# The Artinian property of certain graded generalized local chohomology modules

Document Type : Research Paper

Authors

Department of Mathematics, Soochow University

Abstract

Let $R=\oplus_{n\in \Bbb N_0}R_n$ be a
Noetherian homogeneous ring with local base ring
$(R_0,\frak{m}_0)$, $M$ and $N$ two finitely generated graded
$R$-modules. Let $t$ be the least integer such that
$H^t_{R_+}(M,N)$ is not minimax. We prove that
$H^j_{\frak{m}_0R}(H^t_{R_+}(M,N))$ is Artinian for $j=0,1$. Also,
we show that if ${\rm cd}(R_{+},M,N)=2$ and $t\in \Bbb N_0$, then
$H^t_{\frak{m}_0R}(H^2_{R_+}(M,N))$ is Artinian if and only if
$H^{t+2}_{\frak{m}_0R}(H^1_{R_+}(M,N))$ is Artinian.

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### History

• Receive Date: 26 September 2013
• Revise Date: 06 March 2014
• Accept Date: 13 March 2014
• First Publish Date: 01 April 2015