The Stacks project

Lemma 101.8.4. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Then $|\mathcal{X}|$ is quasi-sober (Topology, Definition 5.8.6).

Proof. We have to prove that every irreducible closed subset $T \subset |\mathcal{X}|$ has a generic point. Choose an affine scheme $U$ and a smooth morphism $f : U \to \mathcal{X}$ such that $f^{-1}(T) \subset |U|$ is nonempty. Since $U$ is Noetherian, the closed subset $f^{-1}(T)$ has finitely many irreducible components (Topology, Lemma 5.9.2). Say $f^{-1}(T) = Z_1 \cup \ldots \cup Z_ n$ is the decomposition into irreducible components. As $f$ is open, the image of $f|_{f^{-1}(T)} : f^{-1}(T) \to T$ contains a nonempty open subset of $T$. Since $T$ is irreducible, this means that $f(f^{-1}(T))$ is dense. Since $T$ is irreducible, it follows that $f(Z_ i)$ is dense for some $i$. Then if $\xi _ i \in Z_ i$ is the generic point we see that $f(\xi _ i)$ is a generic point of $T$. $\square$

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