Cofiniteness with respect to extension of Serre subcategories

Abstract

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