplus H_{\mathfrak{m}_{_k}}^i\left( M \right)$$ for all i < t.

中文翻译：

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$ZD$-modules 和局部上同调上的 $S$-depth

设 R 是诺特环，I 和 J 是 R 的两个理想。设 S 是满足条件 CI 的 R 模范畴的塞尔子范畴，M 是 ZD 模。作为S-depth(I, M)和depth(I, J, M)的概括，M上(I, J)的S-depth定义为$$S{\rm{ - depth}}\ left( {I,J,M} \right) = \inf \left\{ {S{\rm{ - depth}}\left( {\mathfrak{a},M} \right): \mathfrak{a} \in \widetilde{W}\left( {I,J} \right)} \right\}$$ ，并研究了这个概念的一些属性。研究了S-depth(I, J, M)和H(M)的关系，证明了S-depth(I, J, M) = inf{i: H(M) ∉ S}，其中S 是在采用单射外壳下封闭的 Serre 子类别。提供了一些条件，即在这些条件下，关于一对理想的局部上同调模块与普通的局部上同调模块重合。对于所有 i < t，令 SuppRH (M) 是 Max(R) 的有限子集，其中 M 是任意 R 模，t 是整数。结果表明，存在不同的极大理想 $$\mathfrak{m}_1,\mathfrak{m}_2,...\mathfrak{m}_k\in{W(I,J)}$$ 使得 $$ H_{I,J}^i\left( M \right) \cong H_{\mathfrak{m}{_1}}^i\left( M \right) \oplus H_{\mathfrak{m}{_2}} ^i\left( M \right) \oplus \ldots \oplus H_{\mathfrak{m}_{_k}}^i\left( M \right)$$ 对于所有 i < t。