The Stacks project

Lemma 10.105.5. Let $A$ be a Noetherian universally catenary ring. Any $A$-algebra essentially of finite type over $A$ is universally catenary.

Proof. If $B$ is a finite type $A$-algebra, then $B$ is Noetherian by Lemma 10.31.1. Any finite type $B$-algebra is a finite type $A$-algebra and hence catenary by our assumption that $A$ is universally catenary. Thus $B$ is universally catenary. Any localization of $B$ is universally catenary by Lemma 10.105.4 and this finishes the proof. $\square$

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