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# Artinianness of Certain Graded Local Cohomology Modules

Published:2011-03-15
Printed: Mar 2012
• Amir Mafi,
Department of Mathematics, University of Kurdistan, P.O. Box: 416, Sanandaj, Iran
• Hero Saremi,
Department of Mathematics, Islamic Azad University, Sanandaj Branch, Sanandaj, Iran
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## Abstract

We show that if $R=\bigoplus_{n\in\mathbb{N}_0}R_n$ is a Noetherian homogeneous ring with local base ring $(R_0,\mathfrak{m}_0)$, irrelevant ideal $R_+$, and $M$ a finitely generated graded $R$-module, then $H_{\mathfrak{m}_0R}^j(H_{R_+}^t(M))$ 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 $\operatorname{cd}(R_+,M)=2$, then for each $i\in\mathbb{N}_0$, $H_{\mathfrak{m}_0R}^i(H_{R_+}^2(M))$ is Artinian if and only if $H_{\mathfrak{m}_0R}^{i+2}(H_{R_+}^1(M))$ is Artinian, where $\operatorname{cd}(R_+,M)$ is the cohomological dimension of $M$ with respect to $R_+$. This improves some results of R. Sazeedeh.
 Keywords: graded local cohomology, Artinian modules
 MSC Classifications: 13D45 - Local cohomology [See also 14B15] 13E10 - Artinian rings and modules, finite-dimensional algebras