Cofiniteness with respect to Extension of Serre Subcategories at Small Dimensions

Abstract

Let $A$ denote a commutative Noetherian ring, $\mathfrak{a}$ be an ideal of $A$ , $\mathcal{X}$ be an arbitrary dense subcategory of $A$ -modules, and $\mathcal{N}$ be the full subcategory of finite $A$ -modules. In this paper, we study $\mathcal{N}\star \mathcal{X}$ - $\mathfrak{c}$ -cofinite modules with respect to the extension subcategory $\mathcal{N}\star \mathcal{X}$ when $\dim A/\mathfrak{c}\le 2$ . We also study $\mathfrak{c}$ -cofiniteness with respect to a new dimension.