Integrally closed rings in birational extensions of two-dimensional regular local rings

TL;DR

This paper investigates when integrally closed rings between a two-dimensional regular local ring and its localization are characterized by valuation overrings, linking algebraic properties to geometric intersection behavior.

Contribution

It establishes criteria connecting local valuation-based determination of rings with properties of blow-ups and completions, especially for regular local rings and regular parameters.

Findings

Local determination by valuation overrings is equivalent to properties of exceptional prime divisors.

When the base ring is analytically normal, the property extends from the ring to its completion.

In specific cases, the integrally closed ring is determined by a single valuation, and extension primes are pairwise comaximal.

Abstract

Let $D$ be an integrally closed local Noetherian domain of Krull dimension 2, and let $f$ be a nonzero element of $D$ such that $fD$ has prime radical. We consider when an integrally closed ring $H$ between $D$ and $D_f$ is determined locally by finitely many valuation overrings of $D$. We show such a local determination is equivalent to a statement about the exceptional prime divisors of normalized blow-ups of $D$, and, when $D$ is analytically normal, this property holds for $D$ if and only if it holds for the completion of $D$. This latter fact, along with MacLane's notion of key polynomials, allows us to prove that in some central cases where $D$ is a regular local ring and $f$ is a regular parameter of $D$, then $H$ is determined locally by a single valuation. As a consequence, we show that if $H$ is also the integral closure of a finitely generated $D$-algebra, then the exceptional prime ideals of the extension $H/D$ are comaximal. Geometrically, this translates into a statement about intersections of irreducible components in the closed fiber of the normalization of a proper birational morphism.