Global construction of the normalization of Stein spaces
type: Item
instance of: scholarly article
title: Global construction of the normalization of Stein spaces
zbMATH Open document ID: 0676.32002
author: Geneviève Pourcin
author: Sandra Hayes
published in: L'Enseignement Mathématique. 2e Série
publication date: 1988-01-01T00:00:00Z
review text: Let X be a Stein space, \(\tilde X\) its normalization. Stein spaces X can be completely described by the algebra \({\mathcal O}(X)\) of global holomorphic functions. The question is can the normalization \(\tilde X\) of X be constructed just from the holomorphic functions on X. Or equivalently whether the algebra \(\tilde {\mathcal O}(X)\) of all weakly holomorphic functions on X can be derived from the algebra \({\mathcal O}(X)\) of holomorphic functions on X. The main results of the paper are as follows: if X is irreducible \(\tilde {\mathcal O}(X)\) is the topological closure of the integral closure \(\tilde {\mathcal O}(X)\) of \({\mathcal O}(X)\). An example is given when \(\tilde {\mathcal O}(X)\) is not topologically closed and X is locally irreducible. \(\tilde {\mathcal O}(X)\) can also be obtained by taking the intersection of the localizations \(S_ x^{-1}\tilde {\mathcal O}(X)\) of \(\tilde {\mathcal O}(X)\) with respect to \(S_ x:=\{g\in {\mathcal O}(X):\) g(x)\(\neq 0\}\) for every \(x\in X\).
Mathematics Subject Classification ID: 32E10
Mathematics Subject Classification ID: 32C20
zbMATH DE Number: 4107182
zbMATH Keywords: Stein space
zbMATH Keywords: normalization
reviewed by: S. Ivashkovich
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4107182