Remark 10.164.8. The property of being “universally catenary” does not descend; not even along étale ring maps. In Examples, Section 109.18 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.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)