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Tag 08BH

Chapter 70: More on Groupoids in Spaces > Section 70.9: Properties of groups over fields and groupoids on fields

Lemma 70.9.4. In Situation 70.9.2 assume $R$ is a decent space. Then $R$ is a separated algebraic space. In Situation 70.9.1 assume that $G$ is a decent algebraic space. Then $G$ is separated algebraic space.

Proof. We first prove the second assertion. By Groupoids in Spaces, Lemma 69.6.1 we have to show that $e : S \to G$ is a closed immersion. This follows from Decent Spaces, Lemma 59.13.5.

Next, we prove the second assertion. To do this we may replace $B$ by $S$. By the paragraph above the stabilizer group scheme $G \to U$ is separated. By Groupoids in Spaces, Lemma 69.28.2 the morphism $j = (t, s) : R \to U \times_S U$ is separated. As $U$ is the spectrum of a field the scheme $U \times_S U$ is affine (by the construction of fibre products in Schemes, Section 25.17). Hence $R$ is separated, see Morphisms of Spaces, Lemma 58.4.9. $\square$

    The code snippet corresponding to this tag is a part of the file spaces-more-groupoids.tex and is located in lines 691–697 (see updates for more information).

    \begin{lemma}
    \label{lemma-group-scheme-over-field-separated}
    In Situation \ref{situation-groupoid-on-field}
    assume $R$ is a decent space. Then $R$ is a separated algebraic space.
    In Situation \ref{situation-group-over-field} assume that
    $G$ is a decent algebraic space. Then $G$ is separated algebraic space.
    \end{lemma}
    
    \begin{proof}
    We first prove the second assertion. By Groupoids in Spaces,
    Lemma \ref{spaces-groupoids-lemma-group-scheme-separated}
    we have to show that $e : S \to G$ is a closed immersion.
    This follows from Decent Spaces, Lemma
    \ref{decent-spaces-lemma-finite-residue-field-extension-finite}.
    
    \medskip\noindent
    Next, we prove the second assertion. To do this we may replace $B$ by $S$.
    By the paragraph above the stabilizer group scheme $G \to U$ is separated. By
    Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-diagonal}
    the morphism $j = (t, s) : R \to U \times_S U$ is separated.
    As $U$ is the spectrum of a field the scheme
    $U \times_S U$ is affine (by the construction of fibre products in
    Schemes, Section \ref{schemes-section-fibre-products}).
    Hence $R$ is separated, see
    Morphisms of Spaces, Lemma
    \ref{spaces-morphisms-lemma-separated-over-separated}.
    \end{proof}

    Comments (1)

    Comment #2802 by Evan Warner on September 11, 2017 a 2:38 pm UTC

    Typo: second paragraph should begin "Next, we prove the first assertion."

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