Schemes of Finite Expansion and Universally Closed Curves
Matthias Johann Steiner
TL;DR
The paper generalizes the classical contravariant correspondence between normal proper integral curves and transcendence degree $1$ extensions of a field $k$ to the broader category of normal integral universally closed curves by introducing and developing morphisms of finite expansion. It defines algebras and schemes of finite expansion, proves dimension and valuation-theoretic results (notably that normal 1-dimensional finite expansion domains are Prüfer and that closed points correspond to valuation rings), and establishes a functorial bridge from function fields to universally closed curves. The main result is a contravariant equivalence between normal integral universally closed curves over $k$ and field extensions of $k$ with $ ext{trdeg}_k=1$, mirroring and extending the classical curve–function-field correspondence to non-finitely generated extensions. This work broadens the categorical understanding of curves, enabling new applications in settings beyond finite type and linking valuation-theoretic data directly to geometric points on universally closed curves via a precise, fully faithful correspondence.
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 $k$.