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Keywords:
local cohomology module; Matlis dual functor, filter regular sequence
local cohomology module; Matlis dual functor, filter regular sequence
Summary:
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 $\mathfrak a M\neq M$ and ${\rm cd}(\mathfrak a,M) - {\rm grade}(\mathfrak a,M)\leq 1$, where ${\rm cd}(\mathfrak a,M)$ is the cohomological dimension of $M$ with respect to $\mathfrak a$ and ${\rm grade}(\mathfrak a,M)$ is the $M$-grade of $\mathfrak a$. Let $D(-) := {\rm Hom}_R(-,E)$ 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 {eqnarray*} 0 \longrightarrow & H^{n-2}_{\mathfrak a}(D(H^{n-1}_{\mathfrak a}(M))) \longrightarrow H^{n}_{\mathfrak a}(D(H^{n}_{\mathfrak a}(M))) \longrightarrow D(M) \\ \longrightarrow & H^{n-1}_{\mathfrak a}(D(H^{n-1}_{\mathfrak a}(M))) \longrightarrow H^{n+1}_{\mathfrak a}(D(H^{n}_{\mathfrak a}(M))) \\ \longrightarrow & H^{n}_{\mathfrak a}(D(H^{n-1}_{(x_1, \ldots ,x_{n-1})}(M))) \longrightarrow H^{n}_{\mathfrak a}(D(H^{n-1}_\mathfrak (M))) \longrightarrow \ldots , \end {eqnarray*} where $n:={\rm cd}(\mathfrak a,M)$ is a non-negative integer, $x_1, \ldots ,x_{n-1}$ is a regular sequence in $\mathfrak a$ on $M$ and, for an $R$-module $L$, $H^i_{\mathfrak a}(L)$ is the $i$th local cohomology module of $L$ with respect to $\mathfrak a$.
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