[Corollary 2.3, Heinzer-Rotthaus-Wiegand]

Lemma 15.109.6. Let $(A, \mathfrak m)$ be a Noetherian local ring with geometrically normal formal fibres. Then

1. $A^ h$ is universally catenary, and

2. if $A$ is unibranch (for example normal), then $A$ is universally catenary.

Proof. By Lemma 15.108.8 the number of branches of $A$ and $A^\wedge$ are the same, hence Lemma 15.108.2 applies. Then for any minimal prime $\mathfrak q \subset A^ h$ we see that $A^\wedge /\mathfrak q A^\wedge$ has a unique minimal prime. Thus $A^ h$ is formally catenary (by definition) and hence universally catenary by Proposition 15.109.5. If $A$ is unibranch, then $A^ h$ has a unique minimal prime, hence $A^\wedge$ has a unique minimal prime, hence $A$ is formally catenary and we conclude in the same way. $\square$

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