Topologically defined classes of Prüfer domains

Carmelo Antonio Finocchiaro and K. Alan Loper

Vol. 345 (2026), No. 1, 85–104

DOI: 10.2140/pjm.2026.345.85

Abstract

Let $D$ be an integral domain and let $\mathrm{\Delta }$ be a subspace of the Zariski–Riemann space $\mathfrak{𝔛}(D)$ consisting of the valuation overrings of $D$. Our aim is to provide topological conditions on $\mathrm{\Delta }$ making ${\bigcap <!--FUNCTION\; APPLICATION-->}_{V\in \mathrm{\Delta }}V$ a Prüfer domain. Some of the results obtained can be seen as a topological version of a well-known theorem by Nagata stating that the intersection of finitely many valuation domains of a given field is Prüfer. A key role will be played by the constructible topology on $\mathfrak{𝔛}(D)$.

Keywords

Zariski–Riemann space, Prüfer domain, spectral space, constructible topology

Mathematical Subject Classification

Primary: 13A15, 13F05, 13F30

Milestones

Received: 24 December 2025

Revised: 29 June 2026

Accepted: 14 July 2026

Published: 25 September 2026