
Remark 10.158.8. The property of being “universally catenary” does not descend; not even along étale ring maps. In Examples, Section 102.16 there is a construction of a finite ring map $A \to B$ with $A$ local Noetherian and not universally catenary, $B$ semi-local with two maximal ideals $\mathfrak m$, $\mathfrak n$ with $B_{\mathfrak m}$ and $B_{\mathfrak n}$ regular of dimension $2$ and $1$ respectively, and the same residue fields as that of $A$. Moreover, $\mathfrak m_ A$ generates the maximal ideal in both $B_{\mathfrak m}$ and $B_{\mathfrak n}$ (so $A \to B$ is unramified as well as finite). By Lemma 10.147.11 there exists a local étale ring map $A \to A'$ such that $B \otimes _ A A' = B_1 \times B_2$ decomposes with $A' \to B_ i$ surjective. This shows that $A'$ has two minimal primes $\mathfrak q_ i$ with $A'/\mathfrak q_ i \cong B_ i$. Since $B_ i$ is regular local (since it is étale over either $B_{\mathfrak m}$ or $B_{\mathfrak n}$) we conclude that $A'$ is universally catenary.

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