# The Stacks Project

## Tag 0455

Lemma 69.28.2. Let $B \to S$ be as in Section 69.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $G \to U$ be the stabilizer group algebraic space.

1. The following are equivalent
1. $j : R \to U \times_B U$ is separated,
2. $G \to U$ is separated, and
3. $e : U \to G$ is a closed immersion.
2. The following are equivalent
1. $j : R \to U \times_B U$ is locally separated,
2. $G \to U$ is locally separated, and
3. $e : U \to G$ is an immersion.
3. The following are equivalent
1. $j : R \to U \times_B U$ is quasi-separated,
2. $G \to U$ is quasi-separated, and
3. $e : U \to G$ is quasi-compact.

Proof. The group algebraic space $G \to U$ is the base change of $R \to U \times_B U$ by the diagonal morphism $U \to U \times_B U$, see Lemma 69.15.1. Hence if $j$ is separated (resp. locally separated, resp. quasi-separated), then $G \to U$ is separated (resp. locally separated, resp. quasi-separated). See Morphisms of Spaces, Lemma 58.4.4. Thus (a) $\Rightarrow$ (b) in (1), (2), and (3).

Conversely, if $G \to U$ is separated (resp. locally separated, resp. quasi-separated), then the morphism $e : U \to G$, as a section of the structure morphism $G \to U$ is a closed immersion (resp. an immersion, resp. quasi-compact), see Morphisms of Spaces, Lemma 58.4.7. Thus (b) $\Rightarrow$ (c) in (1), (2), and (3).

If $e$ is a closed immersion (resp. an immersion, resp. quasi-compact) then by the result of Lemma 69.28.1 (and Spaces, Lemma 56.12.3, and Morphisms of Spaces, Lemma 58.8.3) we see that $\Delta_{R/U \times_B U}$ is a closed immersion (resp. an immersion, resp. quasi-compact). Thus (c) $\Rightarrow$ (a) in (1), (2), and (3). $\square$

The code snippet corresponding to this tag is a part of the file spaces-groupoids.tex and is located in lines 3196–3221 (see updates for more information).

\begin{lemma}
\label{lemma-diagonal}
Let $B \to S$ be as in Section \ref{section-notation}.
Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.
Let $G \to U$ be the stabilizer group algebraic space.
\begin{enumerate}
\item The following are equivalent
\begin{enumerate}
\item $j : R \to U \times_B U$ is separated,
\item $G \to U$ is separated, and
\item $e : U \to G$ is a closed immersion.
\end{enumerate}
\item The following are equivalent
\begin{enumerate}
\item $j : R \to U \times_B U$ is locally separated,
\item $G \to U$ is locally separated, and
\item $e : U \to G$ is an immersion.
\end{enumerate}
\item The following are equivalent
\begin{enumerate}
\item $j : R \to U \times_B U$ is quasi-separated,
\item $G \to U$ is quasi-separated, and
\item $e : U \to G$ is quasi-compact.
\end{enumerate}
\end{enumerate}
\end{lemma}

\begin{proof}
The group algebraic space $G \to U$ is the base change of $R \to U \times_B U$
by the diagonal morphism $U \to U \times_B U$, see
Lemma \ref{lemma-groupoid-stabilizer}. Hence if
$j$ is separated (resp.\ locally separated, resp.\ quasi-separated),
then $G \to U$ is separated (resp.\ locally separated, resp.\ quasi-separated).
See
Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-base-change-separated}.
Thus (a) $\Rightarrow$ (b) in (1), (2), and (3).

\medskip\noindent
Conversely, if $G \to U$ is separated
(resp.\ locally separated, resp.\ quasi-separated), then the morphism
$e : U \to G$, as a section of the structure morphism $G \to U$ is a closed
immersion (resp.\ an immersion, resp.\ quasi-compact), see
Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-section-immersion}.
Thus (b) $\Rightarrow$ (c) in (1), (2), and (3).

\medskip\noindent
If $e$ is a closed immersion (resp.\ an immersion, resp.\ quasi-compact)
then by the result of
Lemma \ref{lemma-diagram-diagonal}
(and
Spaces, Lemma \ref{spaces-lemma-base-change-immersions}, and
Morphisms of Spaces,
Lemma \ref{spaces-morphisms-lemma-base-change-quasi-compact})
we see that $\Delta_{R/U \times_B U}$ is a closed
immersion (resp.\ an immersion, resp.\ quasi-compact).
Thus (c) $\Rightarrow$ (a) in (1), (2), and (3).
\end{proof}

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