Azumaya algebras, are (etale) twisted forms of matrix rings. These objects are of great utility because they give rise to Brauer classes. Fifty years ago, Grothendieck asked whether every cohomological Brauer class has a corresponding Azumaya algebra. This question is still open even for smooth separated threefolds over the complex numbers!
One says a scheme (or Algebraic stack) X satisfies the resolution property if every coherent sheaf is the quotient of a vector bundle. The work of Totaro and Gross explains that this property holds iff X admits a very special quotient stack presentation. However, whether or not separated Algebraic stacks have this property remains a difficult question.
The goal of our talk will be to explain
(1) Why these two questions are deeply intertwined,
(2) New results regarding the existence of Azumaya Algebras and
(3) How we can use results as in (2) to show large classes of algebraic stacks satisfy the resolution property.