Descent via Tannaka duality

Abstract

Given a diagram of schemes, we can ask if a geometric object over one of them can be built from descent data (usually objects of the same type over the various other schemes in the diagram, together with compatibility isomorphisms). Using the language of moduli stacks, we can rephrase this as follows: saying that descent problems for a given diagram have essentially unique solutions amounts to saying that the diagram in question is a (bicategorical) colimit diagram in a certain 2-category of stacks. In this paper we use generalized Tannaka duality to explicitly compute certain colimits in the 2-category of Adams stacks. Using this we extend recent results of Bhatt from algebraic spaces to Adams stacks and a result of Hall-Rydh to non-noetherian rings. We conclude the paper with a global version of the Beauville-Laszlo theorem, which states that a large class of schemes and stacks can be built by gluing the open complement of an effective Cartier divisor with the infinitesimal neighbourhood of the divisor.