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The \'etale local structure of algebraic stacks

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arxiv 1912.06162 v4 pith:EWAGH2O4 submitted 2019-12-12 math.AG

Jarod Alper , Jack Hall , David Rydh This is my paper · ORCID
classification math.AG
keywords stacksalgebraicresultsetalelinearlyprovereductivestack
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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.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Root stack valuative criterion for good moduli spaces

    math.AG 2025-07 accept novelty 7.0 of 10

    The authors prove a root stack valuative criterion for good moduli spaces and reductive gerbes, enabling root-extension of generic points and several arithmetic and moduli applications.

  2. Moduli spaces for $\Theta$-strata and non-reductive quotients

    math.AG 2025-05 conditional novelty 7.0 of 10

    A new proof of the U-theorem that extends it from complex projective varieties to arbitrary affine Noetherian bases, including all characteristics and families, by proving moduli-space theorems for theta-strata.

  3. Moduli of truncated shtukas and displays

    math.AG 2025-06 conditional novelty 6.0 of 10

    Truncated shtukas and displays are classified by quotient stacks of loop groups by display groups, with explicit cutoff bounds N0 = 2C+1 beyond which truncation determines the full object.

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