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 nitely 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 nitely 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 ber of the normalization of a proper birational morphism.

Olberding, B., Tartarone, F. (2013). Integrally closed rings in birational extensions of two-dimensional regular local rings. MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 155(1), 101-127 [10.1017/S030500411300008X].

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

TARTARONE, FRANCESCA
2013-01-01

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 nitely 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 nitely 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 ber of the normalization of a proper birational morphism.
2013
Olberding, B., Tartarone, F. (2013). Integrally closed rings in birational extensions of two-dimensional regular local rings. MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 155(1), 101-127 [10.1017/S030500411300008X].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/136450
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