Lemma 64.14.5. Notation and assumptions as in Lemma 64.14.3. Assume $G$ is finite. Then

1. if $U \to S$ is quasi-separated, then $U/G$ is quasi-separated over $S$, and

2. if $U \to S$ is separated, then $U/G$ is separated over $S$.

Proof. In the proof of Lemma 64.13.1 we saw that it suffices to prove the corresponding properties for the morphism $j : R \to U \times _ S U$. If $U \to S$ is quasi-separated, then for every affine open $V \subset U$ which maps into an affine of $S$ the opens $g(V) \cap V$ are quasi-compact. It follows that $j$ is quasi-compact. If $U \to S$ is separated, the diagonal $\Delta _{U/S}$ is a closed immersion. Hence $j : R \to U \times _ S U$ is a finite coproduct of closed immersions with disjoint images. Hence $j$ is a closed immersion. $\square$

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