Cofiniteness of general local cohomology modules for small dimensions (Q2828672)

Let \(R\) be a commutative Noetherian ring with identity and \(M\) a finitely generated \(R\)-module. Let \(\Phi\) be a system of ideals of \(R\). This paper studies some finiteness properties of the generalized local cohomology modules \(H_{\Phi}^i(M)\). Note that for an ideal \(\mathfrak a\) of \(R\), the usual local cohomology functors \(H_{\mathfrak a}^i(-)\) correspond to the system of ideals \(\{\mathfrak{a}^n\}_{n\in \mathbb{N}}\).NEWLINENEWLINEFor an \(R\)-module \(X\), set NEWLINE\[NEWLINE\Gamma_{\Phi}(X):=\{x\in X| \;Ix=0 \;\text{ for some } I\in \Phi \}.NEWLINE\]NEWLINE For each non-negative integer \(i\), functor \(H_{\Phi}^i(-)\) is defined to be the \(i\)-th right derived functor of \(\Gamma_{\Phi}(-)\).NEWLINENEWLINERecall that an \(R\)-module \(X\) is called weakly Laskerian if every quotient module of \(X\) has finitely many associated primes. Also, an \(R\)-module \(X\) is called minimax if it possesses a finitely generated submodule \(Y\) such that the quotient module \(X/Y\) is Artinian.NEWLINENEWLINESuppose that \(t\in \mathbb{N}\) is such that the \(R\)-module \(H_{\Phi}^i(M)\) is weakly Laskerian for all \(i<t\). Let \(N\) be a minimax submodule of \(H_{\Phi}^t(M)\) and \(I\in {\Phi}\). The authors show that the \(R\)-modules \(\mathrm{Ext}_R^i(R/I,H_{\Phi}^j(M))\), \(\mathrm{Ext}_R^1(R/I,H_{\Phi}^t(M)/N)\) and \(\mathrm{Hom}_R(R/I,H_{\Phi}^t(M)/N)\) are finitely generated for all \(i\geq 0\) and \(0\leq j<t\).