The cohomological Brauer group of a torsion $\mathbb{G}_{m}$-gerbe

TL;DR

This paper establishes that the cohomological Brauer group of a torsion $ ext{G}_m$-gerbe over a scheme is isomorphic to the quotient of the scheme's Brauer group by the subgroup generated by the gerbe's class, extending Gabber's theorem.

Contribution

It generalizes Gabber's theorem by describing the Brauer group of a torsion $ ext{G}_m$-gerbe as a quotient of the base scheme's Brauer group.

Findings

$ ext{Br}'( ext{G})$ is isomorphic to $ ext{Br}'(S)$ modulo the subgroup generated by $[ ext{G}]$.

The result applies to torsion classes in the cohomological Brauer group.

Analogy with Brauer-Severi schemes is established.

Abstract

Let $S$ be a scheme and let $\pi : \mathcal{G} \to S$ be a $\mathbb{G}_{m,S}$-gerbe corresponding to a torsion class $[\mathcal{G}]$ in the cohomological Brauer group $\mathrm{Br}'(S)$ of $S$. We show that the cohomological Brauer group $\mathrm{Br}'(\mathcal{G})$ of $\mathcal{G}$ is isomorphic to the quotient of $\mathrm{Br}'(S)$ by the subgroup generated by the class $[\mathcal{G}]$. This is analogous to a theorem proved by Gabber for Brauer-Severi schemes.