  We study twisted vector bundles of infinite rank on gerbes, giving a new spin
on Grothendieck's famous problem on the equality of the Brauer group and
cohomological Brauer group. We show that the relaxed version of the question
has an affirmative answer in many, but not all, cases, including for any
algebraic space with the resolution property and any algebraic space obtained
by pinching two closed subschemes of a projective scheme. We also discuss some
possible theories of infinite rank Azumaya algebras, consider a new class of
"very positive" infinite rank vector bundles on projective varieties, and show
that an infinite rank vector bundle on a curve in a surface can be lifted to
the surface away from finitely many points.
