On the Matlis duals of local cohomology modules and modules of generalized fractions (Q968057)

Let \((R,\mathfrak{m})\) be a commutative noetherian local ring and \(M\) a nonzero finitely generated \(R\)-module. Let \(\mathfrak{a}\) be a proper ideal of \(R\) and \(D(-):=\Hom_R(-,E_R(R/\mathfrak{m}))\) denote the Matlis duality functor. Recall that for each nonnegative integer \(i\), the \(i\)-th local cohomology module of \(M\) with respect to \(\mathfrak{a}\) is defined by \(H_{\mathfrak{a}}^i(M):={\varinjlim}_n \mathrm{Ext}_R^i(R/\mathfrak{a^n},M)\). Let \(n>0\) be an integer and \(\underline{x}:=x_1,x_2,\ldots ,x_n\) an \(M\)-regular sequence contained in \(\mathfrak{a}\). The main result of the paper asserts that \(H_{\underline{x}R}^n(D(H_{\mathfrak{a}}^n(M)))\) is a homeomorphic image of \(D(M)\). In particular, this implies that \(H_{\underline{x}R}^n(D(H_{\mathfrak{a}}^n(M)))\) is artinain.