Brauer Groups of Algebraic Stacks and GIT-Quotients

TL;DR

This paper investigates the Brauer groups of algebraic stacks and GIT quotients, revealing their dependence on the Brauer group of the underlying scheme and analyzing various moduli stacks and coarse moduli spaces.

Contribution

It provides new insights into how the Brauer groups of quotient stacks and moduli stacks relate to the Brauer groups of the underlying schemes.

Findings

Brauer groups of stacks depend on the underlying scheme's Brauer group.

Analysis of Brauer groups for moduli stacks of principal G-bundles.

Identification of Brauer groups of coarse moduli spaces with GIT-quotients.

Abstract

In this paper we consider the Brauer groups of algebraic stacks and GIT quotients: the only algebraic stacks we consider in this paper are quotient stacks [X/G], where X is a smooth scheme of finite type over a field k, and G is a linear algebraic group over k and acting on X, as well as various moduli stacks of principal G-bundles on a smooth projective curve X, associated to a reductive group G. We also consider the Brauer groups of the corresponding coarse moduli spaces, which most often identify with the corresponding GIT-quotients. One conclusion that we seem to draw then is that the Brauer groups (or their $\ell$-primary torsion parts, for a fixed prime $\ell$ different from char(k)) of the corresponding stacks and coarse moduli spaces depend strongly on the Brauer group of the given scheme X.