Brauer group

The Brauer group has been introduced in mathematics in order to classify associative division algebras over a given body that have the K center. This is an abelian group whose elements are equivalence classes of certain algebras. In the literature it is therefore also called Brauer Algebrenklassengruppe. It is named after Richard Brauer algebraists.

Construction

A central simple algebra over a finite body is an associative algebra, which is a single ring (ie, a ring, the single -sided ideals are trivial ) and the center of which is straight. For example, the complex numbers a central simple algebra over itself, but not over the real numbers, as its center than is whole and consequently greater. By a theorem of Frobenius finite dimensional associative division algebras are the center with just the real numbers and the quaternions.

Are central simple algebras and two, then one can form their tensor product as algebra. It can be shown that the tensor itself is a central simple algebra again.

With the tensor product as a link, the central simple algebras thus form a monoid. In order to derive obtain a group, one applies the theorem of Artin - Wedderburn, which allows you to write any central simple algebra as a matrix ring over an associative division algebra. If we differentiate now only after the division algebra, but not on the values ​​of, it is a group out of the ring. Formally, this means that we define an equivalence relation and identify with one another and for all natural numbers. The neutral element is the equivalence class of, the inverse element of the equivalence class of the algebra is the equivalence class of counter- algebra which only differs to that multiplication is reversed. It is namely a central simple algebra, the equation, where the degree of over is.

The resulting group is called the Brauer group of the body and is denoted by.

Examples

The Brauer group of an algebraically closed field is the trivial group with only the neutral element, as is the Brauer group of a finite field.

The Brauer group of real numbers is cyclic of order 2, since as mentioned above, up to isomorphism, only two different associative division algebras are over, which have as center: itself and the quaternions. In particular and, while the latter is the ring of the 4 × 4 real matrices.

From the set of Tsen (after Chiungtze Tsen 1933) it follows that the Brauer group of a function field in one variable over an algebraically closed field is also trivial.

Applications

In another theory, one determines the Brauer group of local body, for each non- Archimedean local field is canonically isomorphic to. The results obtained can be applied to global body. This provides access to class field theory, the first time made ​​it possible to derive global class field theory from the local; historical development was reversed. Common applications for the Brauer group even with Diophantine equations.

The transition from local to global body is as follows, the Brauer group of a global field is determined by the exact sequence

Given the direct sum over all ( Archimedean and non- Archimedean ) completions of is formed and the mapping is given by by addition, while we summarize the Brauer group of real numbers than on. The group on the right is the Brauer group of the class formation of the Idel - class associated.

One can represent the Brauer group with the help of Galoiskohomologie, it is. Here is the separable closure of unnecessary perfect body. If it is perfect, is not this the same as the algebraic degree, otherwise the Galois group must be defined to give meaning.

A generalization by means of the theory of Azumaya algebras was introduced in algebraic geometry by Grothendieck.