Lemma 78.6.1. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Then $G \to B$ is separated (resp. quasi-separated, resp. locally separated) if and only if the identity morphism $e : B \to G$ is a closed immersion (resp. quasi-compact, resp. an immersion).
78.6 Properties of group algebraic spaces
In this section we collect some simple properties of group algebraic spaces which hold over any base.
Proof. We recall that by Morphisms of Spaces, Lemma 67.4.7 we have that $e$ is a closed immersion (resp. quasi-compact, resp. an immersion) if $G \to B$ is separated (resp. quasi-separated, resp. locally separated). For the converse, consider the diagram
\[ \xymatrix{ G \ar[r]_-{\Delta _{G/B}} \ar[d] & G \times _ B G \ar[d]^{(g, g') \mapsto m(i(g), g')} \\ B \ar[r]^ e & G } \]
It is an exercise in the functorial point of view in algebraic geometry to show that this diagram is cartesian. In other words, we see that $\Delta _{G/B}$ is a base change of $e$. Hence if $e$ is a closed immersion (resp. quasi-compact, resp. an immersion) so is $\Delta _{G/B}$, see Spaces, Lemma 65.12.3 (resp. Morphisms of Spaces, Lemma 67.8.4, resp. Spaces, Lemma 65.12.3). $\square$
Lemma 78.6.2. 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 67.38.10 (resp. Morphisms of Spaces, Lemma 67.27.2) 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. $\square$
Lemma 78.6.3. 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.
There exists a maximal open subspace $U \subset B$ such that $G_ U \to U$ is unramified and formation of $U$ commutes with base change.
There exists a maximal open subspace $U \subset B$ such that $G_ U \to U$ is locally quasi-finite and formation of $U$ commutes with base change.
Proof. By Morphisms of Spaces, Lemma 67.38.10 (resp. Morphisms of Spaces, Lemma 67.27.2) there is a maximal open subspace $W \subset G$ such that $W \to B$ is unramified (resp. locally quasi-finite). Moreover formation of $W$ commutes with base change. By Lemma 78.6.2 we see that $U = e^{-1}(W)$ in either case. $\square$
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