Ferrand's pushouts for algebraic spaces

TL;DR

This paper generalizes Ferrand's pushouts from schemes to algebraic spaces, enabling new developments in valuation theory, Riemann-Zariski spaces, and a novel proof of Nagata's compactification theorem.

Contribution

It introduces Ferrand's pushouts for algebraic spaces, expanding their applicability beyond schemes and facilitating further theoretical advancements.

Findings

Defined Ferrand's pushouts for algebraic spaces

Extended valuation rings and Riemann-Zariski spaces to algebraic spaces

Provided a new proof of Nagata's compactification theorem

Abstract

We extend Ferrand's results about pushouts of schemes to the category of algebraic spaces. We call the corresponding class of pushouts Ferrand's pushouts. They will be used in our sequel works to extend the notions of valuation rings and Riemann-Zariski spaces to the category of algebraic spaces, and to obtain a new proof of Nagata's compactification theorem for algebraic spaces.