A converse of Ax-Grothendieck theorem for étale endomorphisms of normal schemes

Abstract

Given an \'etale endomorphism of a normal irreducible Noetherian and simply connected scheme, we prove that if the endomorphism is surjective then it is injective. The proof is based on Liu's construction of a Galois cover out of a surjective \'etale morphism. If we give up of the surjectivity hypothesis and suppose the endomorphism is separated, then we prove that the induced field extension is Galois. In the case of an \'etale endomorphism of the affine space over an algebraically closed field of characteristic zero, Campbell's theorem implies that the assumption of surjectivity is superfluous.