# The Stacks Project

## Tag 06P6

Lemma 69.6.1. 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 58.4.7 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 56.12.3 (resp. Morphisms of Spaces, Lemma 58.8.3, resp. Spaces, Lemma 56.12.3). $\square$

The code snippet corresponding to this tag is a part of the file spaces-groupoids.tex and is located in lines 300–307 (see updates for more information).

\begin{lemma}
\label{lemma-group-scheme-separated}
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).
\end{lemma}

\begin{proof}
We recall that by
Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-section-immersion}
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 \ref{spaces-lemma-base-change-immersions}
(resp.\ Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-base-change-quasi-compact},
resp.\ Spaces, Lemma \ref{spaces-lemma-base-change-immersions}).
\end{proof}

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