Galois structure of homogeneous coordinate rings

Suppose G is a finite group acting on a projective scheme X over a commutative Noetherian ring R. We study the RG-modules H-0(X, F circle times L-n) when n >= 0, and F and L are coherent G-sheaves on X such that L is an ample line bundle. We show that the classes of these modules in the Grothendieck group G(0)(RG) of all finitely generated RG-modules lie in a finitely generated subgroup. Under various hypotheses, we show that there is a finite set of inde-composable RG-modules such that each H-0(X, F circle times L-n) is a direct sum of these indecomposables, with multiplicities given by generalized Hilbert polynomials for n >> 0.