A Note on Graded Components of Local Cohomology Modules at the Earliest Level of Non-Artinianess

Let M be a finitely generated graded R-module where R is a Noetherian homogeneous ring with local base ring (Ro, mo) and R+, the irrelevant ideal of R. Let a(M)=sup{j ϵin No| H^i R+(M) is Artinian for all i j .We prove that if α(M) infty then H^α(M) R+ (M) is tame. Some strategies to establish tameness for graded modules in general will be discussed.