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

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An Azumaya algebra[1] [2] over a commutative ring   is an  -algebra   obeying any of the following equivalent conditions:

  1. There exists an  -algebra   such that the tensor product of  -algebras   is Morita equivalent to  .
  2. The  -algebra   is Morita equivalent to  , where   is the opposite algebra of  .
  3. The center of   is  , and   is separable.
  4.   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

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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

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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

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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[3]  , there is also an associated cyclic algebra for the quotient field extension  .

Brauer group of a ring

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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[1]: 3  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

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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

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Over a local non-archimedean field  , such as the p-adic numbers  , 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.

n-torsion for a field

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Recall that there is the Kummer sequence[6]

 

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

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The Galois symbol, or norm-residue symbol, is a map from the  -torsion Milnor K-theory group   to the etale cohomology group  , denoted by

 [6]

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

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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

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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

 [1]: 6 

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])

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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

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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

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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

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References

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  1. ^ a b c Milne, James S. (1980). Étale cohomology (PDF). Princeton, N.J.: Princeton University Press. ISBN 0-691-08238-3. OCLC 5028959. Archived from the original (PDF) on 21 June 2020.
  2. ^ Borceux, Francis; Vitale, Enrico (2002). "Azumaya categories" (PDF). Applied Categorical Structures. 10: 449–467.
  3. ^ meaning it is an integral variety when extended to the algebraic closure of its base field
  4. ^ Serre, Jean-Pierre. (1979). Local Fields. New York, NY: Springer New York. ISBN 978-1-4757-5673-9. OCLC 859586064.
  5. ^ "Lectures on Cohomological Class Field Theory" (PDF). Archived (PDF) from the original on 22 June 2020.
  6. ^ a b Srinivas, V. (1994). "8. The Merkurjev-Suslin Theorem". Algebraic K-Theory (Second ed.). Boston, MA: Birkhäuser Boston. pp. 145–193. ISBN 978-0-8176-4739-1. OCLC 853264222.
Brauer group and Azumaya algebras
Division algebras