Frobenius generation for algebraic stacks

TL;DR

This paper studies how the Frobenius morphism can generate generators for derived categories of algebraic stacks in positive characteristic, providing bounds and conditions for classical or strong generation.

Contribution

It introduces new methods to produce generators via Frobenius pushforwards and establishes bounds for Deligne--Mumford stacks.

Findings

Frobenius pushforwards can generate derived categories in many cases.

Bounds are provided for the number of Frobenius iterations needed.

The work applies to algebraic stacks in positive characteristic.

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.