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

Let S be a Noetherian scheme, φ : X → 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 φ 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 φ 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 φ is flat. The same is true if S is excellent, φ is proper, and Y is Noetherian.