Narrow Results

The purpose of the present paper is to continue the study of modules cofinite and weakly cofinite with respect to an ideal $𝔞$ of a Noetherian ring $R$. It is shown that an $R$-module $M$ is cofinite with respect to $𝔞$, if and only if, ${Ext}_R^i(R/𝔞,M)$ is finitely generated for all $i\le \mathrm{\text{cd}}(𝔞,M)+1$, whenever $dimR/𝔞=1$. In addition, we show that if $M$ is finitely generated and $H_{𝔞}^i(M)$ are weakly Laskerian for all $i\le t-1$, then $H_{𝔞}^i(M)$ are $𝔞$-cofinite for all $i\le t-1$ and for any minimax submodule $K$ of $H_{𝔞}^t(M)$, the $R$-modules ${Hom}_R(R/𝔞,H_{𝔞}^t(M)/K)$ and ${Ext}_R^1(R/𝔞,H_{𝔞}^t(M)/K)$ are finitely generated, where $t$ is a non-negative integer. Finally, we explore a criterion for weakly cofiniteness of modules with respect to an ideal of dimension one. Namely, for such ideals it suffices that the two first $Ext$-modules in the definition for weakly cofiniteness are weakly Laskerian. As an application of this result, we deduce that the category of all $𝔞$-weakly cofinite modules over $R$ forms a full Abelian subcategory of the category of modules.