On the topology of valuation-theoretic representations of integrally closed domains

Abstract

Let $F$ be a field. For each nonempty subset $X$ of the Zariski-Riemann space of valuation rings of $F$, let ${A}(X) = \bigcap_{V \in X}V$ and ${J}(X) = \bigcap_{V \in X}{\mathfrak M}_V$, where ${\mathfrak M}_V$ denotes the maximal ideal of $V$. We examine connections between topological features of $X$ and the algebraic structure of the ring ${A}(X)$. We show that if ${J}(X) \ne 0$ and $A(X)$ is a completely integrally closed local ring that is not a valuation ring of $F$, then there is a subspace $Y$ of the space of valuation rings of $F$ that is perfect in the patch topology such that ${A}(X) = {A}(Y)$. If any countable subset of points is removed from $Y$, then the resulting set remains a representation of $A(X)$. Additionally, if $F$ is a countable field, the set $Y$ can be chosen homeomorphic to the Cantor set. We apply these results to study properties of the ring ${A}(X)$ with specific focus on topological conditions that guarantee ${A}(X)$ is a Prüfer domain, a feature that is reflected in the Zariski-Riemann space when viewed as a locally ringed space. We also classify the rings ${A}(X)$ where $X$ has finitely many patch limit points, thus giving a topological generalization of the class of Krull domains, one that also includes interesting Prüfer domains. To illustrate the latter, we show how an intersection of valuation rings arising naturally in the study of local quadratic transformations of a regular local ring can be described using these techniques.