# Gerbes for trigonalizable group schemes

## TL;DR

This paper proves that certain algebraic stacks with specific properties are global quotient stacks and have the resolution property, extending previous results to more general settings involving trigonalizable group schemes.

## Contribution

It establishes that gerbes banded by trigonalizable group schemes admit faithful vector bundles and are quotient stacks, broadening the class of stacks known to have these properties.

## Key findings

- Gerbes banded by trigonalizable group schemes admit faithful vector bundles.
- Such stacks are proven to be global quotient stacks.
- The results extend previous work from schemes over fields to mixed characteristic stacks.

## Abstract

We prove that smooth, separated Deligne--Mumford stacks in mixed characteristic with quasi-projective coarse moduli space are global quotient stacks and satisfy the resolution property. This builds on work of Kresch and Vistoli and of Bragg, Hall, and Matthur which proves the case when the stack is over a base field, as well as work of Gabber and de Jong which proves the same holds for a $\mu_n$-gerbe over a scheme with an ample line bundle. The key technical input is a result that gerbes banded by so-called trigonalizable group schemes admit faithful vector bundles and are quotient stacks.