Irredundant intersections of valuation overrings of two-dimensional Noetherian domains

Abstract

Let D be a two-dimensional Noetherian domain, let R be an overring of D, and let Σ and Γ be collections of valuation overrings of D. We consider circumstances under which ( ⋂ V ∈ Σ V ) ∩ R = ( ⋂ W ∈ Γ W ) ∩ R implies that Σ = Γ . We show that if R is integrally closed, these representations are “strongly” irredundant, and every member of Σ ∪ Γ has Krull dimension 2, then Σ = Γ . If in addition Σ and Γ are Noetherian subspaces of the Zariski–Riemann space of the quotient field of D (e.g. if Σ and Γ have finite character), then the restriction that the members of Σ ∪ Γ have Krull dimension 2 can be omitted. An example shows that these results do not extend to overrings of three-dimensional Noetherian domains.