Lemma 10.116.5. Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $K/k$ 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.
$\square$

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