Affine schemes and topological closures in the Zariski-Riemann space of valuation rings

TL;DR

This paper studies the topological and scheme-theoretic structure of the Zariski-Riemann space of valuation rings over a field, characterizing affine subspaces within this space using various topologies.

Contribution

It provides a detailed analysis of the topologies on the Zariski-Riemann space and characterizes the affine subspaces as locally ringed subspaces, connecting valuation theory with scheme theory.

Findings

Characterization of affine subspaces in the Zariski-Riemann space.

Analysis of Zariski, inverse, and patch topologies on valuation spaces.

Representation of the space as a projective limit of schemes.

Abstract

Let $F$ be a field, let $D$ be a subring of $F$, and let ${\mathfrak{X}}$ be the Zariski-Riemann space of valuation rings containing $D$ and having quotient field $F$. We consider the Zariski, inverse and patch topologies on ${\mathfrak{X}}$ when viewed as a projective limit of projective integral schemes having function field contained in $F$, and we characterize the locally ringed subspaces of ${\mathfrak{X}}$ that are affine schemes.