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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 $ {\rm 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.

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