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 over K, with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer.
The Brauer group arose out of attempts to classify division algebras over a field. It can also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras, or equivalently using projective bundles.
Construction
A
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 n > 1).
We obtain an
Given central simple algebras A and B, one can look at their tensor product A ⊗ B as a K-algebra (see
As a result, the isomorphism classes of CSAs over K form a
The Brauer group of any field is a torsion group. In more detail, define the period of a central simple algebra A over K to be its order as an element of the Brauer group. Define the index of A to be the degree of the division algebra that is Brauer equivalent to A. Then the period of A divides the index of A (and hence is finite).[1]
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.
- K is a finite field (Wedderburn's theorem).[2] Equivalently, every finite division ring is commutative.
- K is the function field of an algebraic curve over an algebraically closed field (Tsen's theorem).[3] More generally, the Brauer group vanishes for any C1 field.
- K is an algebraic extension of Q containing all roots of unity.[2]
- 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: R itself and the quaternion algebra H.[4] Since H ⊗ H ≅ M(4, R), the class of H has order two in the Brauer group.
- Let K be a non-Archimedean local field, meaning that K is complete under a discrete valuation with finite residue field. Then Br(K) is isomorphic to Q/Z.[5]
Severi–Brauer varieties
Another important interpretation of the Brauer group of a field K is that it classifies the projective varieties over K that become isomorphic to projective space over an algebraic closure of K. Such a variety is called a Severi–Brauer variety, and there is a one-to-one correspondence between the isomorphism classes of Severi–Brauer varieties of dimension n − 1 over K and the central simple algebras of degree n over K.[6]
For example, the Severi–Brauer varieties of dimension 1 are exactly the
The conic is isomorphic to the projective line P1 over K if and only if the corresponding quaternion algebra is isomorphic to the matrix algebra M(2, K).
Cyclic algebras
For a positive integer n, let K be a field in which n is invertible such that K contains a primitive nth root of unity ζ. For nonzero elements a and b of K, the associated cyclic algebra is the central simple algebra of degree n over K defined by
Cyclic algebras are the best-understood central simple algebras. (When n is not invertible in K or K does not have a primitive nth root of unity, a similar construction gives the cyclic algebra (χ, a) associated to a cyclic Z/n-extension χ of K and a nonzero element a of K.[8])
The Merkurjev–Suslin theorem in algebraic K-theory has a strong consequence about the Brauer group. Namely, for a positive integer n, let K be a field in which n is invertible such that K contains a primitive nth root of unity. Then the subgroup of the Brauer group of K killed by n is generated by cyclic algebras of degree n.[9] Equivalently, any division algebra of period dividing n is Brauer equivalent to a tensor product of cyclic algebras of degree n. Even for a prime number p, there are examples showing that a division algebra of period p need not be actually isomorphic to a tensor product of cyclic algebras of degree p.[10]
It is a major open problem (raised by
The period-index problem
For any central simple algebra A over a field K, the period of A divides the index of A, and the two numbers have the same prime factors.[13] The period-index problem is to bound the index in terms of the period, for fields K of interest. For example, if A is a central simple algebra over a local field or global field, then Albert–Brauer–Hasse–Noether showed that the index of A is equal to the period of A.[11]
For a central simple algebra A over a field K of transcendence degree n over an algebraically closed field, it is conjectured that ind(A) divides per(A)n−1. This is true for n ≤ 2, the case n = 2 being an important advance by de Jong, sharpened in positive characteristic by de Jong–Starr and Lieblich.[14]
Class field theory
The Brauer group plays an important role in the modern formulation of class field theory. If Kv is a non-Archimedean local field, local class field theory gives a canonical isomorphism invv : Br(Kv) → Q/Z, the Hasse invariant.[2]
The case of a global field K (such as a
where S is the set of all places of K and the right arrow is the 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.
