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Artinianness of Certain Graded Local Cohomology Modules

Published online by Cambridge University Press:  20 November 2018

Amir Mafi  and
Hero Saremi
Amir Mafi
Affiliation:
Department of Mathematics, University of Kurdistan, P.O. Box: 416, Sanandaj, Iran andInstitute for Research in Fundamental Science (IPM), P.O. Box 19395-5746, Tehran, Iran e-mail: a_mafi@ipm.ir
Hero Saremi
Affiliation:
Department of Mathematics, Islamic Azad University, Sanandaj Branch, Sanandaj, Iranh saremi@iausdj.ac.irherosaremi@yahoo.com
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Abstract

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We show that if $R\,=\,{{\oplus }_{n\in \mathbb{N}0}}\,{{R}_{n}}$ is a Noetherian homogeneous ring with local base ring $({{R}_{0}},\,{{m}_{0}})$, irrelevant ideal ${{R}_{+}}$, and $M$ a finitely generated graded $R$-module, then $H_{{{m}_{0}}R}^{j}\,(H_{R+}^{t}\,(M))$ is Artinian for $j\,=\,0,\,1$ where $t\,=\,\inf ${$i\in {{\mathbb{N}}_{0}}:H_{R+}^{i}(M)$ is not finitely generated}. Also, we prove that if $\text{cd(}{{R}_{+}},M)\,=\,2$, then for each $i\,\in \,{{\mathbb{N}}_{0}},\,H_{{{m}_{0}}R}^{i}\,(H_{R+}^{2}\,(M))$ is Artinian if and only if $H_{{{m}_{0}}R}^{i+2}(H_{R+}^{1}(M))$ is Artinian, where $ \text{cd(}{{R}_{+}},\,M)$ is the cohomological dimension of $M$ with respect to ${{R}_{+}}$. This improves some results of R. Sazeedeh.

Keywords

13D4513E10graded local cohomologyArtinian modules
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 1 , 01 March 2012 , pp. 153 - 156
Copyright
Copyright © Canadian Mathematical Society 2012

References

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