Czechoslovak Mathematical Journal, Vol. 70, No. 1, pp. 1-7, 2020
Matlis dual of local cohomology modules
Batoul Naal, Kazem Khashyarmanesh
Received March 13, 2018. Published online September 5, 2019.
Abstract: 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 ${\rm cd}(\mathfrak a,M) - {\rm grade}(\mathfrak a,M) \leq1$, 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 $\rightarrow H^{n-2}_{\mathfrak a}(D(H^{n-1}_{\mathfrak a}(M))) \rightarrow H^n_{\mathfrak a}(D(H^n_{\mathfrak a}(M))) \rightarrow D(M)
\rightarrow H^{n-1}_{\mathfrak a}(D(H^{n-1}_{\mathfrak a}(M))) \rightarrow H^{n+1}_{\mathfrak a}(D(H^n_{\mathfrak a}(M))) \rightarrow H^n_{\mathfrak a}(D(H^{n-1}_{(x_1, \ldots,x_{n-1})}(M)))
\rightarrow H^n_{\mathfrak a}(D(H^{n-1}_{\mathfrak a}(M))) \rightarrow \ldots$,
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 $\mathfrak a$.
Keywords: local cohomology module; Matlis dual functor, filter regular sequence
Affiliations: Batoul Naal, Kazem Khashyarmanesh (corresponding author), Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O.Box 1159-91775, Mashhad, Iran, e-mail: naalbatoul@yahoo.com, khashyar@ipm.ir
