2019/12/12 by Jarod Alper, Alper, Jarod, Jack Hall +3 · 9 citations
Mathematics · #14D23 (Primary) #14L15 #14L24 #14L30 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1912.06162
openalex publication_date 2019/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.