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Keywords:
local cohomology; ${\rm FD}_{\leq n}$ modules; cofinite modules; cominimax modules
local cohomology; ${\rm FD}_{\leq n}$ modules; cofinite modules; cominimax modules
Summary:
Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$. Let $t\in \mathbb {N}_0$ be an integer and $M$ an $R$-module such that ${\rm Ext}^i_R(R/I,M)$ is minimax for all $i\leq t+1$. We prove that if $H^{i}_{I}(M)$ is ${\rm FD}_{\leq 1}$ (or weakly Laskerian) for all $i<t$, then the $R$-modules $H^{i}_{I}(M)$ are $I$-cominimax for all $i<t$ and ${\rm Ext}^i_R(R/I,H^{t}_{I}(M))$ is minimax for $i=0,1$. Let $N$ be a finitely generated $R$-module. We prove that ${\rm Ext}^j_R(N,H^{i}_{I}(M))$ and ${\rm Tor}^R_{j}(N,H^{i}_I(M))$ are $I$-cominimax for all $i$ and $j$ whenever $M$ is minimax and $H^{i}_{I}(M)$ is ${\rm FD}_{\leq 1}$ (or weakly Laskerian) for all $i$.
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