Base change of invariant subrings

Abstract

Let $R$ be a Dedekind domain, $G$ an affine flat $R$-group scheme, and $B$ a flat $R$-algebra on which $G$ acts. Let $A \rightarrow B^{G}$ be an $R$-algebra map. Assume that $A$ is Noetherian. We show that if the induced map $K \otimes A \rightarrow (K \otimes B)^{K \otimes G}$ is an isomorphism for any algebraically closed field $K$ which is an $R$-algebra, then $S \otimes A \rightarrow (S \otimes B)^{S \otimes G}$ is an isomorphism for any $R$-algebra $S$.