# The Stacks Project

## Tag: 06QF

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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