Lieven Le Bruyn

"Smooth Orders over Surfaces"

Claudio's paper "A formal inverse to the Cayley-Hamilton equation" allows us to define 'smooth' orders in central simple algebras in a sensible way. I will consider the question whether there is an order- version of desingularization, that is, does every central simple algebra (over the functionfield of a projective variety) contain a smooth order? I will show that this cannot be the case, even over surfaces. On the positive side I will show that any central simple algebra over the functionfield of a surface contains an order having at most finitely many (noncommutative) singularities, all locally of quantum-plane type. The proof uses an etale local structure-result on smooth orders (in arbitrary dimension) and results of M. Artin on maximal orders over surfaces. If time allows it, I will also describe the fibers of the corresponding Brauer-Severi varieties.