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Tag 06PL

Chapter 87: Criteria for Representability > Section 87.18: When is a quotient stack algebraic?

Lemma 87.18.3. Let $S$ be a scheme and let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Endow $B$ with the trivial action of $G$. Then the quotient stack $[B/G]$ is an algebraic stack if and only if $G$ is flat and locally of finite presentation over $B$.

Proof. If $G$ is flat and locally of finite presentation over $B$, then $[B/G]$ is an algebraic stack by Theorem 87.17.2.

Conversely, assume that $[B/G]$ is an algebraic stack. By Lemma 87.18.2 and because the action is trivial, we see there exists an algebraic space $B'$ and a morphism $B' \to B$ such that (1) $B' \to B$ is a surjection of sheaves and (2) the projections $$ B' \times_B G \times_B B' \to B' $$ are flat and locally of finite presentation. Note that the base change $B' \times_B G \times_B B' \to G \times_B B'$ of $B' \to B$ is a surjection of sheaves also. Thus it follows from Descent on Spaces, Lemma 65.7.1 that the projection $G \times_B B' \to B'$ is flat and locally of finite presentation. By (1) we can find an fppf covering $\{B_i \to B\}$ such that $B_i \to B$ factors through $B' \to B$. Hence $G \times_B B_i \to B_i$ is flat and locally of finite presentation by base change. By Descent on Spaces, Lemmas 65.10.13 and 65.10.10 we conclude that $G \to B$ is flat and locally of finite presentation. $\square$

    The code snippet corresponding to this tag is a part of the file criteria.tex and is located in lines 3211–3218 (see updates for more information).

    \begin{lemma}
    \label{lemma-BG-algebraic}
    Let $S$ be a scheme and let $B$ be an algebraic space over $S$.
    Let $G$ be a group algebraic space over $B$.
    Endow $B$ with the trivial action of $G$.
    Then the quotient stack $[B/G]$ is an algebraic stack
    if and only if $G$ is flat and locally of finite presentation over $B$.
    \end{lemma}
    
    \begin{proof}
    If $G$ is flat and locally of finite presentation over $B$, then
    $[B/G]$ is an algebraic stack by
    Theorem \ref{theorem-flat-groupoid-gives-algebraic-stack}.
    
    \medskip\noindent
    Conversely, assume that $[B/G]$ is an algebraic stack. By
    Lemma \ref{lemma-group-quotient-algebraic}
    and because the action is trivial, we see
    there exists an algebraic space $B'$ and a morphism
    $B' \to B$ such that (1) $B' \to B$ is a surjection
    of sheaves and (2) the projections
    $$
    B' \times_B G \times_B B' \to B'
    $$
    are flat and locally of finite presentation. Note that the base change
    $B' \times_B G \times_B B' \to G \times_B B'$ of $B' \to B$
    is a surjection of sheaves also. Thus it follows from
    Descent on Spaces, Lemma \ref{spaces-descent-lemma-curiosity}
    that the projection $G \times_B B' \to B'$ is flat and locally
    of finite presentation. By (1) we can find an fppf covering
    $\{B_i \to B\}$ such that $B_i \to B$ factors through $B' \to B$.
    Hence $G \times_B B_i \to B_i$ is flat and locally of finite presentation
    by base change. By
    Descent on Spaces, Lemmas
    \ref{spaces-descent-lemma-descending-property-flat} and
    \ref{spaces-descent-lemma-descending-property-locally-finite-presentation}
    we conclude that $G \to B$ is flat and locally of finite presentation.
    \end{proof}

    Comments (1)

    Comment #2757 by Ariyan Javanpeykar on August 3, 2017 a 3:34 pm UTC

    Slogan: Gerbes are algebraic if and only if the associated groups are flat and locally of finite presentation.

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