On the Cofinite Properties of Generalized Local Cohomology Modules
Abstract
Let $A$ be a commutative Noetherian ring with non-zero identity and let $I$ be an ideal of $A$. Consider $F$ as a finitely generated $A$-module and $C$ as an arbitrary $A$-module. In this context, for a non-negative integer $n$, we establish that if the local cohomology modules $\operatorname{H}^{i}_{I}(C)$ are $(\operatorname{FD}_{< n}, I)$-cofinite over $A$ for all integers $i$ and if the collection of such $(\operatorname{FD}_{< n}, I)$-cofinite $A$-modules constitutes an Abelian category, then the generalized local cohomology modules $\operatorname{H}^{i}_{I}(F, C)$ are likewise $(\operatorname{FD}_{< n}, I)$-cofinite over $A$ for all integers $i$.
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