On the geometry of Prüfer intersections of valuation rings

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 ${\mathbb{P}}_D^1$

13 pages, to appear in Pacific Journal of Mathematics

References in corpus (1)

• The index of an algebraic variety

Cited by in corpus (2)

• Prüfer intersection of valuation domains of a field of rational functions
• Realization of spaces of commutative rings