A note on \'etale endomorphisms of normal schemes

TL;DR

This paper investigates conditions under which surjective étale endomorphisms of certain normal schemes are necessarily injective, focusing on additional hypotheses related to étale covers and Galois properties.

Contribution

It establishes that under specific conditions, surjective étale endomorphisms of normal, simply connected schemes are injective, extending understanding of étale morphisms in algebraic geometry.

Findings

Surjective étale endomorphisms are injective under extra hypotheses.

Galois étale covers can eliminate some assumptions for injectivity.

Conditions for étale endomorphisms to be automorphisms are clarified.

Abstract

We prove that under some extra hypothesis, given an \'etale 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 \'etale cover out of a surjective \'etale morphism. We show that in some cases the surjectivity hypothesis can be removed if the intended \'etale cover is Galois.