Lemma 10.124.5. Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Suppose there is a quasi-finite $k$-algebra map $k[t_1, \ldots , t_ n] \subset S$. Then $\dim (S) \leq n$.

** A quasi-finite cover of affine n-space has dimension at most n. **

**Proof.**
By Lemma 10.113.1 the dimension of any local ring of $k[t_1, \ldots , t_ n]$ is at most $n$. Thus the result follows from Lemma 10.124.4.
$\square$

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