The Stacks project

Lemma 21.31.4. Let $f : X \to Y$ be a morphism of $\textit{LC}$. If $f$ is proper and surjective, then $\{ f : X \to Y\} $ is a qc covering.

Proof. Let $y \in Y$ be a point. For each $x \in X_ y$ choose a quasi-compact neighbourhood $E_ x \subset X$. Choose $x \in U_ x \subset E_ x$ open. Since $f$ is proper the fibre $X_ y$ is quasi-compact and we find $x_1, \ldots , x_ n \in X_ y$ such that $X_ y \subset U_{x_1} \cup \ldots \cup U_{x_ n}$. We claim that $f(E_{x_1}) \cup \ldots \cup f(E_{x_ n})$ is a neighbourhood of $y$. Namely, as $f$ is closed (Topology, Theorem 5.17.5) we see that $Z = f(X \setminus U_{x_1} \cup \ldots \cup U_{x_ n})$ is a closed subset of $Y$ not containing $y$. As $f$ is surjective we see that $Y \setminus Z$ is contained in $f(E_{x_1}) \cup \ldots \cup f(E_{x_ n})$ as desired. $\square$

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