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 { R n } of local quadratic transforms of R along this valuation domain. We consider the situation where the sequence { R n } n ≥ 0 is infinite, and examine ideal-theoretic properties of the integrally closed local domain S = ⋃ n ≥ 0 R n . Among the set of valuation overrings of R , there exists a unique limit point V for the sequence of order valuation rings of the R n . 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 .