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Frobenius generation for algebraic stacks

Pat Lank, Fei Peng

TL;DR

This work extends Frobenius-generation techniques from schemes to algebraic stacks in characteristic p by introducing F-finiteness for stacks and employing étale dévissage. It proves that for concentrated F-finite DM stacks with separated diagonal, the Frobenius pushforward of a suitable Perf_Z generator yields a classical generator for Db_coh,Z, with stronger conclusions for Deligne–Mumford stacks. A key contribution is a method to bound the number of Frobenius iterates needed, via étale-local invariants, enabling concrete generation bounds in terms of local data. The results apply to stacky curves and toric stacks, relate to Kunz-type regularity criteria, and provide a framework for explicit generator constructions in prime characteristic, impacting moduli theory and characteristic p geometry.

Abstract

This work investigates the Frobenius morphism on derived categories associated with algebraic stacks in positive characteristic. Particularly, we show that in many cases sufficiently many Frobenius pushforwards of a compact generator produce a classical or strong generator for the bounded derived category of coherent sheaves. In the case of Deligne--Mumford stacks, we can bound the number of iterates required.

Frobenius generation for algebraic stacks

TL;DR

This work extends Frobenius-generation techniques from schemes to algebraic stacks in characteristic p by introducing F-finiteness for stacks and employing étale dévissage. It proves that for concentrated F-finite DM stacks with separated diagonal, the Frobenius pushforward of a suitable Perf_Z generator yields a classical generator for Db_coh,Z, with stronger conclusions for Deligne–Mumford stacks. A key contribution is a method to bound the number of Frobenius iterates needed, via étale-local invariants, enabling concrete generation bounds in terms of local data. The results apply to stacky curves and toric stacks, relate to Kunz-type regularity criteria, and provide a framework for explicit generator constructions in prime characteristic, impacting moduli theory and characteristic p geometry.

Abstract

This work investigates the Frobenius morphism on derived categories associated with algebraic stacks in positive characteristic. Particularly, we show that in many cases sufficiently many Frobenius pushforwards of a compact generator produce a classical or strong generator for the bounded derived category of coherent sheaves. In the case of Deligne--Mumford stacks, we can bound the number of iterates required.

Paper Structure

This paper contains 20 sections, 23 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Frobenius generation
  3. Bounding iterates
  4. Preliminaries
  5. Generation
  6. Algebraic stacks
  7. Finiteness reminder
  8. Associated triangulated categories
  9. Affine pointed
  10. Concentratedness
  11. Perfect complexes
  12. Supports
  13. Approximation by compacts
  14. Thomason condition
  15. Finiteness of Frobenius
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\mathcal{X}$ be a concentrated $F$-finite Deligne--Mumford stack with separated diagonal. Then for any $Z\subseteq |\mathcal{X}|$ closed and $e \gg 0$, there is a $G\in \operatorname{Perf}_Z (\mathcal{X})$ such that $\mathbf{R} F_\ast^e G$ is classical generator for $D^b_{\operatorname{coh},Z}(

Theorems & Definitions (57)

  • Theorem 1.1: see \ref{['thm:Frobenius_generation']}
  • Proposition 1.2
  • Definition 3.1
  • Example 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • proof
  • Definition 3.5
  • Remark 3.6
  • ...and 47 more