# Quotient Scheme of a Proper Scheme

Let $$X$$ be a scheme proper over $$\mathbb{Z}$$, $$G$$ a finite group acting on $$X$$. Suppose that the quotient scheme $$Y := X/G$$ is well-defined.

Should $$Y$$ be then proper over $$\mathbb{Z}$$ as well?

• By Stacks 01W6 this is equivalent to $Y$ separated. I know examples of nonseparated quotients when quotienting by a $\Bbb C^\times$ action but I can't remember right now whether this can happen or not when taking the quotient by a finite group. Mar 19, 2020 at 5:04
• I don't think this can go wrong when a finite group acts. For example if $X$ is a Hausdorff topological space and $G$ is a finite group acting on $X$, then the quotient space $X/G$ is automatically Hausdorff. Mar 19, 2020 at 10:36