Lemma 78.9.5. Let S be a scheme. Let B be an algebraic space over S. Let G be a group algebraic space over B. Let X be a pseudo G-torsor over B. Assume G and X locally of finite type over B.
If G \to B is unramified, then X \to B is unramified.
If G \to B is locally quasi-finite, then X \to B is locally quasi-finite.
Proof. Proof of (1). By Morphisms of Spaces, Lemma 67.38.10 we reduce to the case where B is the spectrum of a field. If X is empty, then the result holds. If X is nonempty, then after increasing the field, we may assume X has a point. Then G \cong X and the result holds.
The proof of (2) works in exactly the same way using Morphisms of Spaces, Lemma 67.27.2. \square
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