Abstract :
Let image be a Noetherian homogeneous ring with local base ring image and irrelevant ideal R+, let M be a finitely generated graded R-module. In this paper we show that image is Artinian and image is Artinian for each i in the case where R+ is principal. Moreover, for the case where image, we prove that, for each image, image is Artinian if and only if image is Artinian. We also prove that image is Artinian, where image and c is the cohomological dimension of M with respect to R+. Finally we present some examples which show that image and image need not be Artinian.
