Weil divisors and symbolic algebras
type: Item
instance of: scholarly article
title: Weil divisors and symbolic algebras
zbMATH Open document ID: 0658.14006
DOI: 10.1215/S0012-7094-88-05707-9
published in: Duke Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: Let R be a normal analytic local ring, f: \(Y\to X=Spec(R)\) be a projective resolution of singularities, a divisor \(D\in Pic(Y)\) maps to zero in \(R^ 2f_*{\mathbb{Z}}\), and \(E=f_*(D)\). The author proves the following result: \(\oplus_{n\geq 0}{\mathcal O}_ X(nE) \) is finitely generated if and only if D is torsion in Pic(Y). In particular, if every Weil divisor \({\mathcal O}_ X(E)\) on X satisfies the condition that \(\oplus_{n\geq 0}{\mathcal O}_ X(nE) \) is finitely generated as an \({\mathcal O}_ X\)-algebra, then \(R^ 1f_*{\mathcal O}_ Y=0\). If R is algebraizable, then R is \(S_ 3.\) In addition, the author proves that there exists a 3-dimensional affine local ring R with a Weil divisor D on \(X=Spec(R)\) such that \(\oplus_{n\geq 0}{\mathcal O}_ X(nD) \) is not a finitely generated \({\mathcal O}_ X\)-algebra.
reviewed by: Tong Wenting
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 13H99
zbMATH DE Number: 4075255
zbMATH Keywords: analytic local ring
zbMATH Keywords: resolution of singularities
author: Steven Dale Cutkosky
MaRDI profile type: Publication
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full work available at URL: s0012-7094-88-05707-9
OpenAlex ID: W2023386250
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description: scientific article; zbMATH DE number 4075255