In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In the field of Mathematics called Abstract algebra, a division algebra is roughly speaking an Algebra over a field in which division is possible In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division It is an abelian group with elements isomorphism classes of division algebras over K, such that the center is exactly K. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective The term center or centre is used in various contexts in Abstract algebra to denote the set of all those elements that commute with all other elements The group is named for the algebraist Richard Brauer. Richard Dagobert Brauer ( February 10, 1901 &ndash April 17, 1977) was a leading German and American Mathematician
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Construction of the Brauer group
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. In Ring theory and related areas of Mathematics a central simple algebra ( CSA) over a field K (also called a Brauer algebra In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Abstract algebra, a simple ring is a non-zero ring that has no ideal besides the Zero ideal and itself The term center or centre is used in various contexts in Abstract algebra to denote the set of all those elements that commute with all other elements 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. Ferdinand Georg Frobenius ( October 26, 1849 – August 3, 1917) was a German Mathematician, best-known for his contributions
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). In Mathematics, the Tensor product of two ''R''-algebras is also an R -algebra in a natural way 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. In Abstract algebra the matrix ring M( n, R) is the set of all n × n matrices over an arbitrary ring In Mathematics, particularly Abstract algebra, an algebraic closure of a field K is an Algebraic extension of K that is
Given this closure property for CSAs, they form a monoid under tensor product. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation To get a group, apply the Artin-Wedderburn theorem (Wedderburn's part, in fact), to express any CSA as a M(n,D) for some division algebra D. In Abstract algebra, the Artin–Wedderburn theorem is a Classification theorem for semisimple rings. Joseph Henry Maclagan Wedderburn ( 2 February 1882 Forfar Angus, Scotland – 9 October 1948, Princeton New Jersey If we look just at D, rather than the value of n, the monoid becomes a group. 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 an equivalence relation; and the equivalence 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 Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real The term center or centre is used in various contexts in Abstract algebra to denote the set of all those elements that commute with all other elements In other words, for a CSA A we have A ⊗ Aop = M(n2,F), where n is the degree of A over F. (This provides a substantial reason for caring about the notion of an opposite algebra: it provides the inverse in the Brauer group. )
Examples
The Brauer group for an algebraically closed field or a finite field is the trivial group with only the identity element. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements In Mathematics, a trivial group is a group consisting of a single element
The Brauer group Br(R) of the real number field R is a cyclic group of order two: there are just two types of division algebras, R and the quaternion algebra H. In Mathematics, the real numbers may be described informally in several different ways In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician The product in the Brauer group is based on the tensor product: the statement that H has order two in the group is equivalent to the existence of an isomorphism of R-algebras: H ⊗ H ≅ M(4,R), where the RHS is the ring of 4×4 real matrices. In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector In Mathematics, LHS is informal shorthand for the left-hand side of an Equation.
Tsen's theorem implies that the Brauer group of a function field in one variable over an algebraically closed field vanishes. In mathematics Tsen's theorem states that a function field K of an Algebraic curve over an algebraically closed field is Quasi-algebraically closed.
Further theory
In the further theory, the Brauer group of a local field is computed (it turns out to be canonically isomorphic to Q/Z for any local field, of characteristic 0 or characteristic p) and the results are applied to global fields. In Mathematics, a local field is a special type of field that is a Locally compact Topological field with respect to a non-discrete topology In Mathematics, the term global field refers to either of the following a number field, i This gives one approach to class field theory, which was the first approach that allowed global class field theory to be derived from local class field theory; historically it had been the other way around at first. In Mathematics, class field theory is a major branch of Algebraic number theory. It also has been applied to Diophantine equations. In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only More precisely, the Brauer group Br(K) of a global field K is given by the exact sequence
where the direct sum in the middle is over all (archimedean and non-archimedean) completions of K and the map to 
is addition, where we interpret the Brauer group of the reals as (1/2)Z/Z. The group Q/Z on the right is really the "Brauer group" of the class formation of idele classes associated to K. In mathematics a class formation is a structure used to organize the various Galois groups and modules that appear in Class field theory.
In the general theory the Brauer group is expressed by factor sets; and expressed in terms of Galois cohomology via
Here, not assuming K to be a perfect field, Ks is the separable closure. In Mathematics, Galois cohomology is the study of the Group cohomology of Galois modules that is the application of Homological algebra to In Mathematics, an Algebraic field extension L / K is separable if it can be generated by adjoining to K a set each of whose elements In Mathematics, particularly Abstract algebra, an algebraic closure of a field K is an Algebraic extension of K that is When K is perfect this is the same as an algebraic closure; otherwise the Galois group must be defined in terms of Ks/K even to make sense. In Mathematics, particularly Abstract algebra, an algebraic closure of a field K is an Algebraic extension of K that is
A generalisation via the theory of Azumaya algebras was introduced in algebraic geometry by Grothendieck. In Mathematics, an Azumaya algebra is a generalization of Central simple algebras to R -algebras where R need not be a field. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany