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

Abstract

Let $A$ be the ring of integers of global field $K$. Let $G \subseteq GL_2(A)$ be a finite group. Let $G$ act linearly on $R = A[X,Y]$ (fixing $A$). Let $R^G$ be the ring of invariants. In the equi-characteristic case we prove $R^G$ 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 $R^G$ is Cohen-Macaulay. We prove a similar case if for all primes $p$ dividing $|G|$ the prime $p$ is un-ramified in $K$.

Theorems & Definitions (61)

• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Conjecture 1.7
• Remark 2.2
• Proposition 2.3
• proof
• Remark 2.4
• Proposition 2.5