Severi–Brauer variety

They represent (at least if K is a perfect field) Galois cohomology classes in H1(G(Ks/K),PGLn), where PGLn is the projective linear group, and n is one more than the dimension of the variety V. As usual in Galois cohomology, we often leave the

Here H2(GL1) is identified with the Brauer group of K, while the kernel is trivial because H1(GLn) = {1} by an extension of Hilbert's Theorem 90.

[3][4] Therefore, Severi–Brauer varieties can be faithfully represented by Brauer group elements, i.e. classes of central simple algebras.

Lichtenbaum showed that if X is a Severi–Brauer variety over K then there is an exact sequence Here the map δ sends 1 to the Brauer class corresponding to X.

The associated linear system defines the d-dimensional embedding of X over a splitting field L.[5]