On the Artinianness of Graded Local Cohomology Modules

Let R = ⨁n≥ 0 Rn be a homogeneous noetherian ring with local base ring $(R_0, \frak{m}_0)$, and N a finitely generated graded R-module. Let $H_{R_+}^i(N)$ be the i-th local cohomology module of N with respect to R+ := ⨁n > 0 Rn. Let t be the largest integer such that $H_{R_+}^t(N)$ is not minimax. We prove that $H_{R_+}^i(N)$ is $\frak{m}_0$-coartinian for any i > t, and $H_{R_+}^t(N)/\mathfrak{m}_0H_{R_+}^t(N)$ is artinian. Let s be the first integer such that $H_{R_+}^s(N)$ is not minimax. We show that for any i ≤ s, the graded module $\Gamma_{\mathfrak{m}_0}(H_{R_+}^i(N))$ is artinian.