Cominimaxness with respect to ideals of dimension one

Abstract

Let $R$ be a commutative Noetherian ring, $\fa$ be an ideal of $R$ and $M$ be an $R$-module. It is shown that if $\Ext^i_R(R/\fa,M)$ is minimax for all $i\leq \dim M$, then the $R$-module $\Ext^i_R(N,M)$ is minimax for all $i\geq 0$ and for any finitely generated $R$-module $N$ with $\Supp_R(N) \subseteq V (\fa)$ and $\dim N \leq 1$. As a consequence of this result we obtain that for any $\fa$-torsion $R$-module $M$ that $\Ext^i_R(R/\fa, M)$ is minimax for all $i\leq \dim M$, all Bass numbers and all Betti numbers of $M$ are finite. This generalizes \cite[Corollary 2.7]{BNS2015}. Also, some equivalent conditions for the cominimaxness of local cohomology modules with respect to ideals of dimension at most one are given.