Czechoslovak Mathematical Journal, Vol. 69, No. 1, pp. 75-86, 2019


Cominimaxness of local cohomology modules

Moharram Aghapournahr

Received April 3, 2017.   Published online May 21, 2018.

Abstract:  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 ${\rm FD}_{\leq1}$ (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 ${\rm FD}_{\leq1}$ (or weakly Laskerian) for all $i$.
Keywords:  local cohomology; ${\rm FD}_{\leq n}$ modules; cofinite modules; cominimax modules
Classification MSC:  13D45, 13E10, 13C05
DOI:  10.21136/CMJ.2018.0161-17

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Affiliations:   Moharram Aghapournahr, Department of Mathematics, Faculty of Science, Arak University, Arak, 38156-8-8349, Iran, e-mail: m-aghapour@araku.ac.ir


 
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