# Brauer group

In mathematics, the **Brauer group** of a field *K* is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over *K* and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is named after the algebraist Richard Brauer. The group may also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras.

## Construction

A central simple algebra (CSA) over a field *K* is a finite-dimensional associative *K*-algebra *A*, which is a simple ring, and for which the center is exactly *K*. Note that CSAs are in general *not* division algebras, though CSAs can be used to classify division algebras.

For example, the complex numbers **C** form a CSA over themselves, but not over **R** (the center is **C** itself, hence too large to be CSA over **R**). The finite-dimensional division algebras with center **R** (that means the dimension over **R** is finite) are the real numbers and the quaternions by a theorem of Frobenius, while any matrix ring over the reals or quaternions – M(*n*,**R**) or M(*n*,**H**) – is a CSA over the reals, but not a division algebra (if ).

We obtain an equivalence relation on CSAs over *K* by the Artin–Wedderburn theorem (Wedderburn's part, in fact), to express any CSA as a M(*n*,*D*) for some division algebra *D*. If we look just at *D*, that is, if we impose an equivalence relation identifying M(*m*,*D*) with M(*n*,*D*) for all integers *m* and *n* at least 1, we get the **Brauer equivalence** and the **Brauer classes**.

Given central simple algebras *A* and *B*, one can look at the their tensor product *A* ⊗ *B* as a *K*-algebra (see tensor product of R-algebras). It turns out that this is always central simple. A slick way to see this is to use a characterisation: a central simple algebra over *K* is a *K*-algebra that becomes a matrix ring when we extend the field of scalars to an algebraic closure of *K*.

Given this closure property for CSAs, they form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse class to that of an algebra *A* is the one containing the opposite algebra *A*^{op} (the opposite ring with the same action by *K* since the image of *K* → *A* is in the center of *A*). In other words, for a CSA *A* we have *A* ⊗ *A*^{op} = M(*n*^{2},*K*), where *n* is the degree of *A* over *K*. (This provides a substantial reason for caring about the notion of an opposite algebra: it provides the inverse in the Brauer group.)

## Examples

- In the following cases, every finite-dimensional central division algebra over a field
*K*is*K*itself, so that the Brauer group Br(*K*) is trivial:

*K*is an algebraically closed field:^{[1]}more generally, this is true for any pseudo algebraically closed field^{[2]}or quasi-algebraically closed field.^{[3]}

*K*is a finite field (Wedderburn's theorem);^{[1]}^{[4]}

*K*is the function field of an algebraic curve over an algebraically closed field (Tsen's theorem);^{[4]}

- An algebraic extension of
**Q**containing all roots of unity.^{[4]}

- An algebraic extension of

- The Brauer group Br(
**R**) of the field**R**of real numbers is the cyclic group of order two. There are just two non-isomorphic real division algebras with center**R**: the algebra**R**itself and the quaternion algebra**H**.^{[5]}Since**H**⊗**H**≅ M(4,**R**), the class of**H**has order two in the Brauer group. More generally, any real closed field has Brauer group of order two.^{[1]}

*K*is complete under a discrete valuation with finite residue field. Br(*K*) is isomorphic to**Q**/**Z**.^{[5]}

## Brauer group and class field theory

The notion of Brauer group plays an important role in the modern formulation of the class field theory. If *K*_{v} is a non-archimedean local field, the Hasse invariants gives a canonical isomorphism inv_{v}: Br(*K*_{v}) → **Q**/**Z** constructed in local class field theory.^{[6]}^{[7]}^{[8]} An element of the Brauer group of order *n* can be represented by a cyclic division algebra of dimension *n*^{2}.^{[9]}

The case of a global field *K* is addressed by the global class field theory. If *D* is a central simple algebra over *K* and *v* is a valuation then *D* ⊗ *K*_{v} is a central simple algebra over *K*_{v}, the local completion of *K* at *v*. This defines a homomorphism from the Brauer group of *K* into the Brauer group of *K*_{v}. A given central simple algebra *D* splits for all but finitely many *v*, so that the image of *D* under almost all such homomorphisms is 0. The Brauer group Br(*K*) fits into an exact sequence^{[5]}^{[10]}

where *S* is the set of all valuations of *K* and the right arrow is the direct sum of the local invariants: the Brauer group of the real numbers is identified with (1/2)**Z**/**Z**. The injectivity of the left arrow is the content of the Albert–Brauer–Hasse–Noether theorem. Exactness in the middle term is a deep fact from the global class field theory. The group **Q**/**Z** on the right may be interpreted as the "Brauer group" of the class formation of idele classes associated to *K*.

## Properties

- Base change from a field
*K*to an extension field*L*gives a*restriction*map from Br(*K*) to Br(*L*). The kernel is the group Br(*L*/*K*) of classes of*K*-algebras that split over*L*. - The Brauer group of any field is a torsion group.
^{[11]}

## General theory

For an arbitrary field *K*, the Brauer group may be expressed in terms of Galois cohomology as follows:^{[12]}

Here, *K*^{s} is the separable closure of *K*, which coincides with the algebraic closure when *K* is a perfect field. Note that every finite dimensional central simple algebra has a separable splitting field.^{[13]}

The isomorphism of the Brauer group with a Galois cohomology group can be described as follows. If *D* is a division algebra over *K* of dimension *n*^{2} containing a Galois extension *L* of degree *n* over *K*, then the subgroup of elements of *D** that normalize *L* is an extension of the Galois group Gal(*L*/*K*) by the nonzero elements *L** of *L*, so corresponds to an element of H^{2}(Gal(*L*/*K*), *L**).

A generalisation of the Brauer group to the case of commutative rings was introduced by Maurice Auslander and Oscar Goldman,^{[14]} and more generally for schemes by Alexander Grothendieck. In their approach, central simple algebras over a field are replaced with Azumaya algebras.^{[15]}

## See also

## Notes

- ↑
^{1.0}^{1.1}^{1.2}Lorenz (2008) p.164 - ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ Serre (1979) p.161
- ↑
^{4.0}^{4.1}^{4.2}Serre (1979) p.162 - ↑
^{5.0}^{5.1}^{5.2}Serre (1979) p.163 - ↑ Lorenz (2008) p.232
- ↑ Serre (1967) p.137
- ↑ Shatz (1972) p.155
- ↑ Lorenz (2008) p.226
- ↑ Gille & Szamuely (2006) p.159
- ↑ Lorenz (2008) p.194
- ↑ Serre (1979) pp.157-159
- ↑ Jacobson (1996) p.93
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ Saltman (1999) p.21

## References

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## Further reading

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