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 $\mathrm{H}^0(X,\mathcal{F} \otimes \mathcal{L}^n)$ 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 $\mathrm{H}^0(X,\mathcal{F} \otimes \mathcal{L}^n)$ is a direct sum of these indecomposables, with multiplicities given by generalized Hilbert polynomials for $n >> 0$.