The Stacks project

Lemma 77.6.2. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Assume $G \to B$ is locally of finite type. Then $G \to B$ is unramified (resp. locally quasi-finite) if and only if $G \to B$ is unramified (resp. quasi-finite) at $e(b)$ for all $b \in |B|$.

Proof. By Morphisms of Spaces, Lemma 66.38.10 (resp. Morphisms of Spaces, Lemma 66.27.2) there is a maximal open subspace $U \subset G$ such that $U \to B$ is unramified (resp. locally quasi-finite) and formation of $U$ commutes with base change. Thus we reduce to the case where $B = \mathop{\mathrm{Spec}}(k)$ is the spectrum of a field. Let $g \in G(K)$ be a point with values in an extension $K/k$. Then to check whether or not $g$ is in $U$, we may base change to $K$. Hence it suffices to show

\[ G \to \mathop{\mathrm{Spec}}(k)\text{ is unramified at }e \Leftrightarrow G \to \mathop{\mathrm{Spec}}(k)\text{ is unramified at }g \]

for a $k$-rational point $g$ (resp. similarly for quasi-finite at $g$ and $e$). Since translation by $g$ is an automorphism of $G$ over $k$ this is clear. $\square$


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