Extension functors of local cohomology modules (Q1758777)

Let \(R\) be a commutative Noetherian ring with nonzero identity and \(\mathfrak a\) be an ideal of \(R\). Let \(N\) be a finitely generated \(\mathfrak a\)-torsion \(R\)-module, \(X\) an \(R\)-module and \(\mathcal{S}\) a Serre class of \(R\)-modules. In this paper, for any two fixed nonnegative integers \(s,t\), the authors investigated the following problems: 1) Determining when the two \(R\)-modules \(\mathrm{Ext}^{s+t}_R(N,X)\) and \(\mathrm{Ext}^{s}_R(N,H_{\mathfrak a}^t(X))\) are in \(\mathcal{S}\); and 2) Determining some conditions under which these two \(R\)-modules are isomorphic. They established some results regarding each of these problems.