“Geometric quotients are algebraic schemes” based on Fogarty’s idea

Abstract

Let $S$ be a Noetherian scheme, $\varphi : X \to Y$ a surjective $S$-morphism of $S$-schemes, with $X$ of finite type over $S$. We discuss what makes $Y$ of finite type. First, we prove that if $S$ is excellent, $Y$ is reduced, and $\varphi$ is universally open, then $Y$ is of finite type. We apply this to understand Fogarty’s theorem in “Geometric quotients are algebraic schemes, Adv. Math. 48 (1983), 166-171” for the special case that the group scheme $G$ is flat over the Noetherian base scheme $S$ and that the quotient map is universally submersive. Namely, we prove that if $G$ is a flat $S$-group scheme of finite type acting on $X$ and $\varphi$ is its universal strict orbit space, then $Y$ is of finite type ($S$ need not be excellent. Geometric fibers of $G$ can be disconnected and non-reduced). Utilizing the technique used there, we also prove that $Y$ is of finite type if $\varphi$ is flat. The same is true if $S$ is excellent, $\varphi$ is proper, and $Y$ is Noetherian.