Brauer group
1 Algebraic view
Let be a field. The Brauer group of is the set of all equivalence classes
of central simple algebras over , where two central simple algebras and are equivalent
if there exists a division ring over and natural numbers
such that (resp. ) is isomorphic
to the ring of (resp. ) matrices with coefficients in .
The group operation in is given by tensor product
: for any two central simple algebras over , their
product
in is the central simple algebra . The identity element
in is the class of itself, and the inverse
of a central simple algebra is
the opposite algebra
defined by reversing
the order of the multiplication operation
of .
2 Cohomological view
The Brauer group of is naturally isomorphic to the second Galois cohomology group . See http://www.math.harvard.edu/ elkies/M250.01/index.htmlhttp://www.math.harvard.edu/ 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 |