Cominimaxness with respect to ideals of dimension one
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Let $R$ denote a commutative Noetherian (not necessarily local) ring and let $I$ be an ideal of $R$ of dimension one. The main purpose of this note is to show that the category $\mathscr{M}(R, I)_{com}$ of $I$-cominimax $R$-modules forms an Abelian subcategory of the category of all $R$-modules. This assertion is a generalization of the main result of Melkersson in \cite{Me}. As an immediate consequence of this result we get some conditions for cominimaxness of local cohomology modules for ideals of dimension one. Finally, it is shown that the category $\mathcal{C}_{B}^1(R)$ of all $R$-modules of dimension at most one with finite Bass numbers forms an Abelian subcategory of the category of all $R$-modules.
Abstract
Let $R$ denote a commutative Noetherian (not necessarily local) ring and let $I$ be an ideal of $R$ of dimension one. The main purpose of this note is to show that the category $\mathscr{M}(R, I)_{com}$ of $I$-cominimax $R$-modules forms an Abelian subcategory of the category of all $R$-modules. This assertion is a generalization of the main result of Melkersson in \cite{Me}. As an immediate consequence of this result we get some conditions for cominimaxness of local cohomology modules for ideals of dimension one. Finally, it is shown that the category $\mathcal{C}_{B}^1(R)$ of all $R$-modules of dimension at most one with finite Bass numbers forms an Abelian subcategory of the category of all $R$-modules.
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