Lemma 78.27.2 (06PF)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 78: Groupoids in Algebraic Spaces
Section 78.27: Gerbes and quotient stacks
Lemma 78.27.2 (cite)

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Lemma 78.27.2. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$ . Let $G$ be a group algebraic space over $B$ . Endow $B$ with the trivial action of $G$ . The morphism

$[B/G] \longrightarrow \mathcal{S}_ B$

(Lemma 78.20.2) turns $[B/G]$ into a gerbe over $B$ .

Proof. Immediate from Lemma 78.27.1 as the morphisms $B \to B$ and $B \times _ B G \to B$ are surjective as morphisms of sheaves. $\square$

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