Un adjoint

TL;DR

This paper introduces a universal surjective morphism from a scheme T to an étale, separated S-scheme, constructed via quotients by specific equivalence relations, with properties derived from these quotient structures.

Contribution

It defines a new universal surjective morphism in algebraic geometry using quotients by open and closed equivalence relations, expanding the understanding of such morphisms.

Findings

The morphism is surjective and universal.

Properties are derived from quotient constructions.

The approach uses equivalence relations with open and closed graphs.

Abstract

Given a scheme S and a flat morphism T \to S of finite presentation we define a surjective S-morphism to an {\'e}tale and separated S-scheme, which is universal in an obvious sense. Properties of this morphism are deduced from a thorough uses of quotients by equivalence relations whose graph is open and closed.