Schemes of Finite Expansion and Universally Closed Curves

TL;DR

This paper generalizes the classical equivalence between normal proper integral curves and certain field extensions to a broader category of universally closed separated integral schemes of dimension one, using morphisms of finite expansion.

Contribution

It introduces a new category of normal integral universally closed curves and establishes an equivalence with field extensions of transcendence degree one, extending classical results.

Findings

Generalization of the categorical equivalence to broader schemes

Introduction of morphisms of finite expansion as a key technique

Identification of properties similar to proper integral curves in the new category

Abstract

In algebraic geometry there is a well-known categorical equivalence between the category of normal proper integral curves over a field $k$ and the category of finitely generated field extensions of $k$ of transcendence degree $1$. In this paper we generalize this equivalence to the category of normal quasi-compact universally closed separated integral $k$-schemes of dimension $1$ and the category of field extensions of $k$ of transcendence degree $1$. Our key technique are morphisms of finite expansion which can be considered as relaxation of morphisms of finite type. Since the schemes in the generalized category have many properties similar to normal proper integral curves, we call them normal integral universally closed curves over $k$.