Ideal theory of infinite directed unions of local quadratic transforms (Q504326)

Let \((R, \mathfrak{m})\) denote a regular local ring of dimension at least 2. For each valuation domain birationally dominating \(R\) there exists a unique sequence \(\{R_n\}\) of local quadratic transforms of \(R\) along this valuation domain. The authors study the case where the sequence \(\{ R_n \}_{n \geq 0}\) is infinite, and examine ideal-theoretic properties of the integrally closed local domain \(S = \cup_{n \geq 0} R_n\). It is called the Shannon extension in honour of \textit{D. Shannon}'s paper (see [Am. J. Math. 95, 294--320 (1973; Zbl 0271.14003)]). In this article the nature of Shannon extensions, with special emphasis on the ideal theory and representation of such rings is investigated. Assume that \(S\) is a Shannon extension then there exists a unique minimal proper Noetherian overring \(T\) of \(S\) and a valuation overring \(V\) of \(R\) such that \(S = T \cap V\) and the ring \(T\) is a localization of one of the regular Noetherian domains \(R_i\). The valuation domain \(V\) of \(S\) is the unique limit point in the patch topology of the order valuation rings of the regular local rings \(R_i\). Moreover there is a discussion of the complete integral closure of a Shannon extension. Among others there is a description when a Shannon extension \(S\) of a regular local ring \(R\) is a valuation ring. Namely this holds if and only if one of the equivalent conditions is satisfied: (i) \(S\) has only finitely many height 1 prime ideals. (ii) Either (a) \(\dim S = 1\) or (b) \(\dim S = 2\) and the valuation ring \(V\) of \(S\) has value group \({\mathbb{Z}} \oplus G \), where \(G\) is a subgroup of \({\mathbb{Q}}\) and the direct sum is ordered lexicographically. The article generalizes and extends various research papers starting with \textit{S. Abhyankar} (see [Am. J. Math. 78, 321--348 (1956; Zbl 0074.26301)]) and is also related to more recent research (see e.g. [\textit{A. Granja}, J. Algebra 280, No. 2, 699--718 (2004; Zbl 1098.13026)]).