On Brauer groups and embedding problems over function fields
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.
scientific article; zbMATH DE number 4150329