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 $\HH^0(X,$\mathcal{F}$ \otimes $\mathcal{L}$^n)$when$n \ge 0$,and$$\mathcal{F}$$and$$\mathcal{L}$$ are coherent $G$-sheaves on $X$ such that $\mathcal{L}$ 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 indecomposable $RG$-modules such that each ${\text{HH}}^0(X,\mathcal{F}\otimes {\mathcal{L}}^n)$ is a direct sum of these indecomposables, with multiplicites given by generalized Hilbert polynomials for $n>>0$.