Lemma 10.130.8. 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 10.130.7 and the definitions. $\square$

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