Integrally closed rings in birational extensions of two-dimensional regular local rings

Abstract

Link to this article: http://journals.cambridge.org/abstract_S030500411300008XHow to cite this article: BRUCE OLBERDING and FRANCESCA TARTARONE (2013). Integrally closed rings in birational extensions of two-dimensional regular local rings. AbstractLet D be an integrally closed local Noetherian domain of Krull dimension 2, and let f be a nonzero element of D such that f D has prime radical. We consider when an integrally closed ring H between D and D f is determined locally by finitely many valuation overrings of D. We show such a local determination is equivalent to a statement about the exceptional prime divisors of normalized blow-ups of D and, when D is analytically normal, this property holds for D if and only if it holds for the completion of D. This latter fact, along with MacLane's notion of key polynomials, allows us to prove that in some central cases where D is a regular local ring and f is a regular parameter of D, then H is determined locally by a single valuation. As a consequence, we show that if H is also the integral closure of a finitely generated D-algebra, then the exceptional prime ideals of the extension H/D are comaximal. Geometrically, this translates into a statement about intersections of irreducible components in the closed fiber of the normalization of a proper birational morphism.

Topological Aspects of Irredundant Intersections of Ideals and Valuation Rings

“…2 The introduction of finite character rank one C-representations to generalize the theory of Krull domains is due to Heinzer and Ohm [30]. This allows for considerable more flexibility in applying results to settings in which one considers, say, integrally closed rings between A and some integrally closed overring C. Even when A is a two-dimensional Noetherian domain and C is chosen a PID, the analysis of the integrally closed rings between A and C is quite subtle; see for example [1,6,38,42,46]. Regardless of the choice of A and C, Heinzer and Ohm [30,Corollary 1.4] prove that finite character rank one C-representations remain as well behaved as in the classical case of C = F: If C is a ring and A has a C-representation consisting of rank one valuation rings, then A has a unique irredundant finite character representation consisting of rank one valuation rings.…”

“…A feature throughout Sections 4 and 5 that is afforded by the abstract approach of spectral representations is that intersections of valuation rings can be considered relative to a subset of the ambient field. The motivation for this comes from the articles [1,30,38,42,46]. In these studies, one considers integrally closed domains A between a given domain and overring, e.g., between Z[T ] and Q [T ].…”