Locally free twisted sheaves of infinite rank

TL;DR

This paper investigates twisted vector bundles of infinite rank on gerbes, exploring their properties and implications for the Brauer group problem, and introduces new classes of infinite rank bundles with specific lifting properties.

Contribution

It provides new insights into the Brauer group problem for infinite rank sheaves and introduces the concept of very positive infinite rank vector bundles.

Findings

Affirmative results for the Brauer group equality in certain cases

Existence of infinite rank vector bundles that can be lifted off finitely many points

Discussion of potential theories for infinite rank Azumaya algebras

Abstract

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.