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Some results on the local cohomology of minimax modules. (English) Zbl 1340.13009
Let \(R\) be a commutative Noetherian ring with identity. Let \(I\) be an ideal of \(R\) and \(X\) an \(R\)-module. This paper studies the finiteness properties of the local cohomology modules \[ H_I^i(X):=\underset {n} {\varinjlim}\text{Ext}^i_R\left(\frac {R}{I^n},X\right);\;\;i\in \mathbb {N}_0. \] Recall that \(X\) is said to be minimax if it has a finitely generated submodule \(Y\) such that \(X/Y\) is an Artinian \(R\)-module. Also, the \(R\)-module \(X\) is said to be weakly Laskerian if each quotient module of \(X\) has finitely many associated prime ideals.
Let \(M\) be a minimax \(R\)-module such that \(\dim_R \text{Supp}_R(H^i_I(M))\leq 1\) for all integers \(i\). The authors show that \(\text{Ext}_R^j(R/I, H^i_I(M))\) is minimax for all integers \(i\) and \(j\).
Assume that \(R\) is local and \(t\) is a non-negative integer. Let \(N\) be a weakly Laskerian \(R\)-module such that \(\dim_R \text{Supp}_R(H^i_I(N)) \leq 2\) for all \(i<t\). The authors prove that \(\text{Ext}_R^j(R/I,H^i_I(N))\) and \(\text{Hom}_R(R/I,H^t_I(N))\) are weakly Laskerian for all \(i<t\) and all \(j\geq 0\).

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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