The Stacks project

77.6 Properties of group algebraic spaces

In this section we collect some simple properties of group algebraic spaces which hold over any base.

Lemma 77.6.1. 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).

Proof. We recall that by Morphisms of Spaces, Lemma 66.4.7 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 64.12.3 (resp. Morphisms of Spaces, Lemma 66.8.4, resp. Spaces, Lemma 64.12.3). $\square$

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. $\square$

Lemma 77.6.3. 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.

  1. 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.

  2. 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 66.38.10 (resp. Morphisms of Spaces, Lemma 66.27.2) 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 77.6.2 we see that $U = e^{-1}(W)$ in either case. $\square$


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