• On the Finiteness Properties of Matlis Duals of Local Cohomology Modules

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      https://www.ias.ac.in/article/fulltext/pmsc/118/02/0197-0206

    • Keywords

       

      Local cohomology modules; cofinite modules; associated primes; coassociated primes; filter regular sequences; Matlis duality functor.

    • Abstract

       

      Let 𝑅 be a complete semi-local ring with respect to the topology defined by its Jacobson radical, $\mathfrak{a}$ an ideal of 𝑅, and 𝑀 a finitely generated 𝑅-module. Let $D_R(-):=\mathrm{Hom}_R(-,E)$, where 𝐸 is the injective hull of the direct sum of all simple 𝑅-modules. If 𝑛 is a positive integer such that $\mathrm{Ext}^j_R(R/\mathfrak{a},D_R(H^t_{\mathfrak{a}}(M)))$ is finitely generated for all $t>n$ and all $j\geq 0$, then we show that $\mathrm{Hom}_R(R/\mathfrak{a},D_R(H^n_{\mathfrak{a}}(M)))$ is also finitely generated. Specially, the set of prime ideals in $\mathrm{Coass}_R(H^n_{\mathfrak{a}}(M))$ which contains $\mathfrak{a}$ is finite.

      Next, assume that $(R,\mathfrak{m})$ is a complete local ring. We study the finiteness properties of $D_R(H^r_{\mathfrak{a}}(R))$ where 𝑟 is the least integer 𝑖 such that $H^i_{\mathfrak{a}}(R)$ is not Artinian.

    • Author Affiliations

       

      K Khashyarmanesh1 F Khosh-Ahang1

      1. Department of Mathematics, Ferdowsi University of Mashhad, Centre of Excellence in Analysis on Algebraic Structures (CEAAS), P.O. Box 1159-91775, Mashhad, Iran
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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