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- W4300846892 abstract "The purpose of the present paper is to continue the study of modules cofinite and weakly cofinite with respect to an ideal $frak a$ of a Noetherian ring $R$. It is shown that an $R$-module $M$ is cofinite with respect to $frak a$, if and only if, $Ext^i_R(R/frak a,M)$ is finitely generated for all $ileq {rm cd}(frak a,M)+1$, whenever $dim R/frak a=1$. In addition, we show that if $M$ is finitely generated and $H^i_{frak a}(M)$ are weakly Laskerian for all $ileq t-1$, then $H^i_{frak a}(M)$ are ${frak a}$-cofinite for all $ileq t-1$ and for any minimax submodule $K$ of $H^{t}_{frak a}(M)$, the $R$-modules $Hom_R(R/{frak a}, H^{t}_{frak a}(M)/K)$ and $Ext^{1}_R(R/{frak a}, H^{t}_{frak a}(M)/K)$ 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 ${frak a}$-weakly cofinite modules over $R$ forms a full Abelian subcategory of the category of modules." @default.
- W4300846892 created "2022-10-04" @default.
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- W4300846892 date "2017-03-02" @default.
- W4300846892 modified "2023-09-30" @default.
- W4300846892 title "Modules cofinite and weakly cofinite with respect to an ideal" @default.
- W4300846892 doi "https://doi.org/10.48550/arxiv.1703.00766" @default.
- W4300846892 hasPublicationYear "2017" @default.
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