The Stacks project

Lemma 68.25.3. Let $S$ be a locally Noetherian and universally catenary scheme. Let $X$ be an algebraic space over $S$ such that $X$ is decent and such that the structure morphism $X \to S$ is locally of finite type. Then $X$ is catenary.

Proof. The question is local on $S$ (use Topology, Lemma 5.11.5). Thus we may assume that $S$ has a dimension function, see Topology, Lemma 5.20.4. Then we conclude that $|X|$ has a dimension function by Lemma 68.25.2. Since $|X|$ is sober (Proposition 68.12.4) we conclude that $|X|$ is catenary by Topology, Lemma 5.20.2. $\square$

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