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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