The Cohen-Macaulay property of invariant rings over ring of integers of a global field-II

Abstract: Let $A$ be the ring of integers of a number field $K$. Let $G \subseteq GL_3(A)$ be a finite group. Let $G$ act linearly on $R = A[X,Y, Z]$ (fixing $A$) and let $S = R^G$ be the ring of invariants. Assume the Veronese subring $S^{<m>}$ of $S$ is standard graded. We prove that if for all primes $p$ dividing $|G|$, the Sylow $p$-subgroup of $G$ has exponent $p$ then for all $l \gg 0$ the Veronese subring $S^{<ml>}$ of $S$ is Cohen-Macaulay. We prove a similar result if for all primes $p$ dividing $|G|$, the prime $p$ is unramified in $K$.