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Tag 0455

Chapter 69: Groupoids in Algebraic Spaces > Section 69.28: Separation conditions

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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