General Theory
For an arbitrary field K, the Brauer group may be expressed in terms of Galois cohomology as follows:
Here, Ks is the separable closure of K, which coincides with the algebraic closure when K is a perfect field.
A generalisation of the Brauer group to the case of commutative rings by M. Auslander and O. Goldman, and more generally to schemes, was introduced by Alexander Grothendieck. In their approach, central simple algebras over a field are replaced with Azumaya algebras.
Read more about this topic: Brauer Group
Famous quotes containing the words general and/or theory:
“In communist society, where nobody has one exclusive sphere of activity but each can become accomplished in any branch he wishes, society regulates the general production and thus makes it possible for me to do one thing today and another tomorrow, to hunt in the morning, fish in the afternoon, rear cattle in the evening, criticize after dinner, just as I have a mind, without ever becoming hunter, fisherman, shepherd or critic.”
—Karl Marx (1818–1883)
“There is in him, hidden deep-down, a great instinctive artist, and hence the makings of an aristocrat. In his muddled way, held back by the manacles of his race and time, and his steps made uncertain by a guiding theory which too often eludes his own comprehension, he yet manages to produce works of unquestionable beauty and authority, and to interpret life in a manner that is poignant and illuminating.”
—H.L. (Henry Lewis)