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.
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 Aop (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 ⊗ Aop = M(n2,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.)
- 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:
- 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. 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.
- K is complete under a discrete valuation with finite residue field. Br(K) is isomorphic to Q/Z.
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 Kv is a non-archimedean local field, the Hasse invariants gives a canonical isomorphism invv: Br(Kv) → Q/Z constructed in local class field theory. An element of the Brauer group of order n can be represented by a cyclic division algebra of dimension n2.
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 ⊗ Kv is a central simple algebra over Kv, the local completion of K at v. This defines a homomorphism from the Brauer group of K into the Brauer group of Kv. 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
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.
- 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.
Here, Ks 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.
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 n2 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 H2(Gal(L/K), L*).
A generalisation of the Brauer group to the case of commutative rings was introduced by Maurice Auslander and Oscar Goldman, and more generally for schemes by Alexander Grothendieck. In their approach, central simple algebras over a field are replaced with Azumaya algebras.
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