Brewers group

The brewer's group was introduced in mathematics to classify associative division algebras over a given body that have the center . It is an Abelian group , the elements of which are equivalence classes of certain algebras . In the literature it is therefore also called Brewer's algebra class group. It is named after the algebraist Richard Brauer . ${\ displaystyle K}$${\ displaystyle K}$

construction

A central simple algebra over a field is a finite-dimensional associative -algebra , which is a simple ring (i.e. a ring whose only two-sided ideals are the trivial ones) and whose center is straight . For example, the complex numbers are a central simple algebra about themselves, but not about the real numbers , since their center is whole and therefore greater than . According to a theorem of Frobenius , the finite-dimensional associative division algebras with a center are just the real numbers and the quaternions . ${\ displaystyle K}$ ${\ displaystyle K}$ ${\ displaystyle A}$${\ displaystyle K}$ ${\ displaystyle \ mathbb {C}}$ ${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {C}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$

If and are two central simple algebras, their tensor product can be formed as an -algebra. One can show that the tensor product itself is again a central simple algebra. ${\ displaystyle A}$${\ displaystyle B}$ ${\ displaystyle A \ otimes B}$${\ displaystyle K}$

With the tensor product as a link, the central simple algebras thus form a monoid . To obtain a group from this, one applies the Artin-Wedderburn theorem , which allows every central simple algebra to be written as a matrix ring over an associative division algebra . If one differentiates now only according to the division algebra , but not according to the values ​​of , then the ring becomes a group. Formally, this means that we have an equivalence relation defined and with natural for all numbers and identify each other. The neutral element is the equivalence class of , the inverse element of the equivalence class of algebra is the equivalence class of counter-algebra , which differs from only in that the multiplication is reversed. The equation holds for a central simple algebra , where the degree is over . ${\ displaystyle M (n, D)}$${\ displaystyle D}$${\ displaystyle D}$${\ displaystyle n}$${\ displaystyle M (m, D)}$${\ displaystyle M (n, D)}$${\ displaystyle m}$${\ displaystyle n}$${\ displaystyle K \ cong M (1, K)}$${\ displaystyle A}$ ${\ displaystyle A ^ {\ mathrm {op}}}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle A \ otimes A ^ {\ mathrm {op}} \ cong M (n ^ {2}, K)}$${\ displaystyle n}$${\ displaystyle A}$${\ displaystyle K}$

The resulting group is called the brewer's group of the body and is labeled with . ${\ displaystyle K}$${\ displaystyle \ mathrm {Br} (K)}$

Examples

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

The Brauer group of real numbers is cyclic of order 2, since, as already mentioned above, apart from isomorphism, there are only two different associative division algebras that have as their center : themselves and the quaternions . In particular, and , where the latter is the ring of real 4 × 4 matrices. ${\ displaystyle \ mathrm {Br} (\ mathbb {R})}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {H}}$${\ displaystyle \ mathbb {H} \ cong \ mathbb {H} ^ {\ mathrm {op}}}$${\ displaystyle \ mathbb {H} \ otimes \ mathbb {H} \ cong M (4, \ mathbb {R})}$

From Tsen's theorem (after Chiungtze Tsen 1933) it follows that the brewer's group of a function field in a variable over an algebraically closed field is also trivial.

Applications

In the further theory one determines the brewer group of local bodies , for every non-Archimedean local body it is canonically isomorphic to . The results obtained can be applied to global bodies . This provides an approach to class field theory , which for the first time allowed global class field theory to be derived from local one; historically the development was reversed. The brewer's group is also used in Diophantine equations . ${\ displaystyle \ mathbb {Q} / \ mathbb {Z}}$

The transition from the local to the global body results as follows, the brewing group of a global body is determined by the exact sequence${\ displaystyle \ mathrm {Br} (K)}$${\ displaystyle K}$

${\ displaystyle 0 \ rightarrow \ mathrm {Br} (K) \ rightarrow \ bigoplus _ {v} \ mathrm {Br} (K_ {v}) \ rightarrow \ mathbb {Q} / \ mathbb {Z} \ rightarrow 0}$

given, where the direct sum is formed over all (Archimedean and non-Archimedean) completions of and the mapping to is given by addition, we understand the brewer's group of real numbers as . The group on the right side is the brewer group of class formation of Idel class associated to . ${\ displaystyle K}$${\ displaystyle \ mathbb {Q} / \ mathbb {Z}}$${\ displaystyle {\ frac {1} {2}} \ mathbb {Z} / \ mathbb {Z} \ subset \ mathbb {Q} / \ mathbb {Z}}$${\ displaystyle \ mathbb {Q} / \ mathbb {Z}}$${\ displaystyle K}$

The brewer's group can also be represented with the help of Galois cohomology , it applies . This is the separable closure of the not necessarily perfect body . If is perfect, it matches the algebraic closure , otherwise the Galois group has to be defined using to make sense. ${\ displaystyle \ mathrm {Br} (K) \ cong H ^ {2} (\ mathrm {Gal} (K ^ {\ mathrm {sep}} / K), {K ^ {\ mathrm {sep}}} ^ {\ times})}$${\ displaystyle K ^ {\ mathrm {sep}}}$ ${\ displaystyle K}$${\ displaystyle K}$${\ displaystyle K ^ {\ mathrm {sep}} / K}$

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

literature

• Jürgen Neukirch : Class field theory (= BI university scripts 713 / 713a, ). Improved new edition. Bibliographisches Institut, Mannheim et al. 1969.