Lower Bounds for Local Cohomology Modules with Respect to a Pair of Ideals

Let R be a Noetherian ring, I and J two ideals of R, M an R-module and t an integer. Let S be a Serre subcategory of the category of R-modules satisfying the condition CI, and N be a finitely generated R-module with Supp RN= V (𝔞) for some [Formula: see text]. It is shown that if [Formula: see text] for all i < t and all j < t-i, then [Formula: see text] for all i < t. Let S be the class of all R-modules N with dim R N ≤ k, where k is an integer. It is proved that if [Formula: see text] for all i < t and all [Formula: see text], then [Formula: see text] for all i < t. It follows that [Formula: see text].