The Stacks project

Proof. Let $U$ be a scheme and let $U \to X$ be a surjective étale morphism. Let $\omega _ U^\bullet$ be the dualizing complex on $U$ associated to $\omega _ X^\bullet |_ U$. If $u \in U$ maps to $x \in |X|$, then $\mathcal{O}_{X, \overline{x}}$ is the strict henselization of $\mathcal{O}_{U, u}$. By Dualizing Complexes, Lemma 47.22.1 we see that if $\omega ^\bullet$ is a normalized dualizing complex for $\mathcal{O}_{U, u}$, then $\omega ^\bullet \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}}$ is a normalized dualizing complex for $\mathcal{O}_{X, \overline{x}}$. Hence we see that the dimension function $U \to \mathbf{Z}$ of Duality for Schemes, Lemma 48.2.7 for the scheme $U$ and the complex $\omega _ U^\bullet$ is equal to the composition of $U \to |X|$ with $\delta$. Using the specializations in $|X|$ lift to specializations in $U$ and that nontrivial specializations in $U$ map to nontrivial specializations in $X$ (Decent Spaces, Lemmas 68.12.2 and 68.12.1) an easy topological argument shows that $\delta$ is a dimension function on $|X|$. $\square$