Pr\"ufer algebraic spaces

TL;DR

This paper introduces Pr"ufer algebraic spaces and pairs, generalizing spectra of Pr"ufer rings, and establishes a valuative criterion of universal closedness, advancing the understanding of algebraic space geometry.

Contribution

It defines Pr"ufer algebraic spaces and pairs, introduces valuation algebraic spaces, and sharpens the valuative criterion of universal closedness for algebraic spaces.

Findings

Defined Pr"ufer algebraic spaces and pairs

Introduced valuation algebraic spaces

Established valuative criterion of universal closedness

Abstract

This is the first in a series of two papers concerned with relative birational geometry of algebraic spaces. In this paper, we study Pr\"ufer spaces and Pr\"ufer pairs of algebraic spaces that generalize spectra of Pr\"ufer rings. As a particular case of Pr\"ufer spaces we introduce valuation algebraic spaces, and use them to establish valuative criterion of universal closedness that sharpens the standard criterion. In the sequel paper, we will introduce a version of Riemann-Zariski spaces, and will prove Nagata compactification theorem for algebraic spaces.