J\"org Jahnel
The Brauer-Severi variety associated with a central simple algebra: A survey
Preprint series: Mathematica Gottingensis

MSC:

16K20 Finite-dimensional

14G99 None of the above but in this section

Abstract: 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.
Keywords: central simple algebras, Brauer group, Brauer-Severi varieties, functor of points, non-abelian group cohomology