Lemma 65.14.5. Notation and assumptions as in Lemma 65.14.3. Assume G is finite. Then
if U \to S is quasi-separated, then U/G is quasi-separated over S, and
if U \to S is separated, then U/G is separated over S.
Lemma 65.14.5. Notation and assumptions as in Lemma 65.14.3. Assume G is finite. Then
if U \to S is quasi-separated, then U/G is quasi-separated over S, and
if U \to S is separated, then U/G is separated over S.
Proof. In the proof of Lemma 65.13.1 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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