Local structure of Schelter-Procesi smooth orders

Abstract

In this paper we give the étale local classification of Schelter-Procesi smooth orders in central simple algebras. In particular, we prove that if Δ \Delta is a central simple K K -algebra of dimension n 2 n^2 , where K K is a field of trancendence degree d d , then there are only finitely many étale local classes of smooth orders in Δ \Delta . This result is a non-commutative generalization of the fact that a smooth variety is analytically a manifold, and so has only one type of étale local behaviour.