THE BRAUER-SEVERI VARIETY ASSOCIATED WITH A CENTRAL SIMPLE ALGEBRA: A SURVEY by

— The article describes the one-to-one correspondence between central simple algebras and Brauer-Severi varieties over a field. Non-abelian group cohomology is recalled and Galois descent is worked out in detail. The classifications of central simple algebras as well as of Brauer-Severi varieties by one and the same Galois cohomology set are explained. A whole section is devoted to the discussion of functoriality. Finally, the functor of points of the Brauer-Severi variety associated with a central simple algebra is described in terms of the central simple algebra thus giving a link to another approach to the subject. Introductory remarks This article is devoted to the connection between central simple algebras and Brauer-Severi varieties. Central simple algebras were studied intensively by many mathematicians at the end of the 19th and in the first half of the 20th century. We refer the reader to N. Bourbaki [Bou, Note historique] for a detailed account on the history of the subject and mention only a few important milestones here. The structure of central simple algebras (being finite dimensional over a field K) is fairly easy. They are full matrix rings over division algebras the center of which is equal to K. This was finally discovered by J. H. Maclagan-Wedderburn in 1907 [MWe08] after several special cases had been treated before. T. Molien [Mo] had considered the case of C-algebras already in 1893 and the case of R-algebras had been investigated by E. Cartan [Ca]. J. H. Maclagan-Wedderburn himself had proven the structure theorem for central simple algebras over finite fields in 1905 [MWe05, Di]. In 1929, R. Brauer ([Br], see also [De], [A/N/T]) found the group structure on the set of similarity classes of central simple algebras over a field K using the ideas of E. Noether about crossed products of algebras. He proved, in today’s language, that it is isomorphic to the Galois cohomology group H(Gal(K/K), (Ksep)∗). Further, he discusses the structure of this group in the case of a number field. Relative versions of central simple algebras over base rings instead of fields were introduced for the first time by G. Azumaya [Az] and M. Auslander and O. 2000Mathematics Subject Classification. — Primary 16K20, 14G99, Secondary 16K50, 12G05.