The Artinian property of certain graded generalized local chohomology modules

Let R=⊕n∈N0Rn be a Noetherian homogeneous ring with local base ring (R0,m0), M and N two finitely generated graded R-modules. Let t be the least integer such that HtR+(M,N) is not minimax. We prove that Hjm0R(HtR+(M,N)) is Artinian for j=0,1. Also, we show that if cd(R+,M,N)=2 and t∈N0, then Htm0R(H2R+(M,N)) is Artinian if and only if Ht+2m0R(H1R+(M,N)) is Artinian.