The Stacks project

Lemma 10.167.2. Let $k$ be a field. Let $S$ be a Noetherian $k$-algebra. Let $K/k$ be a finitely generated field extension, and set $S_ K = K \otimes _ k S$. Let $\mathfrak q \subset S$ be a prime of $S$. Let $\mathfrak q_ K \subset S_ K$ be a prime of $S_ K$ lying over $\mathfrak q$. Then $S_{\mathfrak q}$ is Cohen-Macaulay if and only if $(S_ K)_{\mathfrak q_ K}$ is Cohen-Macaulay.

Proof. By Lemma 10.31.8 the ring $S_ K$ is Noetherian. Hence $S_{\mathfrak q} \to (S_ K)_{\mathfrak q_ K}$ is a flat local homomorphism of Noetherian local rings. Note that the fibre

\[ (S_ K)_{\mathfrak q_ K} / \mathfrak q (S_ K)_{\mathfrak q_ K} \cong (\kappa (\mathfrak q) \otimes _ k K)_{\mathfrak q'} \]

is the localization of the Cohen-Macaulay (Lemma 10.167.1) ring $\kappa (\mathfrak q) \otimes _ k K$ at a suitable prime ideal $\mathfrak q'$. Hence the lemma follows from Lemma 10.163.3. $\square$

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