Severi–Brauer variety
In
In dimension one, the Severi–Brauer varieties are conics. The corresponding central simple algebras are the quaternion algebras. The algebra (a, b)K corresponds to the conic C(a, b) with equation
and the algebra (a, b)K splits, that is, (a, b)K is isomorphic to a matrix algebra over K, if and only if C(a, b) has a point defined over K: this is in turn equivalent to C(a, b) being isomorphic to the projective line over K.[1][2]
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(G(Ks/K),PGLn), where PGLn is the
- 1 → GL1 → GLn → PGLn → 1
of
- H1(PGLn) → H2(GL1)
at the level of cohomology. 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.[2]
As a consequence, we see that if the class of X has order d in the Brauer group then there is a
See also
Note
- ^ a b Jacobson (1996), p. 113
- ^ a b Gille & Szamuely (2006), p. 129
- ^ Gille & Szamuely (2006), p. 26
- ^ Berhuy (2010), p. 113
- ^ Gille & Szamuely (2006), p. 131
References
- Zbl 0536.14006
- "Brauer–Severi variety", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Gille, Philippe; Szamuely, Tamás (2006), "Severi–Brauer varieties", Central Simple Algebras and Galois Cohomology, Cambridge Studies in Advanced Mathematics, vol. 101, Zbl 1137.12001
- Berhuy, Grégory (2010), An Introduction to Galois Cohomology and its Applications, London Mathematical Society Lecture Note Series, vol. 377, Zbl 1207.12003
- Zbl 0874.16002
- Saltman, David J. (1999), Lectures on division algebras, Regional Conference Series in Mathematics, vol. 94, Providence, RI: Zbl 0934.16013
- Severi, Francesco (1932), "Un nuovo campo di ricerche nella geometria sopra una superficie e sopra una varietà algebrica", Memorie della Reale Accademia d'Italia (in Italian), 3 (5), Reprinted in volume 3 of his collected works
Further reading
- Knus, Max-Albert; Zbl 0955.16001