\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

Lemma 70.28.2. Let $B \to S$ be as in Section 70.3. 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 70.15.1. 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 59.4.4. 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 59.4.7. 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 70.28.1 (and Spaces, Lemma 57.12.3, and Morphisms of Spaces, Lemma 59.8.4) 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). $\square$


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