Matlis dual of local cohomology modules

Let (R, \(\mathfrak{m}\)) be a commutative Noetherian local ring, \(\mathfrak{a}\) be an ideal of R and M a finitely generated R-module such that \(\mathfrak{a}M \ne M\) and cd(\(\mathfrak{a}\), M) — grade(\(\mathfrak{a}\), M) ⩽ 1, where cd(\(\mathfrak{a}\), M) is the cohomological dimension of M with respect to \(\mathfrak{a}\) and grade(\(\mathfrak{a}\), M) is the M-grade of \(\mathfrak{a}\). Let D(−):= HomR(−, E) be the Matlis dual functor, where E:= E(R/\(\mathfrak{m}\)) is the injective hull of the residue field R/\(\mathfrak{m}\). We show that there exists the following long exact sequence $$\begin{array}{l}{0 \longrightarrow H_{\mathfrak{a}}^{n-2}(D(H_{\mathfrak{a}}^{n-1}(M))) \longrightarrow H_{\mathfrak{a}}^{n}(D(H_{\mathfrak{a}}^{n}(M))) \longrightarrow D(M)} \\ {\quad \longrightarrow H_{\mathfrak{a}}^{n-1}(D(H_{\mathfrak{a}}^{n-1}(M))) \longrightarrow H_{\mathfrak{a}}^{n+1}(D(H_{\mathfrak{a}}^{n}(M)))} \\ {\quad \longrightarrow H_{\mathfrak{a}}^{n}(D(H_{(x_{1},\ldots, x_{n-1})}^{n-1}(M))) \longrightarrow H_{\mathfrak{a}}^{n}(D(H_{(}^{n-1} M))) \longrightarrow \cdots},\end{array}$$ where n:= cd(\(\mathfrak{a}\), M) is a non-negative integer, x1,…, xn−1 is a regular sequence in \(\mathfrak{a}\) on M and, for an R-module L, \(H_{\mathfrak{a}}^{n}(L)\) is the ith local cohomology module of L with respect to \(\mathfrak{a}\).