Received August 26, 2024. Published online April 28, 2025.
Abstract: Let $R$ be a commutative Noetherian ring, $J$ an ideal of $R$, $M$ an $R$-module and let $d$ be a non-negative integer. In this paper we introduce a generalization of the notion of a local cohomology module, which we call a local cohomology module with respect to non-negative integer $d$ and ideal $J$, and study some of its properties. Also, we define $W(d,J)$ and $\widetilde{W}(d,J)$ and present the conditions under which $H^i_{d,J}(M)=0$. Finally, we investigate the concept of ${\rm depth}_{d,J}(M)$ and examine the top vanishing of the $R$-module $H^i_{d,J}(M)$.
Keywords: $(d,J)$-local cohomology; system of ideals; $(d,J)$-depth, $W(d,J)$
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