The Cohen-Macaulay Property of Invariant Rings Over Ring of Integers of a Global Field

Let A be the ring of integers of a global field K. Let \(G \subseteq GL_2(A)\) $G\subseteq G{L}_2(A)$ be a finite group. Let G act linearly on \(R = A[X,Y]\) $R=A[X,Y]$ (fixing A). Let \(R^G\) $R^G$ be the ring of invariants. In the equi-characteristic case we prove \(R^G\) $R^G$ is Cohen-Macaulay. In mixed characteristic case we prove that if for all primes p dividing |G|, the Sylow p-subgroup of G has exponent p then \(R^G\) $R^G$ is Cohen-Macaulay. We prove a similar result if for all primes p dividing |G|, the prime p is unramified in K.