Azumaya-Algebren und Oktavenalgebren auf algebraischen Varietäten

Abstract

We study noncommutative algebras over rings and on algebraic varieties. In the first part we give a criterion for whether or how far a central simple algebra over the function field of a projective variety could be extended as a sheaf of Azumaya algebras onto that variety. Furthermore, we examine the local structure of Azumaya algebras or more generally of maximal orders on surfaces given by the cyclic covering trick. The latter is a method introduced by Chan to construct maximal orders on surfaces, which furthermore have ramification exactly over a given curve. In the second part we study the not associative octonion algebras or more generally composition algebras over rings. We transfer well-known facts from the case of composition algebras over fields to the situation over rings. We examine octonion algebras and maximal orders over discrete valuation rings and prove a generalisation of a result of van der Blij and Springer about the local nature of maximal orders in the case of the rational numbers and of algebraic number fields to all noetherian integrally closed domains. Finally, we introduce a notion of sheaves of octonion algebras and of sheaves of maximal orders in octonion algebras on algebraic varieties.