An Abelian category of weakly cofinite modules
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Let $I$ be an ideal of a commutative Noetherian semi-local ring $R$ and $M$ be an $R$-module. It is shown that if $\dim M\leq 2$ and $\Supp_R M\subseteq V(I)$, then $M$ is $I$-weakly cofinite if (and only if) the $R$-modules $\Hom_R(R/I,M)$ and $\Ext^1_R(R/I,M)$ are weakly Laskerian. As a consequence of this result, it is shown that the category of all $I$-weakly cofinite modules $X$ with $\dim X\leq 2$, forms an Abelian subcategory of the category of all $R$-modules. Finally, it is shown that if $\dim R/I\leq 2$, then for each pair of finitely generated $R$-modules $M$ and $N$ and each pair of the integers $i,j\geq 0$, the $R$-modules $\Tor_i^R(N,H^j_I(M))$ and $\Ext^i_R(N,H^j_I(M))$ are $I$-weakly cofinite.
Abstract
Let $I$ be an ideal of a commutative Noetherian semi-local ring $R$ and $M$ be an $R$-module. It is shown that if $\dim M\leq 2$ and $\Supp_R M\subseteq V(I)$, then $M$ is $I$-weakly cofinite if (and only if) the $R$-modules $\Hom_R(R/I,M)$ and $\Ext^1_R(R/I,M)$ are weakly Laskerian. As a consequence of this result, it is shown that the category of all $I$-weakly cofinite modules $X$ with $\dim X\leq 2$, forms an Abelian subcategory of the category of all $R$-modules. Finally, it is shown that if $\dim R/I\leq 2$, then for each pair of finitely generated $R$-modules $M$ and $N$ and each pair of the integers $i,j\geq 0$, the $R$-modules $\Tor_i^R(N,H^j_I(M))$ and $\Ext^i_R(N,H^j_I(M))$ are $I$-weakly cofinite.
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