Finiteness Properties of Minimax and $\mathfrak{a}$-Minimax Generalized Local Cohomology Modules

Abstract

Let $R$ be a commutative Noetherian ring with nonzero identity, let $\mathfrak{a}$ be an ideal of $R$, and let $M$ and $N$ be two (finitely generated) $R$-modules. We prove that $H_{\mathfrak{a}}^i\left( {M,N} \right)$ is a minimax $\mathfrak{a}$-cofinite $R$-module for all $i < t, t ∈ {{\mathbb{N}}_0}$, if and only if $H_{\mathfrak{a}}^i\left( {M,N} \right)$ is a minimax ${R_{\mathfrak{p}}}$ -module for all $i < t$. We also show that, under certain conditions, $\mathrm{Ho}{{\mathrm{m}}_R}\left( {\frac{R}{\mathfrak{a}},H_{\mathfrak{a}}^t\left( {M,N} \right)} \right)$ is minimax $(t ∈ {{\mathbb{N}}_0})$. Finally, we study necessary conditions for $H_{\mathfrak{a}}^i\left( {M,N} \right)$ to be $\mathfrak{a}$-minimax.