\'Etale extensions of polynomial rings are faithfully flat

Abstract

We apply Ohi's criterion for faithfully flatness of extensions of commutative rings to prove that any \'etale extension $k[Y_1, \ldots, Y_n]\subseteq k[X_1, \ldots, X_n]$ of polynomial rings (each in $n$ indeterminates) over a commutative ring $k$ is faithfully flat. In particular, if $k$ is an algebraically closed field then any \'etale polynomial map $k^{n} \to k^{n}$ is surjective.