The Stacks project

Lemma 77.27.1. Notation and assumption as in Lemma 77.21.1. The morphism of quotient stacks

\[ [f] : [U/R] \longrightarrow [U'/R'] \]

turns $[U/R]$ into a gerbe over $[U'/R']$ if $f : U \to U'$ and $R \to R'|_ U$ are surjective maps of fppf sheaves. Here $R'|_ U$ is the restriction of $R'$ to $U$ via $f : U \to U'$.

Proof. We will verify that Stacks, Lemma 8.11.3 properties (2) (a) and (2) (b) hold. Property (2)(a) holds because $U \to U'$ is a surjective map of sheaves (use Lemma 77.24.1 to see that objects in $[U'/R']$ locally come from $U'$). To prove (2)(b) let $x, y$ be objects of $[U/R]$ over a scheme $T/S$. Let $x', y'$ be the images of $x, y$ in the category $[U'/'R]_ T$. Condition (2)(b) requires us to check the map of sheaves

\[ \mathit{Isom}(x, y) \longrightarrow \mathit{Isom}(x', y') \]

on $(\mathit{Sch}/T)_{fppf}$ is surjective. To see this we may work fppf locally on $T$ and assume that come from $a, b \in U(T)$. In that case we see that $x', y'$ correspond to $f \circ a, f \circ b$. By Lemma 77.22.1 the displayed map of sheaves in this case becomes

\[ T \times _{(a, b), U \times _ B U} R \longrightarrow T \times _{f \circ a, f \circ b, U' \times _ B U'} R' = T \times _{(a, b), U \times _ B U} R'|_ U. \]

Hence the assumption that $R \to R'|_ U$ is a surjective map of fppf sheaves on $(\mathit{Sch}/S)_{fppf}$ implies the desired surjectivity. $\square$


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