Galois structure of homogeneous coordinate rings

Abstract:

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 $n >> 0$.