Isolated points of the Zariski space

TL;DR

This paper characterizes isolated points in the Zariski space of valuation domains and integral domains, revealing conditions under which points are isolated and describing the topology of these spaces.

Contribution

It provides a complete characterization of isolated points in the Zariski space for various classes of domains and valuation rings, including Noetherian and transcendental cases.

Findings

L as a point is isolated iff certain algebraic conditions hold.

Valuation domains of dimension 1 can be isolated under specific conditions.

The set of extensions of a valuation domain to a transcendental extension has no isolated points.

Abstract

Let $D$ be an integral domain and $L$ be a field containing $D$. We study the isolated points of the Zariski space $\mathrm{Zar}(L|D)$, with respect to the constructible topology. In particular, we completely characterize when $L$ (as a point) is isolated and, under the hypothesis that $L$ is the quotient field of $D$, when a valuation domain of dimension $1$ is isolated; as a consequence, we find all isolated points of $\mathrm{Zar}(D)$ when $D$ is a Noetherian domain. We also show that if $V$ is a valuation domain and $L$ is transcendental over $V$ then the set of extensions of $V$ to $L$ has no isolated points.