Severi-Brauer variety
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In mathematics, a Severi-Brauer variety over a field K is an algebraic variety V which becomes isomorphic to projective space over an algebraic closure of K. Examples are conic sections C: provided C is non-singular, it becomes isomorphic to the projective line over any extension field L over which C has a point defined. The name is for Francesco Severi and Richard Brauer.
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
- H1(PGLn)
in the projective linear group, where n is the dimension of V. There is a short exact sequence
- 1 → GL1 → GLn → PGLn → 1
of algebraic groups. This implies a connecting homomorphism
- H1(PGLn) → H2(GL1)
at the level of cohomology. The RHS 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. Therefore the Severi-Brauer varieties can be faithfully represented by Brauer group elements, i.e. classes of central simple algebras.
