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Cominimaxness of local cohomology modules. (English) Zbl 07088770
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 \(\mathrm{Ext}^i_R(R/I,M)\) is minimax for all \(i\leq t+1\). We prove that if \(H^i_I(M)\) is \(\mathrm{FD}_{\leq 1}\) (or weakly Laskerian) for all \(i<t\), then the \(R\)-modules \(H^i_I(M)\) are \(I\)-cominimax for all \(i<t\) and \(\mathrm{Ext}^i_R(R/I,H^t_I(M))\) is minimax for \(i=0,1\). Let \(N\) be a finitely generated \(R\)-module. We prove that \(\mathrm{Ext}^j_R(N,H^i_I(M))\) and \(\mathrm{Tor}^R_j(N,H^i_I(M))\) are \(I\)-cominimax for all \(i\) and \(j\) whenever \(M\) is minimax and \(H^i_I(M)\) is \(\mathrm{FD}_{\leq 1}\) (or weakly Laskerian) for all \(i\).

MSC:
13D45 Local cohomology and commutative rings
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
13C05 Structure, classification theorems for modules and ideals in commutative rings
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