Volume 35 • Number 1 • 2012   • A Note on Graded Components of Local Cohomology Modules at the Earliest Level of Non-Artinianess Chia S. Lim Abstract. Let $M$ be a finitely generated graded $R$-module where $R$ is a Noetherian homogeneous ring with local base ring $(R_{0}, m_{0})$ and $R_{+}$, the irrelevant ideal of $R$. Let $ a(M)= \sup \{j \in \mathbb{N}_{0}| H_{R_{+}}^{i}(M)$ is Artinian for all $i < j \}.$ We prove that if $a(M) < \infty$ then $H_{R_{+}}^{a(M)}(M)$ is tame. Some strategies to establish tameness for graded modules in general will be discussed. 2010 Mathematics Subject Classification: Primary: 13M10; Secondary: 57N10, 57M60. Full text: PDF