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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Artinianess of graded local cohomology modules
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by Reza Sazeedeh PDF
Proc. Amer. Math. Soc. 135 (2007), 2339-2345 Request permission

Abstract:

Let $R=\bigoplus _{n \in \mathbb {N}} R_n$ be a Noetherian homogeneous ring with local base ring $(R_0, \mathfrak {m}_0)$ and let $M$ be a finitely generated graded $R$-module. Let $a$ be the largest integer such that $H_{R_+}^a(M)$ is not Artinian. We will prove that $H_{R_+}^i(M)/\mathfrak {m}_0H_{R_+}^i(M)$ are Artinian for all $i\geq a$ and there exists a polynomial $\widetilde {P}\in \mathbb {Q}[\mathbf {x}]$ of degree less than $a$ such that $\textrm {length}_{R_0}(H_{R_+}^a(M)_n /\mathfrak {m}_0H_{R_+}^a(M)_n) =\widetilde {P}(n)$ for all $n\ll 0$. Let $s$ be the first integer such that the local cohomology module $H_{R_+}^s(M)$ is not ${R_+}-$cofinite. We will show that for all $i\leq s$ the graded module $\Gamma _{\mathfrak {m}_0}(H_{R_+}^i(M))$ is Artinian.
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Additional Information
  • Reza Sazeedeh
  • Affiliation: Department of Mathematics, Urmia University, Urmia, Iran –and– Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran
  • Email: rsazeedeh@ipm.ir
  • Received by editor(s): January 9, 2006
  • Received by editor(s) in revised form: April 6, 2006
  • Published electronically: March 21, 2007
  • Additional Notes: This research was in part supported by a grant from IPM (No. 84130033)
  • Communicated by: Bernd Ulrich
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 135 (2007), 2339-2345
  • MSC (2000): Primary 13D45, 13E10
  • DOI: https://doi.org/10.1090/S0002-9939-07-08794-1
  • MathSciNet review: 2302554