2015/05/24 by William Heinzer, K. Alan Loper, Heinzer, W. +7
Mathematics · #13A15 #13A18 #13H05 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1505.06445
openalex publication_date 2015/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let R 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 = \bigcupn ≥ 0 Rn. 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 archimedian, then the complete integral closure S* of S has the form S* = W ∩ T, where W is the rank 1 valuation overring of V.