Severi–Brauer varieties

Abstract

In Chapter 1 we associated with each quaternion algebra a conic with the property that the conic has a k -point if and only if the algebra splits over k . We now generalize this correspondence to arbitrary dimension: with each central simple algebra A of degree n over an arbitrary field k we associate a projective k -variety X of dimension n – 1 which has a k -point if and only if A splits. Both objects will correspond to a class in H 1 ( G , PGL n ( K )), where K is a Galois splitting field for A with group G . The varieties X arising in this way are called Severi–Brauer varieties; they are characterized by the property that they become isomorphic to some projective space over the algebraic closure. This interpretation will enable us to give another, geometric construction of the Brauer group. Another central result of this chapter is a theorem of Amitsur which states that for a Severi–Brauer variety X with function field k ( X ) the kernel of the natural map Br ( k ) → Br ( k ( X )) is a cyclic group generated by the class of X . This seemingly technical statement (which generalizes Witt's theorem proven in Chapter 1) has very fruitful algebraic applications. At the end of the chapter we shall present one such application, due to Saltman, which shows that all central simple algebras of fixed degree n over a field k containing the n -th roots of unity can be made cyclic via base change to some large field extension of k . Severi–Brauer varieties were introduced in the pioneering paper of Châtelet [1], under the name ‘varietes de Brauer’.