The $\mathrm{Br} = \mathrm{Br}'$ question for some classifying stacks

TL;DR

This paper investigates the Brauer group equality for classifying stacks of certain group schemes, providing examples where the Brauer group does not equal the cohomological Brauer group, thus extending understanding beyond scheme cases.

Contribution

It introduces new examples of classifying stacks where the Brauer group differs from the cohomological Brauer group, highlighting cases not reducible to scheme scenarios.

Findings

Identifies classes of group schemes with $ ext{Br} e ext{Br}'$

Provides examples not arising from scheme cases

Shows limitations of traditional methods for these stacks

Abstract

In this paper we consider the $\mathrm{Br} = \mathrm{Br}'$ question for classifying stacks by various group schemes. These are algebraic stacks that do not necessarily admit a finite flat cover by a scheme for which $\mathrm{Br} = \mathrm{Br}'$ holds, hence are not amenable to the usual argument of pushing forward a twisted vector bundle. We provide two classes of examples satisfying $\mathrm{Br} \ne \mathrm{Br}'$ that do not "arise from" the scheme case.