On linear systems and the divisor class group of a real variety
type: Item
instance of: scholarly article
title: On linear systems and the divisor class group of a real variety
zbMATH Open document ID: 0569.14023
DOI: 10.1007/BF01781076
author: Margherita Roggero
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1983-01-01T00:00:00Z
review text: Let k be a field of characteristic zero and non necessarily algebraically closed. If I is an ideal of the polynomial ring \(k[X_ 1,...X_ n]\), \(Z_ k(I)\) the set of zeroes of I in \(k^ n\), \(I_ k(V)\) the ideal of the polynomials vanishing on \(V\subseteq k^ n\), we call I ''ideal of definition'' (or ''defining ideal'') if \(I=I_ k(Z_ k(I))\). If A is the coordinate domain of an affine or projective algebraic variety, we denote by \(Cl_ k(A)\) the subgroup of Cl(A) \((=divisor\) class group of A) generated by the classes of prime ideals of definition. The main results of the paper are: (i) A is factorial if and only if every prime ideal of definition in A of height 1 is principal. (ii) If \(k={\mathbb{R}} = field\) of real numbers, dim V\(>1\), A normal, then \(Cl(A)=Cl_{{\mathbb{R}}}(A)\). - In other words, the author shows that the rôle played by all prime ideals of height one, in some arguments and in the case of an algebraically closed field k, when k is not algebraically closed, it is played by all prime ideals of definition of height one.
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 13F15
Mathematics Subject Classification ID: 14Pxx
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 3908580
zbMATH Keywords: nonalgebraically closed ground field
zbMATH Keywords: real divisor class group
zbMATH Keywords: factorial coordinate domain
zbMATH Keywords: ideal of definition
reviewed by: Ezio Stagnaro
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3908580