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

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 ${\mathrm{Br}}^{\prime }(S)$ of $S$. We show that the cohomological Brauer group ${\mathrm{Br}}^{\prime }(\mathcal{G})$ of $\mathcal{G}$ is isomorphic to the quotient of ${\mathrm{Br}}^{\prime }(S)$ by the subgroup generated by the class $[\mathcal{G}]$. This is analogous to a theorem proved by Gabber for Brauer-Severi schemes.

18 pages