Cofiniteness with respect to extension of Serre subcategories

Let $R$ be a commutative noetherian ring, $\frak a$ be an ideal of $R$, $\mathcal{S}$ be an arbitrary Serre subcategory of $R$-modules satisfying the condition $C_{\frak a}$ and let $\mathcal{N}$ be the subcategory of finitely generated $R$-modules. In this paper, we define and study $\mathcal{NS}$-$\frak a$-cofinite modules with respect to the extension subcategory $\mathcal{NS}$ as an generalization of the classical notion, namely $\frak a$-cofinite modules. For the lower dimensions, we show that the classical results of $\frak a$-cofiniteness hold for the new notion.

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