The Stacks project

Lemma 99.11.11. 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 99.4.3 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 66.10.8 we see that $T = \mathop{\mathrm{Spec}}(K)$ which proves the lemma. $\square$

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