A connectedness theorem for spaces of valuation rings

Abstract

Let F be a field, let D be a local subring of F, and let ValF(D) be the space of valuation rings of F that dominate D. We lift Zariski's connectedness theorem for fibers of a projective morphism to the Zariski-Riemann space of valuation rings of F by proving that a subring R of F dominating D is local, residually algebraic over D and integrally closed in F if and only if there is a closed and connected subspace Z of ValF(D) such that R is the intersection of the rings in Z. Consequently, the intersection of the rings in any closed and connected subset of ValF(D) is a local ring. In proving this, we also prove a converse to Zariski's connectedness theorem. Our results do not require the rings involved to be Noetherian.