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 Prüfer 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.