This question probably follows from standard geometric invariant theory. If true I'd to know a reference for it.


Given a projective scheme $X\rightarrow S$ over the base $S$. Let's assume a finite group $G$ is acting on $X$ and its quotient is an $S$-scheme $X//G$.


Is the quotient projective or at least proper? (I have seen versions of this over fields but not for arbitrary base.)



Is quotient of projective scheme over arbitrary base by a finite group also projective
I think the reference you want is:


Seshadri, C. S., **Geometric reductivity over arbitrary base**. *Advances in Math*. 26 (1977), no. 3, 225–274.


In particular, I think you will find conditions for a positive answer in Theorem 4 and Remark 10.


Here is part of Remark 10:


"We have been so far principally working over affine base schemes [$S$] for the sake of simplicity. Theorems 3 and 4 generalize immediately to arbitrary base (cf. Remark 4)...Let $X = \mathrm{Proj} B$, and $Y = \mathrm{Proj} B^G$ and $\varphi$ the canonical rational morphism $\varphi: X\to Y$ induced by $B^G \subset B$. Then Theorem 4 remains true in this case and of course to say that $Y$ is projective over $S$ we have to assume that $S$ is locally of finite type over a universally Japanese ring."


