On minimax and generalized local cohomology modules
Let \(\mathfrak a\) be an ideal of a commutative noetherian ring \(R\) and \(M,N\) two finitely generated \(R\)-modules. The paper concerns the finiteness properties of the generalized local cohomology modules \(H_{\mathfrak a}^i(M,N):={\varinjlim}_n\mathrm{Ext}^i_R(M/{\mathfrak{a}}^nM,N)\). Recall that an \(R\)-module \(K\) is said to be minimax if it has a finitely generated submodule such that the quotient by it is Artinian. Let \(t\) be a non-negative integer. The author showed that if \(H_{\mathfrak a}^i(R,N)\) is minimax for all \(i<t\), then \(H_{\mathfrak a}^i(M,N)\) is minimax for all \(i<t\). Also, she proved that if \(H_{\mathfrak a}^i(M,N)\) is minimax for all \(i<t\), then for any minimax submodule \(L\) of \(H_{\mathfrak a}^t(M,N)\), the \(R\)-module \(\Hom_R(R/\mathfrak a,H_{\mathfrak a}^t(M,N)/L)\) is finitely generated.
scientific article; zbMATH DE number 5780183