A connectedness theorem for spaces of valuation rings

TL;DR

This paper extends Zariski's connectedness theorem to the space of valuation rings of a field, showing a correspondence between certain subrings and connected subspaces of valuation rings, without requiring Noetherian conditions.

Contribution

It generalizes Zariski's connectedness theorem to non-Noetherian valuation spaces and establishes a correspondence between specific subrings and connected subspaces.

Findings

A subring is characterized by a connected subspace of valuation rings.

The intersection of rings in a connected subset is a local ring.

A converse to Zariski's connectedness theorem is proved.

Abstract

Let $F$ be a field, let $D$ be a local subring of $F$, and let Val$_F(D)$ 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 Val$_F(D)$ such that $R$ is the intersection of the rings in $Z$. Consequently, the intersection of the rings in any closed and connected subset of Val$_F(D)$ 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.