The canonical embedding of an unramified morphism in an \'etale morphism

TL;DR

This paper proves that any unramified morphism can be uniquely factored into a closed embedding followed by an étale morphism, providing a canonical structure for such morphisms.

Contribution

It introduces a canonical and universal factorization of unramified morphisms into a closed embedding and an étale morphism, clarifying their structure.

Findings

Every unramified morphism admits a canonical factorization.

The factorization involves a closed embedding followed by an étale morphism.

The étale morphism in the factorization may not be separated.

Abstract

We show that every unramified morphism X->Y has a canonical and universal factorization X->E->Y where the first morphism is a closed embedding and the second is etale (but not separated).