Lemma 10.116.5. Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $k \subset K$ be a field extension. Then $\dim (S) = \dim (K \otimes _ k S)$.
Proof. By Lemma 10.115.4 there exists a finite injective map $k[y_1, \ldots , y_ d] \to S$ with $d = \dim (S)$. Since $K$ is flat over $k$ we also get a finite injective map $K[y_1, \ldots , y_ d] \to K \otimes _ k S$. The result follows from Lemma 10.112.4. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
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).
All contributions are licensed under the GNU Free Documentation License.