I have some questions about GIT quotients and extensions of scalars of categorical quotients:
1) Let $X$ be a complex algebraic quasi-affine variety, $G$ an algebraic reductive group over $\mathbb{C}$ acting on $X$ . What are the conditions for the existence of the categorical quotient $X//G$ ?
2) Let $\mathcal{X}$ be a separated scheme over a finitely generated $\mathbb{Z}$ -algebra $R$ such that the inclusion $\phi:R\hookrightarrow\mathbb{C}$ is a flat morphism, $\mathcal{G}$ an algebraic reductive group over $R$ acting on $\mathcal{X}$ . Suppose the categorical quotient $\mathcal{X}//\mathcal{G}$ exists. Denote by $\mathcal{X}_{\phi}$ , $\mathcal{G}_{\phi}$ and $\left(\mathcal{X}//\mathcal{G}\right)_{\phi}$ the schemes obtained by extension of scalars from $\mathcal{X}$ , $\mathcal{G}$ and $\mathcal{X}//\mathcal{G}$ via $\phi$ (i.e., fiber products of schemes via $\phi$ ). Is it true that $\left(\mathcal{X}//\mathcal{G}\right)_{\phi}=\mathcal{X}_{\phi}//\mathcal{G}_{\phi}$ ? I know that this is true when $\mathcal{X}$ is an affine scheme by Lemma 2 in Seshadri's “Geometric reductivity over arbitrary base”. Thanks in advance for any suggestions, answers or comments.
