Brauer group

1 Algebraic view

Let K be a field. The Brauer groupMathworldPlanetmath Br(K) of K is the set of all equivalence classesMathworldPlanetmathPlanetmath of central simple algebras over K, where two central simple algebras A and B are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath if there exists a division ring D over K and natural numbersMathworldPlanetmath n,m such that A (resp. B) is isomorphicPlanetmathPlanetmathPlanetmath to the ring of n×n (resp. m×m) matrices with coefficients in D.

The group operationMathworldPlanetmath in Br(K) is given by tensor productPlanetmathPlanetmath: for any two central simple algebras A,B over K, their productPlanetmathPlanetmathPlanetmathPlanetmath in Br(K) is the central simple algebra AKB. The identity elementMathworldPlanetmath in Br(K) is the class of K itself, and the inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of a central simple algebra A is the opposite algebraPlanetmathPlanetmath Aopp defined by reversing the order of the multiplication operationMathworldPlanetmath of A.

2 Cohomological view

The Brauer group of K is naturally isomorphic to the second Galois cohomology group H2(Gal(Ksep/K),(Ksep)×). See elkies/M250.01/index.html elkies/M250.01/index.html Theorem 12 and succeeding remarks.

Title Brauer group
Canonical name BrauerGroup
Date of creation 2013-03-22 11:49:31
Last modified on 2013-03-22 11:49:31
Owner djao (24)
Last modified by djao (24)
Numerical id 13
Author djao (24)
Entry type Definition
Classification msc 16K50
Defines opposite algebra