## Tag `0DTH`

Chapter 90: Properties of Algebraic Stacks > Section 90.11: Residual gerbes

Lemma 90.11.12. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $x \in |\mathcal{X}|$ with image $y \in |\mathcal{Y}|$. If the residual gerbes $\mathcal{Z}_x \subset \mathcal{X}$ and $\mathcal{Z}_y \subset \mathcal{Y}$ of $x$ and $y$ exist, then $f$ induces a commutative diagram $$ \xymatrix{ \mathcal{X} \ar[d]_f & \mathcal{Z}_x \ar[l] \ar[d] \\ \mathcal{Y} & \mathcal{Z}_y \ar[l] } $$

Proof.Choose a field $k$ and a surjective, flat, locally finitely presented morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{Z}_x$. The morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{Y}$ factors through $\mathcal{Z}_y$ by Lemma 90.11.10. Thus $\mathcal{Z}_x \times_\mathcal{Y} \mathcal{Z}_y$ is a nonempty substack of $\mathcal{Z}_x$ hence equal to $\mathcal{Z}_x$ by Lemma 90.11.4. $\square$

The code snippet corresponding to this tag is a part of the file `stacks-properties.tex` and is located in lines 2849–2862 (see updates for more information).

```
\begin{lemma}
\label{lemma-residual-gerbe-functorial}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
Let $x \in |\mathcal{X}|$ with image $y \in |\mathcal{Y}|$.
If the residual gerbes $\mathcal{Z}_x \subset \mathcal{X}$
and $\mathcal{Z}_y \subset \mathcal{Y}$ of $x$ and $y$ exist,
then $f$ induces a commutative diagram
$$
\xymatrix{
\mathcal{X} \ar[d]_f & \mathcal{Z}_x \ar[l] \ar[d] \\
\mathcal{Y} & \mathcal{Z}_y \ar[l]
}
$$
\end{lemma}
\begin{proof}
Choose a field $k$ and a surjective, flat, locally finitely presented
morphism $\Spec(k) \to \mathcal{Z}_x$. The morphism
$\Spec(k) \to \mathcal{Y}$ factors through $\mathcal{Z}_y$ by
Lemma \ref{lemma-residual-gerbe-points}.
Thus $\mathcal{Z}_x \times_\mathcal{Y} \mathcal{Z}_y$
is a nonempty substack of $\mathcal{Z}_x$
hence equal to $\mathcal{Z}_x$ by Lemma \ref{lemma-monomorphism-into-point}.
\end{proof}
```

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