# The Stacks Project

## Tag 06MW

Lemma 91.11.10. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. Assume that the residual gerbe $\mathcal{Z}_x$ of $\mathcal{X}$ exists. Let $f : \mathop{\mathrm{Spec}}(K) \to \mathcal{X}$ be a morphism where $K$ is a field in the equivalence class of $x$. Then $f$ factors through the inclusion morphism $\mathcal{Z}_x \to \mathcal{X}$.

Proof. Choose a field $k$ and a surjective flat locally finite presentation morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{Z}_x$. Set $T = \mathop{\mathrm{Spec}}(K) \times_\mathcal{X} \mathcal{Z}_x$. By Lemma 91.4.3 we see that $T$ is nonempty. As $\mathcal{Z}_x \to \mathcal{X}$ is a monomorphism we see that $T \to \mathop{\mathrm{Spec}}(K)$ is a monomorphism. Hence by Morphisms of Spaces, Lemma 58.10.8 we see that $T = \mathop{\mathrm{Spec}}(K)$ which proves the lemma. $\square$

The code snippet corresponding to this tag is a part of the file stacks-properties.tex and is located in lines 2802–2809 (see updates for more information).

\begin{lemma}
\label{lemma-residual-gerbe-points}
Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$.
Assume that the residual gerbe $\mathcal{Z}_x$ of $\mathcal{X}$ exists.
Let $f : \Spec(K) \to \mathcal{X}$ be a morphism where $K$ is a field
in the equivalence class of $x$. Then $f$ factors through the inclusion
morphism $\mathcal{Z}_x \to \mathcal{X}$.
\end{lemma}

\begin{proof}
Choose a field $k$ and a surjective flat locally finite presentation
morphism $\Spec(k) \to \mathcal{Z}_x$. Set
$T = \Spec(K) \times_\mathcal{X} \mathcal{Z}_x$. By
Lemma \ref{lemma-points-cartesian}
we see that $T$ is nonempty. As $\mathcal{Z}_x \to \mathcal{X}$
is a monomorphism we see that $T \to \Spec(K)$ is a monomorphism.
Hence by
Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-monomorphism-toward-field}
we see that $T = \Spec(K)$ which proves the lemma.
\end{proof}

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