The Stacks project

Lemma 37.73.6. Let $f : X \to Y$ be a locally quasi-finite morphism. Let $w : X \to \mathbf{Z}$ be a weighting. If $w(x) > 0$ for all $x \in X$, then $f$ is universally open.

Proof. Since the property is preserved by base change, see Lemma 37.73.3, it suffices to prove that $f$ is open. Since we may also replace $X$ by any open of $X$, it suffices to prove that $f(X)$ is open. Let $y \in f(X)$. Choose $x \in X$ with $f(x) = y$. It suffices to prove that $f(X)$ contains an open neighbourhood of $y$ and it suffices to do so after replacing $Y$ by an ├ętale neighbourhood of $y$. By ├ętale localization of quasi-finite morphisms, see Section 37.41, we may assume there is an open neighbourhood $U \subset X$ of $x$ such that $\pi = f|_ U : U \to Y$ is finite. Then $\int _\pi w|_ U$ is locally constant and has positive value at $y$. Hence $\pi (U)$ contains an open neighbourhood of $y$ and the proof is complete. $\square$

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