Brauer-Severi variety

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An algebraic variety over a field that, if considered over the algebraic closure of , becomes isomorphic to a projective space.

The arithmetic properties of such varieties were studied in 1932 by F. Severi; F. Châtelet subsequently discovered a connection between Brauer–Severi varieties and central simple algebras (cf. Central simple algebra) over and the Brauer group.

The simplest non-trivial example of a one-dimensional Brauer–Severi variety is the projective conic section :

on the real projective plane . Over the field of complex numbers this variety is isomorphic to the projective line . The set of all one-dimensional Brauer–Severi varieties, considered up to isomorphism, is in a one-to-one correspondence with the set of projective non-degenerate conic sections (considered up to projective equivalence over ), which is in turn in a one-to-one correspondence with the set of non-isomorphic generalized quaternion algebras over . In the above example the conical section corresponds to the algebra of ordinary quaternions.

In the more-dimensional case, the set of classes of -dimensional Brauer–Severi varieties (i.e. Brauer–Severi varieties distinguished up to -isomorphism) may be identified with the Galois cohomology group where is the projective group of automorphisms of the projective space [3], [4]. This cohomology group describes the classes of -isomorphic central simple -algebras of rank (i.e. forms of the matrix algebra ). The connection between Brauer–Severi varieties and central simple algebras is more explicitly described as follows. To a -algebra of rank one associates the variety of its left ideals of rank , which is defined as a closed subvariety of the Grassmann manifold of all -linear subspaces of dimension in . In certain cases the variety may be defined by norm equations — e.g. in the case of quaternion algebras. The connection between Brauer–Severi varieties and algebras is taken advantage of in the study of both [1], [4].

The most significant properties of Brauer–Severi varieties are the following. A Brauer–Severi variety is -isomorphic to a projective space if and only if it has a point in the field . All Brauer–Severi varieties have a point in some finite separable extension of [1].

The Hasse principle applies to Brauer–Severi varieties defined over an algebraic number field.

The field of rational functions on a Brauer–Severi variety is the splitting field of the corresponding algebra ; moreover, an arbitrary extension of is the splitting field for if and only if has a -point [4].

In the context of the generalization of the concepts of a central simple algebra and the Brauer group to include schemes, the Brauer–Severi varieties were generalized to the concept of Brauer–Severi schemes [2]. Let be a morphism of schemes. A scheme is called a Brauer–Severi scheme if it is locally isomorphic to a projective space over in the étale topology of . A scheme over a scheme is a Brauer–Severi scheme if and only if is a finitely-presented proper flat morphism and if all of its geometrical fibres are isomorphic to projective spaces [2].

References

 [1] F. Châtelet, "Variations sur un thème de H. Poincaré" Ann. Sci. École Norm. Sup. (3) , 61 (1944) pp. 249–300 [2] A. Grothendieck, "Le groupe de Brauer" A. Grothendieck (ed.) J. Giraud (ed.) et al. (ed.) , Dix exposés sur la cohomologie des schémas , North-Holland & Masson (1968) pp. 1–21 [3] J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) [4] P. Roguette, "On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting fields of algebras" Math. Ann. , 150 (1963) pp. 411–439