Étale 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 étale extension $k[Y_1,\dots ,Y_n]\subseteq k[X_1,\dots ,X_n]$ 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 étale polynomial map $k^n\to k^n$ is surjective.

arXiv commentThere is a gap in the last step of the proof of Theorem 1.1. The author would like to thank R. van Dobben de Bruyn and Sean Cotner for bringing this point to the author's attention