\'Etale extensions of polynomial rings are faithfully flat

TL;DR

This paper proves that étale extensions of polynomial rings over a commutative ring are faithfully flat, leading to the surjectivity of étale polynomial maps over algebraically closed fields.

Contribution

It applies Ohi's criterion to establish faithful flatness of étale polynomial ring extensions, a result not previously demonstrated.

Findings

Étale extensions of polynomial rings are faithfully flat.

Étale polynomial maps over algebraically closed fields are surjective.

Provides a new proof using Ohi's criterion.

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.