Unipotent morphisms

TL;DR

This paper develops the theory of unipotent morphisms of algebraic stacks, establishing a local-to-global principle for vector bundles and applying it to torsion gerbes and Deligne-Mumford stacks.

Contribution

It introduces unipotent morphisms of algebraic stacks and proves a descent result, leading to new results on torsion gerbes and the resolution property in positive characteristic.

Findings

A unipotent analogue of Gabber's Theorem is established.

Smooth Deligne-Mumford stacks with quasi-projective coarse spaces satisfy the resolution property in positive characteristic.

A descent result for flags is proved using results of Sch"appi.

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 Sch\"appi.