Cominimaxness of certain general local cohomology modules

Abstract

Let $R$ be a commutative Noetherian ring, $\Phi$ a system of ideals of $R$ and $I\in \Phi$. Let $t\in\Bbb{N}_0$ be an integer and $M$ an $R$-module such that $Ext^i_R(R/I,M)$ is minimax for all $i\leq t+1$. We prove that if the $R$-module $H^{i}_\Phi(M)$ is ${FD_{\leq 1}}$ (or weakly Laskerian) for all $i<t$, then $H^i_{\Phi}(M)$ is $\Phi$-cominimax for all $i<t$ and for any $FD_{\leq 0}$ (or minimax) submodule $N$ of $H^t_{\Phi}(M)$, the $R$-modules $Hom_{R}(R/I,H^t_{\Phi}(M)/N)$ and $Ext^1_{R}(R/I,H^t_{\Phi}(M)/N)$ are minimax. Let $N$ be a finitely generated $R$-module. We also prove that $Ext^j_{R}(N,H^i_\Phi(M))$ and $Tor^R_{j}(N,H^i_{\Phi}(M))$ are $\Phi$-cominimax for all $i$ and $j$ whenever $M$ is minimax and $H^i_{\Phi}(M)$ is ${FD_{\leq 1}}$ (or weakly Laskerian) for all $i$.

Keywords

• Local cohomology,${\rm FD_{\leq n}}$ modules,$ETH$-cominimax modules

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