This tag has label stacks-morphisms-lemma-gerbe-descent and it points to
The corresponding content:
Lemma 70.19.5. Let $$ \xymatrix{ \mathcal{X}' \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ \mathcal{Y}' \ar[r] & \mathcal{Y} } $$ be a fibre product of algebraic stacks. If $\mathcal{Y}' \to \mathcal{Y}$ is surjective, flat, and locally of finite presentation and $\mathcal{X}'$ is a gerbe over $\mathcal{Y}'$, then $\mathcal{X}$ is a gerbe over $\mathcal{Y}$.Proof. Follows immediately from Lemma 70.18.9 and Stacks, Lemma 8.11.7. $\square$
\begin{lemma}
\label{lemma-gerbe-descent}
Let
$$
\xymatrix{
\mathcal{X}' \ar[r] \ar[d] & \mathcal{X} \ar[d] \\
\mathcal{Y}' \ar[r] & \mathcal{Y}
}
$$
be a fibre product of algebraic stacks.
If $\mathcal{Y}' \to \mathcal{Y}$ is surjective, flat, and locally
of finite presentation and $\mathcal{X}'$ is a gerbe over $\mathcal{Y}'$,
then $\mathcal{X}$ is a gerbe over $\mathcal{Y}$.
\end{lemma}
\begin{proof}
Follows immediately from
Lemma \ref{lemma-surjective-flat-locally-finite-presentation}
and
Stacks, Lemma \ref{stacks-lemma-gerbe-descent}.
\end{proof}
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