Brauer Group and Class Field Theory
The notion of Brauer group plays an important role in the modern formulation of the class field theory. If Kv is a non-archimedean local field, there is a canonical isomorphism invv: Br(Kv) → Q/Z constructed in local class field theory. An element of the Brauer group of order n can be represented by a cyclic division algebra of dimension n2.
The case of a global field K is addressed by the global class field theory. If D is a central simple algebra over K and v is a valuation then D ⊗ Kv is a central simple algebra over Kv, the local completion of K at v. This defines a homomorphism from the Brauer group of K into the Brauer group of Kv. A given central simple algebra D splits for all but finitely many v, so that the image of D under almost all such homomorphisms is 0. The Brauer group Br(K) fits into an exact sequence
where S is the set of all valuations of K and the right arrow is the direct sum of the local invariants and the Brauer group of the real numbers is identified with (1/2)Z/Z. The injectivity of the left arrow is the content of the Albert–Brauer–Hasse–Noether theorem. Exactness in the middle term is a deep fact from the global class field theory. The group Q/Z on the right may be interpreted as the "Brauer group" of the class formation of idele classes associated to K.
Read more about this topic: Brauer Group
Famous quotes containing the words group, class, field and/or theory:
“It’s important to remember that feminism is no longer a group of organizations or leaders. It’s the expectations that parents have for their daughters, and their sons, too. It’s the way we talk about and treat one another. It’s who makes the money and who makes the compromises and who makes the dinner. It’s a state of mind. It’s the way we live now.”
—Anna Quindlen (20th century)
“There is only one class in the community that thinks more about money than the rich, and that is the poor. The poor can think of nothing else.”
—Oscar Wilde (1854–1900)
“... many American Jews have a morbid tendency to exaggerate their handicaps and difficulties. ... There is no doubt that the Jew ... has to be twice as good as the average non- Jew to succeed in many a field of endeavor. But to dwell upon these injustices to the point of self-pity is to weaken the personality unnecessarily. Every human being has handicaps of one sort or another. The brave individual accepts them and by accepting conquers them.”
—Agnes E. Meyer (1887–1970)
“The theory of truth is a series of truisms.”
—J.L. (John Langshaw)