More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras, or equivalently using projective bundles.
As a result, the isomorphism classes of CSAs over K form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse of an algebra A is given by its opposite algebra Aop (the opposite ring with the same action by K since the image of K → A is in the center of A).
In more detail, define the period of a central simple algebra A over K to be its order as an element of the Brauer group.
(When n is not invertible in K or K does not have a primitive nth root of unity, a similar construction gives the cyclic algebra (χ, a) associated to a cyclic Z/n-extension χ of K and a nonzero element a of K.[8]) The Merkurjev–Suslin theorem in algebraic K-theory has a strong consequence about the Brauer group.
[12] For any central simple algebra A over a field K, the period of A divides the index of A, and the two numbers have the same prime factors.
[14] The Brauer group plays an important role in the modern formulation of class field theory.
The fact that the sum of all local invariants of a central simple algebra over K is zero is a typical reciprocity law.
There are two ways of defining the Brauer group of a scheme X, using either Azumaya algebras over X or projective bundles over X.
For smooth projective varieties over a field, the Brauer group is an important birational invariant.
Indeed, the finiteness of the Brauer group for surfaces in that case is equivalent to the Tate conjecture for divisors on X, one of the main problems in the theory of algebraic cycles.
[22] For a regular integral scheme of dimension 2 which is flat and proper over the ring of integers of a number field, and which has a section, the finiteness of the Brauer group is equivalent to the finiteness of the Tate–Shafarevich group Ш for the Jacobian variety of the general fiber (a curve over a number field).
[23] The finiteness of Ш is a central problem in the arithmetic of elliptic curves and more generally abelian varieties.
Manin used the Brauer group of X to define the Brauer–Manin obstruction, which can be applied in many cases to show that X has no K-points even when X has points over all completions of K.