On the asymptotic behavior of graded generalized local cohomology modules

Let R =(circle plus)n =0Rn be a graded Noetherian ring with local base ring Ro andlet R+ =(circle plus) =1Rn. Let M and N be finitely generated graded R-modules. Inthis paper, we prove some results on the asymptotic behaviour of the n-th graded components Hr+ (M, N)n of H1+ (M, N) for n -+∞. We also study the tamenessand asymptotical stability of the homogeneous components of Hk+ (M, N) for someis with a specified property.