The \'etale local structure of algebraic stacks

TL;DR

This paper demonstrates that algebraic stacks with affine stabilizers are locally quotient stacks in the étale topology, extending previous results and establishing foundational theorems with various applications.

Contribution

It generalizes the local structure theorem for algebraic stacks with affine stabilizers, including new foundational results and applications to group schemes and group actions.

Findings

Proves étale local quotient structure for stacks with affine stabilizers.

Establishes new foundational results on algebraic stacks, including coherence and effectivity.

Provides applications to structure theorems for group schemes and actions.

Abstract

We prove that an algebraic stack with affine stabilizers over an arbitrary base is \'etale-locally a quotient stack around any point with a linearly reductive stabilizer. This generalizes earlier work by the authors of this article (stacks over algebraically closed fields) and by Abramovich, Olsson and Vistoli (stacks with finite inertia). In addition, we prove a number of foundational results, which are new even over a field. These include various coherent completeness and effectivity results for adic sequences of algebraic stacks. Finally, we give several applications of our results and methods, such as structure theorems for linearly reductive group schemes and generalizations to the relative setting of Sumihiro's theorem on torus actions and Luna's \'etale slice theorem.