  We prove a local-to-global principle for Brauer classes: for any finite
collection of non-trivial Brauer classes on a variety over a field of
transcendence degree at least 3, there are infinitely many specializations
where each class stays non-trivial. This is deduced from a
Grothendieck--Lefschetz-type theorem for Brauer groups of certain smooth
stacks. This also leads to the notion of a Brauertsch field.
