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Extension functors of cominimax modules. (English) Zbl 1360.13044
Let \(R\) be a commutative Noetherian ring with identity and \(I\) an ideal of \(R\). Recall that an \(R\)-module \(X\) is called minimax if it admits a finitely generated submodule \(Y\) such that the \(R\)-module \(X/Y\) is Artinian. An \(R\)-module \(M\) is called \(I\)-cominimax if \(\text{Supp}_RM\subseteq \text{V}(I)\) and \(\text{Ext}_R^i(R/I,M)\) is minimax for all \(i\geq 0\). Finally, an \(R\)-module \(M\) is called \(I\)-weakly cofinite if \(\text{Supp}_RM\subseteq \text{V}(I)\) and for each \(i\geq 0\), any quotient module of \(\text{Ext}_R^i(R/I,M)\) has finitely many associated primes.
Let \(M\) be an \(I\)-cominimax \(R\)-module and \(N\) a finitely generated \(R\)-module. The authors prove that if either \(\dim_RM\leq 1\), or \(\dim_RN\leq 2\), then \(\text{Ext}_R^i(N,M)\) is \(I\)-cominimax for all \(i\geq 0\). As an application, they show that if \(R\) is local and \(M\) is a minimax \(R\)-module such that \(\dim_R(\text{H}_I^i(M))\leq 2\) for all \(i\geq 0\) , then \(\text{Ext}_R^j(N,\text{H}_I^i(M))\) is \(I\)-weakly cofinite for all finitely generated \(R\)-modules \(N\), and all \(i,j\geq 0\).

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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