• On the Matlis Duals of Local Cohomology Modules and Modules of Generalized Fractions

• Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/120/01/0035-0043

• Keywords

Local cohomology module; Matlis dual functor; module of generalized fractions; filter regular sequence.

• Abstract

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring with non-zero identity, $\mathfrak{a}$ a proper ideal of 𝑅 and 𝑀 a finitely generated 𝑅-module with $\mathfrak{a}M\neq M$. Let $D(-):=\mathrm{Hom}_R(-,E)$ be the Matlis dual functor, where $E:=E(R/\mathfrak{m})$ is the injective hull of the residue field $R/\mathfrak{m}$. In this paper, by using a complex which involves modules of generalized fractions, we show that, if $x_1,\ldots,x_n$ is a regular sequence on 𝑀 contained in $\mathfrak{a}$, then $H^n_{(x_1,\ldots,x_n)R}(D(H^n_{\mathfrak{a}}(M)))$ is a homomorphic image of $D(M)$, where $H^i_{\mathfrak{b}}(-)$ is the 𝑖-th local cohomology functor with respect to an ideal $\mathfrak{b}$ of 𝑅. By applying this result, we study some conditions on a certain module of generalized fractions under which $D(H^n_{(x_1,\ldots,x_n)R}(D(H^n_{\mathfrak{a}}(M))))\cong D(D(M))$.

• Author Affiliations

1. Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O. Box 1159–91775, Mashhad, Iran

• Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
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• Editorial Note on Continuous Article Publication

Posted on July 25, 2019