Minimaxness and cofiniteness properties of generalized local cohomology with respect to a pair of ideals
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Let $I$ and $J$ be two ideals of a commutative Noetherian ring $R$ and $M$, $N$ be two non-zero finitely generated $R$-modules. Let $t$ be a non-negative integer such that $H^{i}_{I, J}(N)$ is $(I, J)$-minimax for all $i<t$. It is shown that the generalized local cohomology module $H^{i}_{I, J}(M, N)$ is $(I, J)$-Cofinite minimax for all $i<t$. Also, we prove that the $R$-module $Ext^{j}_{R}(R/I, H^{i}_{I, J}(N))$ is finitely generated for all $i\leq t$ and $j=0, 1$.
Abstract
Let $I$ and $J$ be two ideals of a commutative Noetherian ring $R$ and $M$, $N$ be two non-zero finitely generated $R$-modules. Let $t$ be a non-negative integer such that $H^{i}_{I, J}(N)$ is $(I, J)$-minimax for all $i<t$. It is shown that the generalized local cohomology module $H^{i}_{I, J}(M, N)$ is $(I, J)$-Cofinite minimax for all $i<t$. Also, we prove that the $R$-module $Ext^{j}_{R}(R/I, H^{i}_{I, J}(N))$ is finitely generated for all $i\leq t$ and $j=0, 1$.
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