Integrally closed rings in birational extensions of two-dimensional regular local rings (Q2841502)

Let \(D\) be an integral domain with quotient field \(K\). Call a ring \(R\) an overring of \(D\) if \(D\subseteq R\subseteq K.\) For a nonzero (preferably nonunit) \(f\in D,\) let \(D_{f}=D[1/f].\) Call a morphism \(\pi :X\rightarrow S\) of schemes a modification if \(\pi \) is proper and birational. For \(0\neq f\in D\), denote by \(X_{f}\) the open subscheme of \(X\) consisting of all points \(x\in X\) such that \(f\) is not in the local ring \(O_{X,x}\) of the point \(x.\) The authors of the paper under review prove the following result.NEWLINENEWLINETheorem A. Let (\(D,m\)) be a two-dimensional integrally closed local Noetherian domain, let \(0\neq f\in m\) and let \(n\) be a positive integer. Then the following are equivalent.NEWLINENEWLINE(1) For each modification \(\pi :\) \(X\rightarrow \mathrm{Spec}(D)\), at most \(n\) irreducible components of the closed fiber of the normalization of \(\pi \) meet in any affine open subscheme of \(\overline{X}\) containing \(\overline{ X_{f}}\).NEWLINENEWLINE(2) For every finitely generated \(D\)-subalgebra \(H\) of \(D_{f}\) , at most \(n\) of the height \(1\) prime ideals of the integral closure \(\overline{H}\) of \(D\) lying over \(m\) are contained in any single maximal ideal of \(H\).NEWLINENEWLINEIf also \(\surd fD\) is a prime ideal of \(D\), then (1) and (2) are equivalent toNEWLINENEWLINE(3) For every integrally closed ring \(H\) between \(D\) and \(D_{f}\) and maximal ideal \(M\) of \(H\), there is a representation \(H_{M}=V_{1}\cap \) \(\cap V_{n}\cap (D_{f})_{H\backslash M}\), for some not necessarily distinct valuation overrings \(V_{1},\dots,V_{n}\) of \(H\).NEWLINENEWLINEThe authors then set out to show that the conditions (1)-(3) are equivalent in various other situations ending up showing that the following also holds.NEWLINENEWLINETheorem B. Let \(D\) be a regular local ring of Krull dimension \(2\), and let \(f \) be a regular parameter of \(D\). Suppose that either (a) \(D\) is equicharacteristic, or (b) \(D\) has mixed characteristic and \(f\) is a prime integer in \(D\). Then \(D\), \(f\) and \(n=1\) satisfy the equivalent conditions (1)--(3) of Theorem A.NEWLINENEWLINEThis interesting paper takes you into several directions that may appear, at places, to take you close to algebraic geometry. However, the authors claim at the outset that one of their main motivations for the study was the problem of classifying integrally closed overrings of a Noetherian local Krull domain of Krull dimension \(2\). The reader may look up Olberding's survey article [\textit{B. Olberding}, in: Commutative algebra. Noetherian and non-Noetherian perspectives. New York, NY: Springer. 335--361 (2011; Zbl 1225.13010)] to see the state of the art. To the reviewer the present article looks like an effort at working towards a specific goal while appearing relevant in other areas as well.