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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