Lemma 10.129.7. Let $R$ be a ring. Let $R \to S$ be of finite type. Let $R \to R'$ be any ring map. Set $S' = R' \otimes _ R S$. Denote $f : \mathop{\mathrm{Spec}}(S') \to \mathop{\mathrm{Spec}}(S)$ the map associated to the ring map $S \to S'$. Set $W$ equal to the set of primes $\mathfrak q$ such that the fibre ring $S_{\mathfrak q} \otimes _ R \kappa (\mathfrak p)$, $\mathfrak p = R \cap \mathfrak q$ is Cohen-Macaulay, and let $W'$ denote the analogue for $S'/R'$. Then $W' = f^{-1}(W)$.

**Proof.**
Trivial from Lemma 10.129.6 and the definitions.
$\square$

## 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Your email address will not be published.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)