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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 045I. Beware of the difference between the letter 'O' and the digit '0'.