The Regular property of Invariant Rings over Regular Domains

The main result of this paper is a generalization of the theorem of Chevalley-Shephard-Todd to the rings of invariants of pseudo-reflection groups over regular domains. More precisely, let $A$ be a regular domain and let $K$ be its field of fractions. Let $G\subseteq G{L}_n(A)$ be a finite group. Let $G$ act linearly on $A[X_1,X_2,\dots ,X_n]$ (fixing $A$). Assume that $\mathrm{\mid }G\mathrm{\mid }$ is invertible in $A$. We prove that $G\subseteq G{L}_n(K)$ is generated by pseudo-reflections if and only if $(A[X_1,X_2,\dots ,X_n]{)}^G$ is regular.