Groupe de Brauer des corps de fractions rationnelles à coefficients complexes
type: Item
instance of: scholarly article
title: Groupe de Brauer des corps de fractions rationnelles à coefficients complexes
zbMATH Open document ID: 0555.13002
author: Philippe A. J. Steiner
published in: L'Enseignement Mathématique. 2e Série
publication date: 1984-01-01T00:00:00Z
review text: The author presents the following description of the Brauer group of the rational function fields \(Br({\mathbb{C}}(t_ 1,...,t_ n))\cong \oplus_ f\{H^ 1(X_ g,{\mathbb{Q}}/{\mathbb{Z}})\oplus Div_ 0(X_ g)\oplus_{{\mathbb{Z}}}{\mathbb{Q}}/{\mathbb{Z}}\}\) where f runs through all unitary irreducible polynomials with coefficients in \({\mathbb{C}}(t_ 1,...,t_ i)\) for \(1\leq i<n_ g\). Here, \(X_ g\) is a geometrical model, i.e. a smooth variety having \(K_ g={\mathbb{C}}(t_ 1,...,t_ i)(\alpha)\), where \(\alpha\) is a root of f, as its function field and \(Div_ 0(X_ g)\) is the divisor subgroup consisting of those divisors with trivial first Chern class. Further, the author shows that \(Br({\mathbb{C}}(t_ 1,...,t_ n))\) is a divisible group so isomorphic to \(\oplus_{i\in I}({\mathbb{Q}}/{\mathbb{Z}})\) where I is in one-to-one correspondence with \({\mathbb{C}}\). In particular for all m,n\(\geq 2\) one has \(Br({\mathbb{C}}(x_ 1,...,x_ m))\cong Br({\mathbb{C}}(x_ 1,...,x_ n)).\)
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 3884278
zbMATH Keywords: Brauer group of the rational function fields
reviewed by: Lieven Le Bruyn
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3884278