Lemma 10.105.7 (00NK)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.105: Catenary rings
Lemma 10.105.7 (cite)

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Lemma 10.105.7. Any quotient of a catenary ring is catenary. Any quotient of a Noetherian universally catenary ring is universally catenary.

Proof. Let $A$ be a ring and let $I \subset A$ be an ideal. The description of $\mathop{\mathrm{Spec}}(A/I)$ in Lemma 10.17.7 shows that if $A$ is catenary, then so is $A/I$ . The second statement is a special case of Lemma 10.105.5. $\square$

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