The Stacks project

Lemma 10.167.1. Let $k$ be a field and let $K/k$ and $L/k$ be two field extensions such that one of them is a field extension of finite type. Then $K \otimes _ k L$ is a Noetherian Cohen-Macaulay ring.

Proof. The ring $K \otimes _ k L$ is Noetherian by Lemma 10.31.8. Say $K$ is a finite extension of the purely transcendental extension $k(t_1, \ldots , t_ r)$. Then $k(t_1, \ldots , t_ r) \otimes _ k L \to K \otimes _ k L$ is a finite free ring map. By Lemma 10.112.9 it suffices to show that $k(t_1, \ldots , t_ r) \otimes _ k L$ is Cohen-Macaulay. This is clear because it is a localization of the polynomial ring $L[t_1, \ldots , t_ r]$. (See for example Lemma 10.104.7 for the fact that a polynomial ring is Cohen-Macaulay.) $\square$

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