• Vanishing of the Top Local Cohomology Modules over Noetherian Rings

    • Fulltext


        Click here to view fulltext PDF

      Permanent link:

    • Keywords


      Artinian modules; attached prime ideals; cohomological dimension; formally isolated; local cohomology; secondary representations.

    • Abstract


      Let 𝑅 be a (not necessarily local) Noetherian ring and 𝑀 a finitely generated 𝑅-module of finite dimension 𝑑. Let $\mathfrak{a}$ be an ideal of 𝑅 and $\mathfrak{M}$ denote the intersection of all prime ideals $\mathfrak{p}\in\mathrm{Supp}_R H^d_a(M)$. It is shown that

      $$H^d_a(M)\simeq H^d_{\mathfrak{M}}(M)/\sum\limits_{n\in\mathbb{N}}\langle \mathfrak{M}\rangle(0:_{H^d_{\mathfrak{M}}(M)}a^n),$$

      where for an Artinian 𝑅-module 𝐴 we put $\langle\mathfrak{M}\rangle A=\cap_{n\in\mathbb{N}}\mathfrak{M}^n A$. As a consequence, it is proved that for all ideals $\mathfrak{a}$ of 𝑅, there are only finitely many non-isomorphic top local cohomology modules $H^d_a(M)$ having the same support. In addition, we establish an analogue of the Lichtenbaum–Hartshorne vanishing theorem over rings that need not be local.

    • Author Affiliations


      Kamran Divaani-Aazar1

      1. Department of Mathematics, Az-Zahra University, Vanak, Post Code 19834, Tehran, Iran
    • Dates

  • Proceedings – Mathematical Sciences | News

    • Editorial Note on Continuous Article Publication

      Posted on July 25, 2019

      Click here for Editorial Note on CAP Mode

© 2022-2023 Indian Academy of Sciences, Bengaluru.