Descent for algebraic stacks

TL;DR

The work develops a comprehensive framework for descent of algebraic stacks along fppf morphisms, proving that $2$-descent data are effective and that algebraic and Deligne–Mumford properties descend. It extends classical Galois descent from schemes to stacks by reformulating $2$-descent data as actions of a finite Galois group on the stack, and it establishes an equivalence between descent data and group actions in the stack setting. The paper also clarifies the morphisms between stacks with descent data and provides a concrete example showcasing nontrivial $2$-descent. Overall, this yields a robust reference for descent theory of stacks and a practical bridge between group actions and stack descent with broad implications for moduli problems.

Abstract

We prove that algebraic stacks satisfy 2-descent for fppf coverings. We generalize Galois descent for schemes to stacks, by considering the case where the fppf covering is a finite Galois covering, and reformulating 2-descent data in terms of Galois group actions on the stack.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2: Breen, Giraud
• proof
• Definition 1.3
• Theorem 1.4
• Proposition 1.5
• Theorem 2.1: Grothendieck
• proof
• Lemma 2.2
• proof