The Stacks project

Lemma 10.130.5. Let $R$ be a ring. Let $R \to S$ be flat of finite presentation. The set of primes $\mathfrak q$ such that the fibre ring $S_{\mathfrak q} \otimes _ R \kappa (\mathfrak p)$, with $\mathfrak p = R \cap \mathfrak q$ is Cohen-Macaulay is open and dense in every fibre of $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$.

Proof. The set, call it $W$, is open by Lemma 10.130.4. It is dense in the fibres because the intersection of $W$ with a fibre is the corresponding set of the fibre to which Lemma 10.130.3 applies. $\square$

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