Finiteness of graded local cohomology modules

Let R = ⨁ n ∈ N 0 R n be a Noetherian homogeneous ring with local base ring ( R 0 , m 0 ) and irrelevant ideal R + , let M be a finitely generated graded R -module. In this paper we show that H m 0 R 1 ( H R + 1 ( M ) ) is Artinian and H m 0 R i ( H R + 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 ∈ N 0 , H m 0 R i ( H R + 2 ( M ) ) is Artinian if and only if H m 0 R i + 2 ( H R + 1 ( M ) ) is Artinian. We also prove that H m 0 d ( H R + c ( M ) ) is Artinian, where d = dim ( R 0 ) and c is the cohomological dimension of M with respect to R + . Finally we present some examples which show that H m 0 R 2 ( H R + 1 ( M ) ) and H m 0 R 3 ( H R + 1 ( M ) ) need not be Artinian.