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:
“As a general truth, it is safe to say that any picture that produces a moral impression is a bad picture.”
—Edmond De Goncourt (1822–1896)
“By the “mud-sill” theory it is assumed that labor and education are incompatible; and any practical combination of them impossible. According to that theory, a blind horse upon a tread-mill, is a perfect illustration of what a laborer should be—all the better for being blind, that he could not tread out of place, or kick understandingly.... Free labor insists on universal education.”
—Abraham Lincoln (1809–1865)