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Proceedings of the American Mathematical Society

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



Artinianess of graded local cohomology modules

Author: Reza Sazeedeh
Journal: Proc. Amer. Math. Soc. 135 (2007), 2339-2345
MSC (2000): Primary 13D45, 13E10
Published electronically: March 21, 2007
MathSciNet review: 2302554
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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

Keywords: Graded local cohomology, Artinian module, polynomial
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
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.