The Stacks project

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

Lemma 94.18.3. 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 94.17.2.

Conversely, assume that $[B/G]$ is an algebraic stack. By Lemma 94.18.2 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 71.7.1 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 71.10.13 and 71.10.10 we conclude that $G \to B$ is flat and locally of finite presentation. $\square$


Comments (1)

Comment #2757 by Ariyan Javanpeykar on

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


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