A principal ideal theorem for compact sets of rank one valuation rings

Abstract

Let $F$ be a field, and let Zar$(F)$ be the space of valuation rings of $F$ with respect to the Zariski topology. We prove that if $X$ is a quasicompact set of rank one valuation rings in Zar$(F)$ whose maximal ideals do not intersect to $0$, then the intersection of the rings in $X$ is an integral domain with quotient field $F$ such that every finitely generated ideal is a principal ideal. To prove this result, we develop a duality between (a) quasicompact sets of rank one valuation rings whose maximal ideals do not intersect to $0$, and (b) one-dimensional Pr\"ufer domains with nonzero Jacobson radical and quotient field $F$. The necessary restriction in all these cases to collections of valuation rings whose maximal ideals do not intersect to $0$ is motivated by settings in which the valuation rings considered all dominate a given local ring.