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Tag 06P6

Chapter 69: Groupoids in Algebraic Spaces > Section 69.6: Properties of group algebraic spaces

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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