On Brauer groups and embedding problems over function fields
type: Item
instance of: scholarly article
title: On Brauer groups and embedding problems over function fields
zbMATH Open document ID: 0702.11080
DOI: 10.1016/0021-8693(90)90199-X
author: Jack Sonn
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: Denote by \(Br(F)_ p\) the p-primary part of the Brauer group of a field F. The author proves the following theorem: let k be a global field and p a rational prime different from the characteristic of k. Let \(k_ v\) be the completion of k at the prime v. For each v fix an embedding \(k\to k_ v\). Then the induced map \[ Br(k(t_ 1,...,t_ n))_ p\to \prod_{v}Br(k_ v(t_ 1,...,t_ n))_ p \] is injective, where v runs through all the primes of k, and \(t_ 1,...,t_ n\) are algebraically independent indeterminates. The proof is based on a lemma which states that the Auslander-\(Brumer\)-\(Faddeev\) theorem behaves well under extension of the field of constants. As a consequence, in the last section, it is shown that certain embedding problems defined over k(t) are solvable if and only if they are solvable over \(k_ v(t)\) for all primes v of k.
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 12G99
Mathematics Subject Classification ID: 14F22
zbMATH DE Number: 4150329
zbMATH Keywords: rational function fields
zbMATH Keywords: Brauer group
zbMATH Keywords: embedding problems
reviewed by: Richard Massy
MaRDI profile type: Publication
cites work: Q5573504
cites work: Q3903092
cites work: Q3030914
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full work available at URL: 0021-8693(90)90199-x
OpenAlex ID: W2039981804
description: scientific article; zbMATH DE number 4150329