Brauer Groups of Algebraic Stacks and 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.