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. $\square$

Comment #2757 by Ariyan Javanpeykar on

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

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).