The Zariski-Riemann space of valuation domains associated to pseudo-convergent sequences

TL;DR

This paper explores valuation domains linked to pseudo-convergent sequences in a valuation domain, analyzing their properties, equivalences, and the topology of the associated Zariski-Riemann spaces, especially in rank-one cases.

Contribution

It introduces a detailed study of valuation domains from pseudo-convergent sequences, compares two constructions, and characterizes their equivalences and topological structures.

Findings

Characterization of when valuation domains from pseudo-convergent sequences are equal.

Description of topological properties of the Zariski-Riemann spaces.

Analysis of the two main constructions by Ostrowski and Loper-Werner.

Abstract

Let $V$ be a valuation domain with quotient field $K$. Given a pseudo-convergent sequence $E$ in $K$, we study two constructions associating to $E$ a valuation domain of $K(X)$ lying over $V$, especially when $V$ has rank one. The first one has been introduced by Ostrowski, the second one more recently by Loper and Werner. We describe the main properties of these valuation domains, and we give a notion of equivalence on the set of pseudo-convergent sequences of $K$ characterizing when the associated valuation domains are equal. Then, we analyze the topological properties of the Zariski-Riemann spaces formed by these valuation domains.