Valuation Theory, Riemann Varieties and the Structure of integral Preschemes

TL;DR

This paper broadens valuation theory and Zariski-Riemann varieties to a wider algebraic geometric context, constructing projective limits and using nonseparated Riemann varieties to analyze integral preschemes.

Contribution

It introduces new birational projective limit objects and explores the structure of integral preschemes via nonseparated Riemann varieties, extending classical valuation theory.

Findings

Constructed birational projective limit objects for multiple models

Analyzed the birational structure of integral preschemes

Extended valuation theory beyond classical algebraic contexts

Abstract

In this work we show that the classical subject of general valuation theory and Zariski-Riemann varieties has a much wider scope than commutative algebra and desingularization theory. We construct and investigate birational projective limit objects appropriate for the study of countably many birational models at one time. We use nonseparated Riemann varieties to investigate the birational structure of integral preschemes satisfying the existence condition of the valuative criterion of properness.