Lemma 10.130.7. 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 10.130.6 and the definitions. \square
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