Some Theorems on the Resolution Property and the Brauer map

Abstract

Using formal-local methods, we prove that a separated and normal Deligne-Mumford surface must satisfy the resolution property, this includes the first class of separated algebraic spaces which are not schemes. Our analysis passes through the case of gerbes and an arbitrarily singular Deligne-Mumford curve, each of which we establish independently. Our methods can be extended to give new results on the surjectivity of the Brauer map. For example, we show that on a generically reduced variety, any cohomological Brauer class is represented by an Azumaya algebra away from a closed subset of codimension $\geq 3$. We also investigate generically trivial Brauer classes in high codimension which arise from singularities.