Existence of dicritical divisors
type: Item
instance of: scholarly article
title: Existence of dicritical divisors
zbMATH Open document ID: 1248.13021
author: William J. Heinzer
author: Shreeram S. Abhyankar
published in: American Journal of Mathematics
publication date: 2012-03-29T00:00:00Z
Mathematics Subject Classification ID: 13H05
zbMATH DE Number: 6019025
zbMATH Keywords: two-dimensional regular local rings
zbMATH Keywords: complete ideals
zbMATH Keywords: reduction of ideals
MaRDI profile type: Publication
full work available at URL: ajm.2012.0002
OpenAlex ID: W2064879389
DOI: 10.1353/AJM.2012.0002
Recommended article: Algebraic theory of dicritical divisors
Recommended article: Dicritical divisors and Jacobian problem
Recommended article: More about dicriticals
Recommended article: The effect of quadratic transformations on degree functions
Recommended article: A mixed multiplicity formula for complete ideals in 2-dimensional rational singularities
Recommended article: Powers of simple complete ideals in two-dimensional regular local rings
Recommended article: First neighborhood complete ideals in two-dimensional Muhly local domains
Recommended article: Non-splitting of prime divisors
Recommended article: Existence of dicritical divisors revisited
Recommended article: Factorization of complete ideals
review text: Let \(R\) be a two-dimensional regular local ring with field of quotients \(L\), and let \(D(R)^\Delta\) be the set of all prime divisors of \(R\). Let \(z\in L\); the subset \(\mathfrak D(R,z)\) of \(D(R)^\Delta\) consisting of all \(V\in D(R)^\Delta\) such that \(z\bmod{ M(V)}\in V/M(V)\) is transcendental over \(R/M(R)\) is called the set of dicritical divisors of \(z\) in \(R\). (The analytical (topological) concept of dicritical divisors was developed by several authors (see the first line of the introduction of the paper under review); in [\textit{S. S. Abhyankar}, Inversion and Invariance of Characteristic Terms: Part I. The Legacy of Alladi Ramakrishnan in the Mathematical Sciences. New York, NY: Springer. (2010; Zbl 1200.00039) pp. 93--168] Abhyankar algebracized this concept). For a finite subset \(U\) of \(D(R)^\Delta\), let \(\mathfrak D^*(R,U)\) be the set of all \(z\in L\) such that \(\mathfrak D(R,z)=U\). The main results of this paper are the existence theorem ET and the existence question EQ:NEWLINENEWLINEET: Given a finite subset \(U\) of \(D(R)^\Delta\), there exists \(z\in L^\times\) such that \(U=\mathfrak D(R,z)\). In case the residue field \(R/M(R)\) is infinite, the element \(z\) can be chosen in such a way that it satisfies additional properties.NEWLINENEWLINEEQ: Can you describe the set \(\mathfrak D^*(R,U)\)?NEWLINENEWLINEThe authors give a sketch of the proof of ET in section 3.1; a complete proof of ET is given in section 9 where also EQ is answered.NEWLINENEWLINEThe authors use the theory of complete ideals for two-dimensional regular local rings as developed by Zariski in Appendix 5 of [\textit{O. Zariski} and \textit{P. Samuel}, Commutative algebra. Vol. II. (The University Series in Higher Mathematics.) Princeton, N.J.-Toronto- London-New York: D. Van Nostrand Company, Inc. (1960; Zbl 0121.27801)]; another source for this theory is the book [\textit{I. Swanson} and \textit{C. Huneke}, Integral closure of ideals, rings, and modules. L…
reviewed by: Karl-Heinz Kiyek
description: scientific article; zbMATH DE number 6019025