امیر مافی

The aim of this note is to improve the main‎ ‎results of [{\bf 9}] (Theorems 2.2, 2.3). We show that if‎ ‎$R=\oplus_{n\in\mathbb{N}_0}R_n$ is a Noetherian homogeneous ring‎ ‎with local base ring $(R_0,{\fm}_0)$, irrelevant ideal $R_+$, and‎ ‎$M$ a finitely generated graded $R$-module, then‎ ‎$H_{{\fm}_0R}^j(H_{R_+}^t(M))$ 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 $\Cd(R_+,M)=2$ then, for‎ ‎each $i\in\mathbb{N}_0$, $H_{{\fm}_0R}^i(H_{R_+}^2(M))$ is‎ ‎Artinian if and only if $H_{{\fm}_0R}^{i+2}(H_{R_+}^1(M))$ is‎ ‎Artinian, where $\Cd(R_+,M)$ is the cohomological dimension of $M$‎ ‎with respect to $R_+$.‎