Severi-Brauer variety

In .

Such varieties are of interest not only in diophantine geometry, but also in Galois cohomology. They represent (at least if K is a perfect field) Galois cohomology classes in

: H 1( PGL n )

in the projective linear group, where n is the dimension of a variety of V . There is a short exact sequence

:1 → GL 1 → GL n → PGL n → 1

of algebraic groups. This implies a connecting homomorphism

: H 1( PGL n ) → H 2( GL 1)

at the level of cohomology. The RHS is identified with the Brauer group of K , while the kernel is trivial because

: H 1( GL n ) = {1}

by an extension of Hilbert s Theorem 90. Therefore the Severi-Brauer varieties can be faithfully represented by Brauer group elements, i.e. classes of central simple algebras.