Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where
The notion was developed further in ring theory, and in algebraic geometry, where Alexander Grothendieck made it the basis for his geometric theory of the Brauer group in Bourbaki seminars from 1964–65.
Given a finite cyclic Galois field extension
such that and the following commutation property holds: As a vector space over
with multiplication given by Note that give a geometrically integral variety[3]
, there is also an associated cyclic algebra for the quotient field extension
are similar if there is an isomorphism of rings for some natural numbers
There is another equivalent definition of the Brauer group using Galois cohomology.
is defined as where the colimit is taken over all finite Galois field extensions.
, local class field theory gives the isomorphism of abelian groups:[4]pg 193 This is because given abelian field extensions
there is a short exact sequence of Galois groups and from Local class field theory, there is the following commutative diagram:[5] where the vertical maps are isomorphisms and the horizontal maps are injections.
, there is an associated short exact sequence showing the second etale cohomology group with coefficients in the
, denoted by It comes from the composition of the cup product in etale cohomology with the Hilbert's Theorem 90 isomorphism hence It turns out this map factors through
-modules The long exact sequence yields a map For the unique character with
One of the important structure results about Azumaya algebras is the Skolem–Noether theorem: given a local commutative ring
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
An Azumaya algebra on a scheme X with structure sheaf
-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.
Milne, Étale Cohomology, starts instead from the definition that it is a sheaf
are equivalent if there exist locally free sheaves
(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
The construction of a quaternion algebra over a field can be globalized to
The reason for restricting to the open affine set
is and only if the Hilbert symbol which is true at all but finitely many primes.
can be constructed from this sheaf tensored with an Azumaya algebra
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.