Lemma 10.105.5 (0ECE)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.105: Catenary rings
Lemma 10.105.5 (cite)

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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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