Flat descent for Artin n-stacks

TL;DR

This paper establishes flat descent properties for Artin n-stacks within derived algebraic geometry, showing equivalences between different topologies and providing comparison results for non-abelian cohomology theories.

Contribution

It proves flat descent for Artin n-stacks in derived algebraic geometry, linking etale and fppf topologies and characterizing Artin n-stacks via fppf atlases.

Findings

An n-stack for the etale topology that is an Artin n-stack is also an n-stack for the fppf topology.

An n-stack with an fppf n-atlas and the fppf topology is an Artin n-stack.

Comparison results between fppf and etale non-abelian cohomologies.

Abstract

We prove two flat descent statements for Artin n-stacks. We first show that an n-stack for the etale topology which is an Artin n-stack in the sense of HAGII, is also an n-stack for the fppf topology. Moreover, an n-stack for the fppf topology which possess a fppf n-atlas is an Artin n-stack (i.e. possesses a smooth n-atlas). We deduce from these results some comparison statements between fppf and etale (non-ablelian) cohomolgies. This paper is written in the setting of derived algebraic geometry and its results are also valid for derived Artin n-stacks.