Rees valuations

Swanson, Irena

doi:10.1007/978-1-4419-6990-3_16

Cited by 4 publications

(2 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An ideal I of the local ring D that has only one Rees valuation ring is one-fibered. For more on such ideals, see [25] and [26] and their references. The property of being one-fibered can be expressed also in terms of exceptional prime divisors of an extension.…”

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

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

Olberding¹,

Tartarone²

2013

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

Link to this article: http://journals.cambridge.org/abstract_S030500411300008XHow to cite this article: BRUCE OLBERDING and FRANCESCA TARTARONE (2013). Integrally closed rings in birational extensions of two-dimensional regular local rings. AbstractLet D be an integrally closed local Noetherian domain of Krull dimension 2, and let f be a nonzero element of D such that f D 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.

show abstract

Section: Preliminariesmentioning

confidence: 99%

“…In Proposition 4.1 of [26], Swanson proves that if D is a local Noetherian domain with maximal ideal m of dimension d > 1 and (f 1 , . .…”

Section: Two-dimensional Regular Local Ringsmentioning

confidence: 99%