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Unipotent morphisms

Daniel Bragg, Jack Hall, Siddharth Mathur

TL;DR

The paper develops a theory of unipotent morphisms of algebraic stacks and proves a local-to-global principle for constructing vector bundles via flags, enabling global unipotence from local data. It introduces flag-based descent and Schäppi's Lazard-type results to control pushforwards and to realize vector bundles through $\mathrm{U}(\mathscr{L})$-towers, culminating in a unipotent analogue of Gabber's theorem and a broad resolution-property toolkit. The results yield that $\mathbf{G}_a$-gerbes over stacks with the resolution property have the resolution property, extend to Deligne–Mumford gerbes in positive characteristic, and apply to DM stacks with quasi-projective coarse moduli spaces, thereby advancing the understanding of moduli with unipotent stabilizers. Overall, the work unifies and extends resolution-property phenomena via a robust framework for unipotent morphisms and flags, with implications for Brauer groups, coarse moduli spaces, and faithful moduli spaces.

Abstract

We introduce the theory of unipotent morphisms of algebraic stacks and prove a surprising local to global principle for a class of vector bundles. Two sample applications of our methods are the following: (1) a unipotent analogue of Gabber's Theorem for torsion $\mathbf{G}_m$-gerbes and (2) smooth Deligne-Mumford stacks with quasi-projective coarse spaces satisfy the resolution property in positive characteristic. Our main tool is a descent result for flags, which we prove using results of Schäppi.

Unipotent morphisms

TL;DR

The paper develops a theory of unipotent morphisms of algebraic stacks and proves a local-to-global principle for constructing vector bundles via flags, enabling global unipotence from local data. It introduces flag-based descent and Schäppi's Lazard-type results to control pushforwards and to realize vector bundles through -towers, culminating in a unipotent analogue of Gabber's theorem and a broad resolution-property toolkit. The results yield that -gerbes over stacks with the resolution property have the resolution property, extend to Deligne–Mumford gerbes in positive characteristic, and apply to DM stacks with quasi-projective coarse moduli spaces, thereby advancing the understanding of moduli with unipotent stabilizers. Overall, the work unifies and extends resolution-property phenomena via a robust framework for unipotent morphisms and flags, with implications for Brauer groups, coarse moduli spaces, and faithful moduli spaces.

Abstract

We introduce the theory of unipotent morphisms of algebraic stacks and prove a surprising local to global principle for a class of vector bundles. Two sample applications of our methods are the following: (1) a unipotent analogue of Gabber's Theorem for torsion -gerbes and (2) smooth Deligne-Mumford stacks with quasi-projective coarse spaces satisfy the resolution property in positive characteristic. Our main tool is a descent result for flags, which we prove using results of Schäppi.

Paper Structure

This paper contains 14 sections, 30 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Vector bundles, global quotients, and the resolution property
  3. Schäppi's Theorem
  4. Flags
  5. Unipotent morphisms and groups
  6. Unipotent morphisms
  7. Unipotent groups
  8. Representable R-unipotent morphisms
  9. Geometrically unipotent vs locally unipotent groups
  10. Applications
  11. Unipotent gerbes and the resolution property
  12. Deligne--Mumford Gerbes
  13. Deligne--Mumford stacks in positive characteristic
  14. Faithful moduli spaces

Key Result

Theorem 1

Let $X$ be a quasi-compact scheme that admits an ample line bundle. Let $\mathscr X \to X$ be a $\mathbf{G}_m$-gerbe. If the cohomology class $[\mathscr X]\in\operatorname{H}^2(X,\mathbf{G}_m)$ of $\mathscr X$ is torsion, then $\mathscr X$ is a global quotient stack (see Definition def:globalquotien

Theorems & Definitions (104)

  • Theorem : Gabber
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Example 2.5
  • Definition 2.6
  • ...and 94 more