Azumaya algebra

In mathematics, an Azumaya algebra is a generalization of central simple algebras to "R"-algebras where "R" need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where "R" 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 formula_1 is an formula_1-algebra formula_3 that is finitely generated, faithful, and projective as an formula_1-module, such that the tensor product formula_5 (where formula_6 is the opposite algebra) is isomorphic to the matrix algebra formula_7 via the map sending formula_8 to the endomorphism formula_9 of formula_3.
Examples over a field.
Over a field formula_11, 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 formula_12 for some division algebra formula_13 over formula_11. For example, quaternion algebras provide examples of central simple algebras.
Examples over local rings.
Given a local commutative ring formula_15, an formula_1-algebra formula_3 is Azumaya if and only if A is free of positive finite rank as an R-module, and the algebra formula_18 is a central simple algebra over formula_19, hence all examples come from central simple algebras over formula_19.
Cyclic algebras.
There is a class of Azumaya algebras called cyclic algebras which generate all similarity classes of Azumaya algebras over a field formula_21, hence all elements in the Brauer group formula_22 (defined below). Given a finite cyclic Galois field extension formula_23 of degree formula_24, for every formula_25 and any generator formula_26 there is a twisted polynomial ring formula_27, also denoted formula_28, generated by an element formula_29 such that
and the following commutation property holds:
As a vector space over formula_32, formula_27 has basis formula_34 with multiplication given by
Note that give a geometrically integral variety formula_36, there is also an associated cyclic algebra for the quotient field extension formula_37.
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 formula_1, where rings formula_39 are similar if there is an isomorphism
of rings for some natural numbers formula_41. Then, this equivalence is in fact an equivalence relation, and if formula_42, formula_43, then formula_44, showing
is a well defined operation. This forms a group structure on the set of such equivalence classes called the Brauer group, denoted formula_46. 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 formula_1 is a field.
Brauer group using Galois cohomology.
There is another equivalent definition of the Brauer group using Galois cohomology. For a field extension formula_49 there is a cohomological Brauer group defined as
and the cohomological Brauer group for formula_51 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 formula_51, such as the p-adic numbers formula_54, local class field theory gives the isomorphism of abelian groups:pg 193
This is because given abelian field extensions formula_56 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 there is the Kummer sequence
giving a long exact sequence in cohomology for a field formula_51. Since Hilbert's Theorem 90 implies formula_61, there is an associated short exact sequence
showing the second etale cohomology group with coefficients in the n-th roots of unity formula_63 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 n-torsion Milnor K-theory group formula_65 to the etale cohomology group formula_66, 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 formula_70, whose class for formula_71 is represented by a cyclic algebra formula_72. For the Kummer extension formula_73 where formula_74, take a generator formula_75 of the cyclic group, and construct formula_76. There is an alternative, yet equivalent construction through Galois cohomology and etale cohomology. Consider the short exact sequence of trivial formula_77-modules
The long exact sequence yields a map
For the unique character
with formula_81, there is a unique lift
and
note the class formula_84 is from the Hilberts theorem 90 map formula_85. Then, since there exists a primitive root of unity formula_86, there is also a class
It turns out this is precisely the class formula_88. Because of the Norm residue isomorphism theorem, formula_89 is an isomorphism and the formula_24-torsion classes in formula_91 are generated by the cyclic algebras formula_76.
Skolem-Noether theorem.
One of the important structure results about Azumaya algebras is the Skolem-Noether theorem: given a local commutative ring formula_1 and an Azumaya algebra formula_94, the only automorphisms of formula_3 are inner. Meaning, the following map is surjective:
where formula_97 is the group of units in formula_98 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 formula_99 for some formula_24, and the Čech cohomology group
gives a cohomological classification of such bundles. Then, this can be related to formula_102 using the exact sequence
It turns out the image of formula_104 is a subgroup of the torsion subgroup formula_105.
On a scheme.
An Azumaya algebra on a scheme "X" with structure sheaf formula_106, according to the original Grothendieck seminar, is a sheaf formula_107 of formula_106-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 formula_109 into a 'twisted-form' of the sheaf formula_110. Milne, "Étale Cohomology", starts instead from the definition that it is a sheaf formula_107 of formula_106-algebras whose stalk formula_113 at each point formula_29 is an Azumaya algebra over the local ring formula_115 in the sense given above.
Two Azumaya algebras formula_116 and formula_117 are "equivalent" if there exist locally free sheaves formula_118 and formula_119 of finite positive rank at every point such that 
where formula_121 is the endomorphism sheaf of formula_122. The Brauer group formula_123 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 formula_102.
Example over Spec(Z[1/n]).
The construction of a quaternion algebra over a field can be globalized to formula_125 by considering the noncommutative formula_126-algebra
then, as a sheaf of formula_106-algebras, formula_129 has the structure of an Azumaya algebra. The reason for restricting to the open affine set formula_130 is because the quaternion algebra is a division algebra over the points formula_131 is and only if the Hilbert symbol
which is true at all but finitely many primes.
Example over Pn.
Over formula_133 Azumaya algebras can be constructed as formula_134 for an Azumaya algebra formula_3 over a field formula_11. For example, the endomorphism sheaf of formula_137 is the matrix sheaf
so an Azumaya algebra over formula_133 can be constructed from this sheaf tensored with an Azumaya algebra formula_3 over formula_11, 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.