Submission: 2000, Oct 27
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.
1991 Mathematics Subject Classification: Primary 16K20, 14G99, Secondary 16K50, 12G05
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