Constructing colimits by gluing vector bundles

TL;DR

This paper demonstrates that by focusing on vector bundles, one can construct pushouts and colimits in certain algebraic stacks using a gluing approach, leveraging generalized Tannaka duality and a novel fiber functor existence theorem.

Contribution

It introduces a new method for constructing colimits in algebraic stacks via vector bundle gluing, utilizing generalized Tannaka duality and a novel fiber functor existence criterion.

Findings

Constructs pushouts along arbitrary morphisms of Adams stacks.

Uses generalized Tannaka duality to facilitate colimit construction.

Provides a new criterion for the existence of stacks with atlases based on quasi-coherent sheaves.

Abstract

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. We use this version of Deligne's existence theorem for fiber functors as a novel way of recognizing stacks which have atlases. It differs considerably from Artin's algebraicity results and their generalizations: rather than studying conditions on the functor of points which ensure the existence of an atlas, our theorem identifies conditions on the category of quasi-coherent sheaves of the stack which imply that an atlas exists.