Remark 10.164.8. The property of being “universally catenary” does not descend; not even along étale ring maps. In Examples, Section 110.19 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 10.152.3 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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