Lemma 10.31.8. Let $k$ be a field and let $R$ be a Noetherian $k$-algebra. If $K/k$ is a finitely generated field extension then $K \otimes _ k R$ is Noetherian.
Proof. Since $K/k$ is a finitely generated field extension, there exists a finitely generated $k$-algebra $B \subset K$ such that $K$ is the fraction field of $B$. In other words, $K = S^{-1}B$ with $S = B \setminus \{ 0\} $. Then $K \otimes _ k R = S^{-1}(B \otimes _ k R)$. Then $B \otimes _ k R$ is Noetherian by Lemma 10.31.7. Finally, $K \otimes _ k R = S^{-1}(B \otimes _ k R)$ is Noetherian by Lemma 10.31.1. $\square$
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