Ideal theory of infinite directed unions of local quadratic transforms

Abstract

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.