Ideal theory of infinite directed unions of local quadratic transforms

Let (R,m) be a regular local ring of dimension at least 2. Associated to each valuation domain birationally dominating R, there exists a unique sequence {Rn} of local quadratic transforms of R along this valuation domain. We consider the situation where the sequence {Rn}n≥0 is infinite, and examine...

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Bibliographic Details
Published in:Journal of algebra 2017-03, Vol.474, p.213-239
Main Authors: Heinzer, William, Loper, K. Alan, Olberding, Bruce, Schoutens, Hans, Toeniskoetter, Matthew
Format: Article
Language:English
Subjects:
Complete integral closure
Local quadratic transform
Regular local ring
Valuation ring
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Summary:Let (R,m) be a regular local ring of dimension at least 2. Associated to each valuation domain birationally dominating R, there exists a unique sequence {Rn} of local quadratic transforms of R along this valuation domain. We consider the situation where the sequence {Rn}n≥0 is infinite, and examine ideal-theoretic properties of the integrally closed local domain S=⋃n≥0Rn. Among the set of valuation overrings of R, there exists a unique limit point V for the sequence of order valuation rings of the Rn. We prove the existence of a unique minimal proper Noetherian overring T of S, and establish the decomposition S=T∩V. If S is archimedean, then the complete integral closure S⁎ of S has the form S⁎=W∩T, where W is the rank 1 valuation overring of V.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2016.11.014