The \'etale local structure of algebraic stacks
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.
Cite
@article{arxiv.1912.06162,
title = {The \'etale local structure of algebraic stacks},
author = {Jarod Alper and Jack Hall and David Rydh},
journal= {arXiv preprint arXiv:1912.06162},
year = {2025}
}
Comments
61 pages; v4 additional reorganization and renumbering, together with a number of corrections; v3 reorganization and renumbering of sections, other small comments and corrections; v2 replaced Theorem 1.3 with a variant and added Section 17 on further applications to good moduli spaces