The fact that the sum of all local invariants of a central simple algebra over K is zero is a typical
Galois cohomology
For an arbitrary field K, the Brauer group can be expressed in terms of Galois cohomology as follows:[17]
where Gm denotes the
The isomorphism of the Brauer group with a Galois cohomology group can be described as follows. The automorphism group of the algebra of n × n matrices is the projective linear group PGL(n). Since all central simple algebras over K become isomorphic to the matrix algebra over a separable closure of K, the set of isomorphism classes of central simple algebras of degree n over K can be identified with the Galois cohomology set H1(K, PGL(n)). The class of a central simple algebra in H2(K, Gm) is the image of its class in H1 under the boundary homomorphism
associated to the
The Brauer group of a scheme
The Brauer group was generalized from fields to
There are two ways of defining the Brauer group of a scheme X, using either Azumaya algebras over X or projective bundles over X. The second definition involves projective bundles that are locally trivial in the étale topology, not necessarily in the Zariski topology. In particular, a projective bundle is defined to be zero in the Brauer group if and only if it is the projectivization of some vector bundle.
The cohomological Brauer group of a quasi-compact scheme X is defined to be the torsion subgroup of the étale cohomology group H2(X, Gm). (The whole group H2(X, Gm) need not be torsion, although it is torsion for regular schemes X.[18]) The Brauer group is always a subgroup of the cohomological Brauer group. Gabber showed that the Brauer group is equal to the cohomological Brauer group for any scheme with an ample line bundle (for example, any quasi-projective scheme over a commutative ring).[19]
The whole group H2(X, Gm) can be viewed as classifying the gerbes over X with structure group Gm.
For smooth projective varieties over a field, the Brauer group is a
Relation to the Tate conjecture
Artin conjectured that every
For a regular integral scheme of dimension 2 which is flat and proper over the ring of integers of a number field, and which has a section, the finiteness of the Brauer group is equivalent to the finiteness of the Tate–Shafarevich group Ш for the Jacobian variety of the general fiber (a curve over a number field).[23] The finiteness of Ш is a central problem in the arithmetic of elliptic curves and more generally abelian varieties.
The Brauer–Manin obstruction
Let X be a smooth projective variety over a number field K. The
Notes
- ^ Farb & Dennis 1993, Proposition 4.16
- ^ a b c Serre 1979, p. 162
- ^ Gille & Szamuely 2006, Theorem 6.2.8
- ^ Serre 1979, p. 163
- ^ Serre 1979, p. 193
- ^ Gille & Szamuely 2006, § 5.2
- ^ Gille & Szamuely 2006, Theorem 1.4.2.
- ^ Gille & Szamuely 2006, Proposition 2.5.2
- ^ Gille & Szamuely 2006, Theorem 2.5.7
- ^ Gille & Szamuely 2006, Remark 2.5.8
- ^ a b Pierce 1982, § 18.6
- ^ Saltman 2007
- ^ Gille & Szamuely 2006, Proposition 4.5.13
- ^ de Jong 2004
- ^ Gille & Szamuely 2006, p. 159
- ^ Pierce 1982, § 18.5
- ^ Serre 1979, pp. 157–159
- ^ Milne 1980, Corollary IV.2.6
- ^ de Jong, A result of Gabber
- ^ Colliot-Thélène 1995, Proposition 4.2.3 and § 4.2.4
- ^ Milne 1980, Question IV.2.19
- ^ Tate 1994, Proposition 4.3
- ^ Grothendieck 1968, Le groupe de Brauer III, Proposition 4.5
References
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- MR 2060023
- de Jong, A. J. "A result of Gabber" (PDF).
- MR 1233388.
- Gille, Philippe; Szamuely, Tamás (2006). Central Simple Algebras and Galois Cohomology. Cambridge Studies in Advanced Mathematics. Vol. 101. Cambridge: MR 2266528.
- MR 0244271
- V.A. Iskovskikh (2001) [1994], k "Brauer group of a field k", Encyclopedia of Mathematics, EMS Press
- MR 0559531
- Pierce, Richard (1982). Associative Algebras. MR 0674652.
- Saltman, David J. (1999). Lectures on Division Algebras. Regional Conference Series in Mathematics. Vol. 94. Providence, RI: MR 1692654.
- Saltman, David J. (2007), "Cyclic algebras over p-adic curves", Journal of Algebra, 314 (2): 817–843, S2CID 119160155
- MR 0554237.
- MR 1265523