The Stacks project

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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