Lemma 67.25.2. Let $S$ be a locally Noetherian and universally catenary scheme. Let $\delta : S \to \mathbf{Z}$ be a dimension function. Let $X$ be a decent algebraic space over $S$ such that the structure morphism $X \to S$ is locally of finite type. Let $\delta _ X : |X| \to \mathbf{Z}$ be the map sending $x$ to $\delta (f(x))$ plus the transcendence degree of $x/f(x)$. Then $\delta _ X$ is a dimension function on $|X|$.

Proof. Let $\varphi : U \to X$ be a surjective étale morphism where $U$ is a scheme. Then the similarly defined function $\delta _ U$ is a dimension function on $U$ by Morphisms, Lemma 29.52.3. On the other hand, by the definition of relative transcendence degree in (Morphisms of Spaces, Definition 66.33.1) we see that $\delta _ U(u) = \delta _ X(\varphi (u))$.

Let $x \leadsto x'$ be a specialization of points in $|X|$. by Lemma 67.12.2 we can find a specialization $u \leadsto u'$ of points of $U$ with $\varphi (u) = x$ and $\varphi (u') = x'$. Moreover, we see that $x = x'$ if and only if $u = u'$, see Lemma 67.12.1. Thus the fact that $\delta _ U$ is a dimension function implies that $\delta _ X$ is a dimension function, see Topology, Definition 5.20.1. $\square$

Comment #7203 by 羽山籍真 on

Here $f$ is not defined, it is $f\colon X\rightarrow S$ right?

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