One says a scheme, or an algebraic stack, has the resolution property if every coherent sheaf is the quotient of a locally free sheaf. Although this is a fundamental and widely used property in algebraic geometry, it is still poorly understood. After giving the appropriate definitions, we will explain the two most important sources of non-examples:
(1) affine group schemes G/S which cannot be embedded into GLn but which are forms of embeddable group schemes, and
(2) cohomological Brauer classes which are not represented by Azumaya algebras.
After describing a new way to construct non-trivial vector bundles on schemes and stacks, we introduce the notion of an R-unipotent morphism and characterize it geometrically. We will then present a surprising local to global principle: a locally R-unipotent morphism over a base with enough line bundles is globally R-unipotent. To conclude, we will explain why the unipotent analogues of (1) and (2) above cannot occur. This is joint work with Daniel Bragg and Jack Hall.
