The étale local structure of algebraic stacks
Jarod Alper, Jack Hall, David Rydh
TL;DR
The paper extends the local structure theory for algebraic stacks from algebraically closed fields to arbitrary bases, proving that stacks with affine stabilizers are étale-locally quotient stacks around points with linearly reductive stabilizers. It builds a robust framework around coherent completeness and effectivity of adic sequences, enabling formal neighborhood control, completions, henselizations, and Artin-style algebraization in mixed characteristics. Key contributions include the universality of adequate moduli spaces, structural results for linearly reductive group schemes, refined local-structure theorems, and relative versions of Sumihiro and Luna theorems, with broad applications to equivariant geometry and moduli. The work provides powerful deformation/approximation techniques that bridge infinitesimal and global geometry, offering practical tools for constructing local models, proving algebraicity of various moduli stacks, and understanding representations and actions in mixed characteristic settings.
Abstract
We prove that an algebraic stack with affine stabilizers over an arbitrary base is étale-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 étale slice theorem.