Constructing colimits by gluing vector bundles

Understand

As already observed by Gabriel, coherent sheaves on schemes obtained by gluing affine open subsets can be described by a simple gluing construction.

• An example due to Ferrand shows that this fails in general for pushouts along closed immersions, though the gluing construction still works for flat coherent sheaves.
• We show that by further restricting this gluing construction to vector bundles, we can construct pushouts along arbitrary morphisms (and more general colimits) of certain algebraic stacks called Adams stacks.
• The proof of this fact uses generalized Tannaka duality and a variant of Deligne's argument for the existence of fiber functors which works in arbitrary characteristic.