Nicht endlich erzeugte Primideale in Steinschen Algebren. (Not finitely generated prime ideals in Stein algebras)
type: Item
instance of: scholarly article
title: Nicht endlich erzeugte Primideale in Steinschen Algebren. (Not finitely generated prime ideals in Stein algebras)
zbMATH Open document ID: 0584.32028
DOI: 10.1007/BF01168837
author: Otto Forster
published in: Manuscripta Mathematica
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155060
review text: In this note the author constructs a closed prime ideal I in the ring \({\mathcal O}({\mathbb{C}}^ 3)\) which is not finitely generated. It is known that 3 is the smallest dimension in which this kind of phenomena can occur. I is the ideal of a certain irreducible curve \(Y\subset {\mathbb{C}}^ 3,\) obtained as the image of a proper holomorphic map \(f: {\mathbb{C}}\to {\mathbb{C}}^ 3.\) The construction is performed as follows: After some preliminar results on local ideal, the author considers the planes \(E_ n=\{(z_ 1,z_ 2,z_ 3)\in {\mathbb{C}}^ 3| z_ 1+nz_ 2+n^ 2z_ 3=0\}\) and the union of lines \(T_ k=\cup_{m<n\leq k}E_ m\cap E_ n;\) then he proves that the ideal in \({\mathcal O}_ 0\) of germs of holomorphic functions around the origin, vanishing on \(T_ k\) cannot be generated by less than k elements. Then an irreducible curve Y, obtained as the image of a proper holomorphic map \(f: {\mathbb{C}}\to {\mathbb{C}}^ 3\) is constructed in such a way in a discrete sequence of points \(p_ k\), the germs of Y coincides up to order k (in a suitable sense) with the model \(T_ k\).
Mathematics Subject Classification ID: 32E10
Mathematics Subject Classification ID: 32A38
Mathematics Subject Classification ID: 32H35
Mathematics Subject Classification ID: 13E15
zbMATH DE Number: 3935456
zbMATH Keywords: Stein algebra
zbMATH Keywords: ring of holomorphic functions on Stein spaces
zbMATH Keywords: closed prime ideal
MaRDI profile type: Publication
cites work: Entwicklung analytischer Funktionen auf Riemannschen Flächen
cites work: Zur Theorie der Steinschen Algebren und Moduln
cites work: On Generators of Ideals
cites work: Ideal theory on open Riemann surfaces
description: scientific article; zbMATH DE number 3935456