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Root stack valuative criterion for good moduli spaces

Dori Bejleri, Giovanni Inchiostro, Matthew Satriano

Abstract

We prove a root stack valuative criterion for good moduli space maps and for gerbes for reductive groups under some mild assumptions on the residue characteristic. We give several applications to parahoric extension for torsors, rational points on stacks, gerbes and homogeneous spaces, and the geometry of fibrations.

Root stack valuative criterion for good moduli spaces

Abstract

We prove a root stack valuative criterion for good moduli space maps and for gerbes for reductive groups under some mild assumptions on the residue characteristic. We give several applications to parahoric extension for torsors, rational points on stacks, gerbes and homogeneous spaces, and the geometry of fibrations.
Paper Structure (18 sections, 30 theorems, 16 equations)

This paper contains 18 sections, 30 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Applications
  3. Torsors and parahoric extension
  4. The Lang-Nishimura Theorem for Artin stacks
  5. Gerbes and homogeneous spaces
  6. Applications to fibrations
  7. Conventions.
  8. Acknowledgements
  9. Groups, torsors and gerbes on algebraic stacks
  10. Reductive groups over a stack
  11. Gerbes and torsors over stacks
  12. Preliminary reductions
  13. Passing to the polystable locus
  14. Reduction to the case of a gerbe
  15. Gerbe case
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\mathcal{X}\to X$ be a good moduli space where $\mathcal{X}$ is an Artin stack with affine diagonal and of finite type over a locally Noetherian scheme $S$. Let $R$ be a DVR with fraction field $K$ and residue field $k$. Given a commutative diagram of solid arrows \xymatrix@R=4em@C=4em{ there exists a root stack $\sqrt[n]{\operatorname{Spec} R} \to\operatorname{Spec} R$ and dotted arr

Theorems & Definitions (69)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Proposition 1.1
  • Corollary 1.1
  • Corollary 1.2
  • Remark 1.3
  • Corollary 1.3
  • Remark 1.4
  • ...and 59 more