Multiplicities of graded components of local cohomology modules

Let M be a finitely generated graded module over a Noetherian homogeneous ring R with local base ring image. Then, the nth graded component image of the ith local cohomology module of M with respect to the irrelevant ideal R+ of R is a finitely generated R0-module which vanishes for all nmuch greater-than0. In various situations we show that, for an image-primary ideal image, the multiplicity image of image is antipolynomial in n of degree less than i. In particular we consider the following three cases: (a) i