CHARACTERIZATION OF WEAKLY COFINITE LOCAL COHOMOLOGY MODULES
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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 a characterization for $\lc_{\fa}^{i}(M)$ and $\lc_{\fa}^{i}(X,M)$ to be $\fa$-weakly cofinite for all $i$, whenever one of the following cases holds: (a) $\ara (\fa)\leq 1$, (b) $\dim R/\fa \leq 1$ or (c) $\dim R\leq 2$. We also prove that, if $M$ is a weakly Laskerian $R$-module, then $\lc_{\fa}^{i}(X,M)$ is $\fa$-weakly cofinite for all $i$, whenever $\dim X\leq 2$ or $\dim M\leq 2$ (resp.~$(R,\fm)$ a local ring and $\dim X\leq 3$ or $\dim M\leq 3$). Let $d=\dim M<\infty$, we prove an equivalent condition for top local cohomology module $\lc_{\fa}^{d}(M)$ to be weakly Artinian.
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 a characterization for $\lc_{\fa}^{i}(M)$ and $\lc_{\fa}^{i}(X,M)$ to be $\fa$-weakly cofinite for all $i$, whenever one of the following cases holds: (a) $\ara (\fa)\leq 1$, (b) $\dim R/\fa \leq 1$ or (c) $\dim R\leq 2$. We also prove that, if $M$ is a weakly Laskerian $R$-module, then $\lc_{\fa}^{i}(X,M)$ is $\fa$-weakly cofinite for all $i$, whenever $\dim X\leq 2$ or $\dim M\leq 2$ (resp.~$(R,\fm)$ a local ring and $\dim X\leq 3$ or $\dim M\leq 3$). Let $d=\dim M<\infty$, we prove an equivalent condition for top local cohomology module $\lc_{\fa}^{d}(M)$ to be weakly Artinian.
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