The Stacks project

Lemma 77.9.5. 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 $B$.

  1. If $G \to B$ is unramified, then $X \to B$ is unramified.

  2. 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 66.38.10 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 66.27.2. $\square$

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