# The Stacks Project

## Tag 06MX

Lemma 91.11.11. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. Let $\mathcal{Z}$ be an algebraic stack satisfying the equivalent conditions of Lemma 91.11.3 and let $\mathcal{Z} \to \mathcal{X}$ be a monomorphism such that the image of $|\mathcal{Z}| \to |\mathcal{X}|$ is $x$. Then the residual gerbe $\mathcal{Z}_x$ of $\mathcal{X}$ at $x$ exists and $\mathcal{Z} \to \mathcal{X}$ factors as $\mathcal{Z} \to \mathcal{Z}_x \to \mathcal{X}$ where the first arrow is an equivalence.

Proof. Let $\mathcal{Z}_x \subset \mathcal{X}$ be the full subcategory corresponding to the essential image of the functor $\mathcal{Z} \to \mathcal{X}$. Then $\mathcal{Z} \to \mathcal{Z}_x$ is an equivalence, hence $\mathcal{Z}_x$ is an algebraic stack, see Algebraic Stacks, Lemma 85.12.4. Since $\mathcal{Z}_x$ inherits all the properties of $\mathcal{Z}$ from this equivalence it is clear from the uniqueness in Lemma 91.11.7 that $\mathcal{Z}_x$ is the residual gerbe of $\mathcal{X}$ at $x$. $\square$

The code snippet corresponding to this tag is a part of the file stacks-properties.tex and is located in lines 2824–2835 (see updates for more information).

\begin{lemma}
\label{lemma-residual-gerbe-unique}
Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$.
Let $\mathcal{Z}$ be an algebraic stack satisfying the equivalent conditions of
Lemma \ref{lemma-unique-point-better}
and let $\mathcal{Z} \to \mathcal{X}$ be a monomorphism such that the image
of $|\mathcal{Z}| \to |\mathcal{X}|$ is $x$. Then the residual gerbe
$\mathcal{Z}_x$ of $\mathcal{X}$ at $x$ exists and
$\mathcal{Z} \to \mathcal{X}$ factors as
$\mathcal{Z} \to \mathcal{Z}_x \to \mathcal{X}$ where the first arrow
is an equivalence.
\end{lemma}

\begin{proof}
Let $\mathcal{Z}_x \subset \mathcal{X}$ be the full subcategory corresponding
to the essential image of the functor $\mathcal{Z} \to \mathcal{X}$.
Then $\mathcal{Z} \to \mathcal{Z}_x$ is an equivalence, hence
$\mathcal{Z}_x$ is an algebraic stack, see
Algebraic Stacks, Lemma \ref{algebraic-lemma-equivalent}.
Since $\mathcal{Z}_x$ inherits all the properties of $\mathcal{Z}$ from
this equivalence it is clear from the uniqueness in
Lemma \ref{lemma-residual-gerbe}
that $\mathcal{Z}_x$ is the residual gerbe of $\mathcal{X}$ at $x$.
\end{proof}

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