Local-global principles for the Brauer group
type: Item
instance of: scholarly article
title: Local-global principles for the Brauer group
zbMATH Open document ID: 0846.12007
DOI: 10.1007/BF02568005
author: Götz Wiesend
published in: Manuscripta Mathematica
publication date: 1995-08-29T00:00:00Z
full work available at URL: https://eudml.org/doc/156065
review text: Suppose that \(K\) is a field and that \(I\) is some set of local objects like valuations or places. For each \(i\in I\) let \(K_i\) be some extension field of \(K\) which may be a Henselization or a completion. One says that \(K\) has a local-global principle for the Brauer group with respect to \(I\) if the canonical map \(\text{Br} (K)\to \prod \text{Br} (K_i)\) is injective. For global fields the Hasse-Brauer-Noether theorem is a classical result of this type. More recently, F. Pop obtained results about function fields in one variable over real closed fields or \(p\)-adically closed fields. The author shows by examples that there are fields having no local-global principle. He studies an elementary class of fields with the property that all regular function fields in one variable have a local-global principle. This class of fields includes real closed and \(p\)-adically closed fields.
reviewed by: Niels Schwartz
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 12J12
Mathematics Subject Classification ID: 14H05
zbMATH DE Number: 784052
zbMATH Keywords: local-global principle
zbMATH Keywords: Brauer group
zbMATH Keywords: real closed fields
zbMATH Keywords: \(p\)-adically closed fields
zbMATH Keywords: regular function fields
MaRDI profile type: Publication
cites work: Valuation theory
cites work: On quasi algebraic closure
cites work: Duality theorems for curves over p-adic fields
cites work: Galoissche Kennzeichnung p-adisch abgeschlossener Körper.
cites work: Formally \(p\)-adic fields
cites work: On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting fields of algebras
cites work: Q3856855
cites work: Über ein Gegenbeispiel zum Normensatz
OpenAlex ID: W1993115168
description: scientific article; zbMATH DE number 784052