Azumaya algebra
In mathematics, an Azumaya algebra is a generalization of central simple algebras to -algebras where  need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where  is a commutative local ring. 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. There are now several points of access to the basic definitions. Over a ring 
An Azumaya algebra
 over a commutative ring  is an -algebra  obeying any of the following equivalent conditions:
 There exists an -algebra  such that the tensor product of -algebras  is Morita equivalent to . The -algebra  is Morita equivalent to , where  is the opposite algebra of . The center of  is , and  is separable. is finitely generated, faithful, and projective as an -module, and the tensor product   is isomorphic to  via the map sending  to the endomorphism  of . Examples over a field 
Over a field , Azumaya algebras are completely classified by the Artin–Wedderburn theorem since they are the same as central simple algebras. These are algebras isomorphic to the matrix ring  for some division algebra  over  whose center is just . For example, quaternion algebras provide examples of central simple algebras. Examples over local rings 
Given a local commutative ring , an -algebra  is Azumaya if and only if  is free of positive finite rank as an -module, and the algebra  is a central simple algebra over , hence all examples come from central simple algebras over . Cyclic algebras 
There is a class of Azumaya algebras called cyclic algebras which generate all similarity classes of Azumaya algebras over a field , hence all elements in the Brauer group  (defined below). Given a finite cyclic Galois field extension  of degree , for every  and any generator  there is a twisted polynomial ring , also denoted , generated by an element  such that
 
and the following commutation property holds:
 

As a vector space over ,  has basis  with multiplication given by
 

Note that give a geometrically integral variety , there is also an associated cyclic algebra for the quotient field extension . Brauer group of a ring 
Over fields, there is a cohomological classification of Azumaya algebras using Étale cohomology. In fact, this group, called the Brauer group, can be also defined as the similarity classes of Azumaya algebras over a ring , where rings  are similar if there is an isomorphism
 
of rings for some natural numbers . Then, this equivalence is in fact an equivalence relation, and if , , then , showing
 
is a well defined operation. This forms a group structure on the set of such equivalence classes called the Brauer group, denoted . Another definition is given by the torsion subgroup of the etale cohomology group
 
which is called the cohomological Brauer group. These two definitions agree when  is a field. Brauer group using Galois cohomology 
There is another equivalent definition of the Brauer group using Galois cohomology. For a field extension  there is a cohomological Brauer group defined as
 
and the cohomological Brauer group for  is defined as
 
where the colimit is taken over all finite Galois field extensions. Computation for a local field 
Over a local non-archimedean field , such as the p-adic numbers , local class field theory gives the isomorphism of abelian groups: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:
 
where the vertical maps are isomorphisms and the horizontal maps are injections. n-torsion for a field 
Recall that there is the Kummer sequence
 
giving a long exact sequence in cohomology for a field . Since Hilbert's Theorem 90 implies , there is an associated short exact sequence
 
showing the second etale cohomology group with coefficients in the th roots of unity  is

Generators of n-torsion classes in the Brauer group over a field 
The Galois symbol, or norm-residue symbol, is a map from the -torsion Milnor K-theory group  to the etale cohomology group , 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 , whose class for  is represented by a cyclic algebra . For the Kummer extension  where , take a generator  of the cyclic group, and construct . There is an alternative, yet equivalent construction through Galois cohomology and etale cohomology. Consider the short exact sequence of trivial -modules
 

The long exact sequence yields a map
 

For the unique character
 
with , there is a unique lift
 
and
 
note the class  is from the Hilberts theorem 90 map . Then, since there exists a primitive root of unity , there is also a class
 

It turns out this is precisely the class . Because of the norm residue isomorphism theorem,  is an isomorphism and the -torsion classes in  are 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  (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 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