The Stacks project

Lemma 10.105.4. Any localization of a catenary ring is catenary. Any localization of a Noetherian universally catenary ring is universally catenary.

Proof. Let $A$ be a ring and let $S \subset A$ be a multiplicative subset. The description of $\mathop{\mathrm{Spec}}(S^{-1}A)$ in Lemma 10.17.5 shows that if $A$ is catenary, then so is $S^{-1}A$. If $S^{-1}A \to C$ is of finite type, then $C = S^{-1}B$ for some finite type ring map $A \to B$. Hence if $A$ is Noetherian and universally catenary, then $B$ is catenary and we see that $C$ is catenary too. This proves the lemma. $\square$

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