generated by the cyclic algebras . Skolem-Noether theorem One of the important structure results about Azumaya algebras is the Skolem-Noether theorem: given a local commutative ring and an Azumaya algebra , the only automorphisms of are inner. Meaning, the following map is surjective: where is the group of units in This is important because it directly relates to the cohomological classification of similarity classes of Azumaya algebras over a scheme. In particular, it implies an Azumaya algebra has structure group for some , and the Čech cohomology group gives a cohomological classification of such bundles. Then, this can be related to using the exact sequence It turns out the image of is a subgroup of the torsion subgroup . On a scheme An Azumaya algebra on a scheme X with structure sheaf , according to the original Grothendieck seminar, is a sheaf of -algebras that is étale locally isomorphic to a matrix algebra sheaf; one should, however, add the condition that each matrix algebra sheaf is of positive rank. This definition makes an Azumaya algebra on into a 'twisted-form' of the sheaf . Milne, Étale Cohomology, starts instead from the definition that it is a sheaf of -algebras whose stalk at each point is an Azumaya algebra over the local ring in the sense given above. Two Azumaya algebras and are equivalent if there exist locally free sheaves and of finite positive rank at every point such that where is the endomorphism sheaf of . The Brauer group of X (an analogue of the Brauer group of a field) is the set of equivalence classes of Azumaya algebras. The group operation is given by tensor product, and the inverse is given by the opposite algebra. Note that this is distinct from the cohomological Brauer group which is defined as . Example over Spec(Z[1/n]) The construction of a quaternion algebra over a field can be globalized to by considering the noncommutative -algebra then, as a sheaf of -algebras, has the structure of an Azumaya algebra. The reason for restricting to the open affine set is because the quaternion algebra is a division algebra over the points is and only if the Hilbert symbol which is true at all but finitely many primes. Example over Pn Over Azumaya algebras can be constructed as for an Azumaya algebra over a field . For example, the endomorphism sheaf of is the matrix sheaf so