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

Published online by Cambridge University Press:  12 February 2013

BRUCE OLBERDING  and
FRANCESCA TARTARONE
BRUCE OLBERDING
Affiliation:
Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003-8001, U.S.A. e-mail: olberdin@nmsu.edu
FRANCESCA TARTARONE
Affiliation:
Dipartimento di Matematica, Università degli Studi “Roma Tre”Largo San Leonardo Murialdo 1, Roma, 00146, Italy. e-mail: tfrance@mat.uniroma3.it
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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 Df 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 1 , July 2013 , pp. 101 - 127
Copyright
Copyright © Cambridge Philosophical Society 2013 

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