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[1] over a commutative ring is an -algebra that is finitely generated, faithful, and projective as an -module, such that the tensor product (where is the opposite algebra) is isomorphic to the matrix algebra 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 . 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 A is free of positive finite rank as an R-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[2] , 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[1]:3 of Azumaya algebras over a ring , where rings are similar if there is an isomorphism

of rings for some . 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

[3]pg 193

of abelian groups. 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

[4]

where the vertical maps are isomorphisms and the horizontal maps are injections.

n-torsion for a field

Recall there is the Kummer sequence[5]

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 n-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 n-torsion Milnor K-theory group to the etale cohomology group , denoted by

[5]

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 commutative ring and an Azumaya algebra , the only automorphisms of are inner. Meaning, the map

[6]

sending

is surjective. 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

[1]:6

where is the endomorphism sheaf of . The Brauer group 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 .

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

References

  1. Milne, J. S., 1942- (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.CS1 maint: multiple names: authors list (link)
  2. meaning it is an integral variety when extended to the algebraic closure of its base field
  3. Serre, Jean-Pierre. (1979). Local Fields. New York, NY: Springer New York. ISBN 978-1-4757-5673-9. OCLC 859586064.
  4. "Lectures on Cohomological Class Field Theory" (PDF). Archived (PDF) from the original on 22 June 2020.
  5. 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.
  6. is the group of units in

Brauer group and Azumaya algebras

Division algebras

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