On the geometry of Prüfer intersections of valuation rings

Bruce Olberding

Vol. 273 (2015), No. 2, 353–368

DOI: 10.2140/pjm.2015.273.353

Abstract

Let $F$ be a field, let $D$ be a subring of $F$ and let $Z$ be an irreducible subspace of the space of all valuation rings between $D$ and $F$ that have quotient field $F$. Then $Z$ is a locally ringed space whose ring of global sections is $A={\bigcap }_{V\in Z}V$. All rings between $D$ and $F$ that are integrally closed in $F$ arise in such a way. Motivated by applications in areas such as multiplicative ideal theory and real algebraic geometry, a number of authors have formulated criteria for when $A$ is a Prüfer domain. We give geometric criteria for when $A$ is a Prüfer domain that reduce this issue to questions of prime avoidance. These criteria, which unify and extend a variety of different results in the literature, are framed in terms of morphisms of $Z$ into the projective line ${\mathbb{P}}_D^1$.

Keywords

Prüfer domain, valuation ring, Zariski–Riemann space

Mathematical Subject Classification 2010

Primary: 13F05, 13F30

Secondary: 13B22, 14A15

Milestones

Received: 22 January 2014

Revised: 26 July 2014

Accepted: 19 August 2014

Published: 23 December 2014