A note on étale endomorphisms of normal schemes

Abstract

We prove that under some extra hypothesis, given an étale endomorphism of a normal irreducible Noetherian and simply connected scheme, if the endomorphism is surjective then it is injective. The additional assumption concerns the possibility of constructing an étale cover out of a surjective étale morphism. We show that in some cases the surjectivity hypothesis can be removed if the intended étale cover is Galois.

Theorem 2 of the previous version is incorrect