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 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 .

t-Local Domains and Valuation Domains

“…From [27,Proposition 4.1], we conclude that S has another property of interest. So, if dim(S) = 2 and m S contains a nonzero comparable element then we know that S is a valuation domain (Theorem 4.15 and (5)).…”

“…On the other hand, in the non-Archimedean case, we know the following fact. In the previous situation, the integral domain S/p is a DVR [27,Lemma 3.4], and T = S p , since T = S[1/x] is a ring of fractions of S and p is disjoint from the multiplicative set {x n | n ≥ 0}. Therefore, S is the pullback of S/p with respect to the canonical projection T → T /p, where T /p is a field, coinciding with the residue field S p /pS p (isomorphic to the field of quotients of the integral domain S/p).…”

“…Section 5 has to do with "applications" which are essentially more efficient proofs of known results. We follow the study of the ring called Shannon's quadratic extension in [27] and point out that it is indeed a t-local domain, thus providing a Proof. Let J be a finitely generated (integral) ideal contained in P , the conclusion will follow if we show that J * ⊆ P .…”

Directed unions of local monoidal transforms and GCD domains

“…If dimR > 2, then, examples due to Shannon [18] show that the union may be not a valuation ring. This fact leaded to the study of the structure of the rings of the form S = n R n , which are called quadratic Shannon extension of R [12].…”

“…The tools used in these works are usually those from multiplicative ideal theory. In [12] and [13], the authors call a quadratic Shannon extension S simply a Shannon extension of R.…”