Abelian category of cominimax modules and local cohomology

Abstract

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, $M$ an arbitrary $R$-module and $X$ a finite $R$-module. We prove that the category of $\fa$-cominimax modules is a Melkersson subcategory of $R$-modules whenever $\dim R\leq 1$ and is an Abelian subcategory whenever $\dim R\leq 2$. We prove a characterization theorem for $\lc_{\fa}^{i}(M)$ and $\lc_{\fa}^{i}(X,M)$ to be $\fa$-cominimax for all $i$, whenever one of the following cases holds: (a) $\ara (\fa)\leq 1$, (b) $\dim R/\fa \leq 1$ or (c) $\dim R\leq 2$.