Lemma 10.130.3. Let $k$ be a field. Let $S$ be a finite type $k$ algebra. The set of Cohen-Macaulay primes forms a dense open $U \subset \mathop{\mathrm{Spec}}(S)$.

**Proof.**
The set is open by Lemma 10.130.2. It contains all minimal primes $\mathfrak q \subset S$ since the local ring at a minimal prime $S_{\mathfrak q}$ has dimension zero and hence is Cohen-Macaulay.
$\square$

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