Finiteness of graded local cohomology modules

Let R=⨁n∈N0Rn be a Noetherian homogeneous ring with local base ring (R0,m0) and irrelevant ideal R+, let M be a finitely generated graded R-module. In this paper we show that Hm0R1(HR+1(M)) is Artinian and Hm0Ri(HR+1(M)) is Artinian for each i in the case where R+ is principal. Moreover, for the case where ara(R+)=2, we prove that, for each i∈N0, Hm0Ri(HR+2(M)) is Artinian if and only if Hm0Ri+2(HR+1(M)) is Artinian. We also prove that Hm0d(HR+c(M)) is Artinian, where d=dim(R0) and c is the cohomological dimension of M with respect to R+. Finally we present some examples which show that Hm0R2(HR+1(M)) and Hm0R3(HR+1(M)) need not be Artinian.