Let $(R,\U0001d52a)$ be a commutative Noetherian local ring, $\U0001d51e$ be an ideal of $R$ and $M$ a finitely generated $R$-module such that $\U0001d51eM\ne M$ and $\mathrm{cd}(\U0001d51e,M)-\mathrm{grade}(\U0001d51e,M)\le 1$, where $\mathrm{cd}(\U0001d51e,M)$ is the cohomological dimension of $M$ with respect to $\U0001d51e$ and $\mathrm{grade}(\U0001d51e,M)$ is the $M$-grade of $\U0001d51e$. Let $D(-):={\mathrm{Hom}}_{R}(-,E)$ be the Matlis dual functor, where $E:=E(R/\U0001d52a)$ is the injective hull of the residue field $R/\U0001d52a$. We show that there exists the following long exact sequence $$\begin{array}{cc}\hfill 0\u27f6& {H}_{\U0001d51e}^{n-2}\left(D\left({H}_{\U0001d51e}^{n-1}\left(M\right)\right)\right)\u27f6{H}_{\U0001d51e}^{n}\left(D\left({H}_{\U0001d51e}^{n}\left(M\right)\right)\right)\u27f6D\left(M\right)\\ \hfill \u27f6& {H}_{\U0001d51e}^{n-1}\left(D\left({H}_{\U0001d51e}^{n-1}\left(M\right)\right)\right)\u27f6{H}_{\U0001d51e}^{n+1}\left(D\left({H}_{\U0001d51e}^{n}\left(M\right)\right)\right)\\ \hfill \u27f6& {H}_{\U0001d51e}^{n}\left(D\left({H}_{({x}_{1},...,{x}_{n-1})}^{n-1}\left(M\right)\right)\right)\u27f6{H}_{\U0001d51e}^{n}\left(D\left({H}_{(}^{n-1}M\right)\right))\u27f6...,\end{array}$$
where $n:=\mathrm{cd}(\U0001d51e,M)$ is a non-negative integer, ${x}_{1},...,{x}_{n-1}$ is a regular sequence in $\U0001d51e$ on $M$ and, for an $R$-module $L$, ${H}_{\U0001d51e}^{i}\left(L\right)$ is the $i$th local cohomology module of $L$ with respect...