an Azumaya algebra over can be constructed from this sheaf tensored with an Azumaya algebra over , such as a quaternion algebra. Applications There have been significant applications of Azumaya algebras in diophantine geometry, following work of Yuri Manin. The Manin obstruction to the Hasse principle is defined using the Brauer group of schemes. See also Gerbe Class field theory Algebraic K-theory Motivic cohomology Norm residue isomorphism theorem References Brauer group and Azumaya algebras Milne, John. Etale cohomology. Ch IV Mathoverflow Thread on "Explicit examples of Azumaya algebras" Division algebras Ring theory Scheme theory Algebras